Unified S-Band Telecommunications Techniques for Apollo Volume II Mathematical Models and Analysis

8/7/2019 Unified S-Band Telecommunications Techniques for Apollo Volume II Mathematical Models and Analysis

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NASA

US-B- ~

V. 2c.1

N A S A T E C H N I C A L N O T E N A S A T N

UNIFIED S-BAND TELECOMMUNICATION

TECHNIQUES FOR APOLLO

VO LUM E 11-MATHEMATICAL MODELS

A N D A NA LY SIS

by John H . Puin ter and George Hondros

Manned Spucecruft Center

Houston, Texas

N A T I O N A L A E R O N A U T I C S A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D . C . A P R I L 1 9 6 6



OOb9092

UNIFIED S-BAND TELECOMMUNICATION TECHNIQUES FOR APOLLO

VOLUME I1

MATHEMATICAL MODELS AND ANALYSIS

By John H. P ain ter and George Hondros

Manned Spacecraft Center

Houston, Texas

NATIONAL AERONAUTICs AND SPACE ADMINISTRATION~.

For sale by the C lear inghouse fo r Fede r a l S c ien t i f i c and Tech n ica l In fo r mat ion

Spr ingf ie ld , V irg in ia 22151 - P r i c e $5.00



ABSTRACT This is the second volume in a series done at

Manned Spacecraft Center, documenting comunication

techniques used in the Apollo Unified S-band Telecom

munications and Tracking System. A s stated in the

first volume, NASA TJY D-2208,the present document isconcerned with detailed mathematical modeling of certain

channels in the system. Specifically, this volume pro

vides simple mathematical tools usable for predicting,

approximately, the performance of various communicationsand tracking channels in the system.



FOREWORD

The con t en t of t h i s pub l i ca t i on w a s o r i g i n a l l y p u b l i s h e c as NASAApollo Working Paper N o . 1184, and w a s e n t i t l e d "Unified S-Band Telecommun ication Techniques f o r Apollo, Volume 11, Mathematical Models and

Analysis." Volume I of t h i s s e r i e s , by t he s am e au t ho r s , i s e n t i t l e d"Funct ional Desc ript ion." This pub l ica t ion , which lias become Volume 11,

i s nea r ly id en t i ca l t o the Working Paper ment ioned above , i nco rpo ra t i ngonly a few minor changes i n th e form.

The au tho rs wish t o acknowledge th e t ime and ef f o r t of members of

t he t ec hn i c a l s t a f f s of J e t P ropu l si on L abora t ory , Pasadena, C a l i fo rn i a ,and Motoro la Mi l i t a ry E le c t ro n ic s Div i s ion , Western Center , Sco t t sd a le ,Arizona. The f i r s t d r a f t w a s reviewed by the fol lowing in div idu als fromJ e t P ropu l si on L abora t ory , each performing a sepa ra t e r ev i ew func t i on :

M . KoernerL . CouvilLon R . Titsworth M. B r o c k " R . Toukdarian W . Victor

The fol lowing ind ivi du als from Motorola , In c. reviewed th e f i r s t d r a f t i ni t s e n t i r e t y :

D r . S . C . GuptaT . G. H a l l

A word of thanks i s due t o Messrs . B. D. Martin and M . E a s t e r l i n gof JPL fo r con t inued a s s i s t an ce and i n s p i r a t i oc du r i ng t he p repa ra t i onof this volume.

The au t ho r s w ish t o ded i ca t e t h i s pub l i ca t i on t o G. Barry Graves

who made th e w ri t i ng of t h i s s e r i e s of documents po ss ibl e, and t oTed Freeman, one of t ho se in d iv id ua ls whom t h i s work w a s i n t ended t ob e n e f i t .



Sect ion Page SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . 1 SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Background and Purpose of t h e Document . . . . . . . 5 1.2 Th eo re t ic al Approach . . . . . . . . . . . . . . . . 5

1.2 .1 An aly t i ca l Scope . . . . . . . . . . . . . . 5 1.2.2 Method of Presen ta t ion . . . . . . . . . . . 6 1.2.3 Assumptions . . . . . . . . . . . . . . . . . 6

1.3 System Description . . . . . . . . . . . . . . . . . 7 2 .0 GROUND-TO-SPACECRAFT CHANNEL ANALYSES . . . . . . . . . . 7

2.1 Ca rr ie r Tracking Channel . . . . . . . . . . . . . . 7 2.2 The Voice Chann el . . . . . . . . . . . . . . . . . 8 2.3 The Up-Data Channel . . . . . . . . . . . . . . . . 11

3-0 SPACECRAFT-TO-GROUND PHASE MODULATED C M E L ANALYSES . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 Carrier Tracking Channel . . . . . . . . . . . . . . 13 3.2 Angle Tra ckin g Channel . . . . . . . . . . . . . . . 15 3.3 Ranging Channel . . . . . . . . . . . . . . . . . . 1'5

3.3.1 Clock Loop Threshold . . . . . . . . . . . . 1 6 3.3 .2 Range Code Ac qu is iti on Time . . . . . . . . . 18

3 .4 The F C M Telemetry Channel . . . . . . . . . . . . . 19 3.5 The Voice Channel . . . . . . . . . . . . . . . . . 22 3.6 Biomedical Data Channels . . . . . . . . . . . . . . 25 3.7 The Ehergency Voice Channel . . . . . . . . . . . . 28 3.8 The Emergency Key Channel . . . . . . . . . . . . . 31

i



Section Page

4 .0 SPACECRAFT-TO-mOUND FREQUENCY MODULATED CHANNEL ANALYSES . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1 Carrier Demodulation Channel . . . . . . . . . . . . 33

4.2 Te lev isi on Channel . . . . . . . . . . . . . . . . . 33 4.3 PCM Telem et ry Channel . . . . . . . . . . . . . . . 37 4 .4 Voice Channel . . . . . . . . . . . . . . . . . . . 40 4 .5 Biomedical Data Channel . . . . . . . . . . . . . . 42

APPENDDC A.ANGLE MODULATION . . . . . . . . . ' . . . . . . . . A - 1 A . l Basic Considerat ions

. . . . . . . . . . . . . . . .A - 1

A .2 The Ca rr ie r wi th K S u b c a r r i e r s . . . . . . . . . . A -4 A . 3 The Ca r r i e r wi t h K Su bca rr i ers and Range

Code . . . . . . . . . . . . . . . . . . . . . . . . A -6 APPENDIX B .NOISE . . . . . . . . . . . . . . . . . . . . . . . B-1

B . l The Narrow-Band Gauss ian Random Process . . . . . . B - 1 B.2 Angle Modulated C ar ri e r Pl us Noise . . . . . . . . . B-3 B.3 Transmiss ion of S ig na l Plus Noise Through a

Perfect Band-pas s Limi te r . . . . . . . . . . . . . B-5 B .4 Transm ission of Si gn al Plus Noise Through a

Perfect Product Device . . . . . . . . . . . . . . . B-6 B . 4 . 1 A Nonprelimited Product Detector ' . . . . . . B-6 B.4.2 A Pre l im i ted Produc t De tec to r . . . . . . . . B-7 B .4 .3 A Nonprelimited Product Mixer . . . . . . . . B-9

APPENDIX C.PHASE-LOCKED LOOP THEORY . . . . . . . . . . . . . C - 1 C . l A Ph ys ic al Approach t o th e Phase-locked Loop . . . . C - 1 C.2 The Li ne ar iz ed Model of t h e Phase-locked Loop . . . C - 4

C.2.1 The Closed Loop Tr an sf er Fu nc tio ns . . . . . C - 5 C.2.2 Modulation Trac king Error . . . . . . . . . . C-9 C.2.3 Loop Phase No ise . . . . . . . . . . . . . . C-13 C.2.4 Threshold Pr ed ic tio n . . . . . . . . . . . . C-15

ii ...- ... .- -.rrr..1. ...II.II.11 ... I I 1.1111 111 11111111 1111 11111111 1 1 1 m 1 ~ 1 1 1 1 1 1 11111 II



Sect ion Page

C . 3 Si gna l and N oise C h a ra c t e r i s t i c s o f P re l i m i t ed

Phase-Locked Loops . . . . . . . . . . . . . . . . . C . 3 . 1 L i m i t e r Effects on Loop Parameters . . . . .

C .4 Modulat ion R es t r ic t i v e Loop . . . . . . . . . . . . C.b.1 Loop Noise Bandwidth Above Threshold . . . .

C . 5 P re f i l t e r ed Modula tion Track ing Loops . . . . . . . APPENDIX D.PRODUCT DEMODULATION . . . . . . . . . . . . . . .

D . l Linear Product Demodulator . . . . . . . . . . . . . D . l . l D.1.2 D . 1 . 3 D . 1 . 4

D et ec t i on of S i nuso i da l Subc a r r i e r s . . . . . Detec t ion of Arbitrary Baseband Modu l a t i o n . . . . . . . . . . . . . . . . . . . N oi se C ha rac t e r i s t i c s . . . . . . . . . . . .Output Signal - to -no i se Rat ios . . . . . . . .

D . 1 . 4 . 1 Sub car r ier and band-pass f i l t e r . . . . . . . . . . . . . . .

D.1.4.2 Baseband mod ulation and low-pass f i l t e r . . . . . . . . . .

D.2 Pr el im ite d Prod uct Demodulators . . . . . . . . . . APPENDM E.DEMODULATION WITH MODULATION TRACKING LOOPS

. . . .

E . l D et ec ti on o f S i nus o i da l Subc a r r i e r s and Arbitrary Baseband Modulat ion . . . . . . . . . . .

E .2 N o i s e C h a r a c t e r i s t i c s . . . . . . . . . . . . . . . E.2.1 Low-pass Output F i l t e r . . . . . . . . . . . E.2.2 Band-pass Output F i l t e r . . . . . . . . . . .

E . 3 Output Signa l - to-nois e R at i os . . . . . . . . . . . E. 3 . 1 Su bca rr i e r and Band-pass F i l t e r . . . . . . . E.3.2 Baseband Modulation and Low-pass F i l t e r . . .

APPENDIX F.SPECIALIZED DETECTORS . . . . . . . . . . . . . . . . F . l Range Clock Rece iver and Code Co rr el at or . . . . . . .

F . l . l S i g na l T reat ment . . . . . . . . . . . . . . . F.1.2 Noise Treatment . . . . . . . . . . . . . . . .

iii

C - 1 8

C - 1 9

C - 2 0

C-22

C-24

D - 1

D - 1

D-2

D - 4D - 6D - 6

D-6

D -7

D - 9

E-1 E -3

E-4

E-5E-6

E - 8

E - 8

E-10

F-1

F-1

F-2F-5



Sect ion Page F.1.3 Signal - to -no i se Rat ios . . . . . . . . . . . F-10 F.1.4 Receiver Threshold . . . . . . . . . . . . . F-12 F.1.5 Range Code Ac qu is it io n Time . . . . . . . . F-13

F.2 PC M Telemetry Subcarrier Demodulator . . . . . . . F-18 F.2.1 Output Data Treatment . . . . . . . . . . . F-19 F.2.2 Ref ere nce Loop Treatm ent . . . . . . . . . . F-20

F.3 The Residual Carrier Tracking Receiver (Ground) . . . . . . . . . . . . . . . . . . . . . F-22

F.4 The Res idua l Car ri er Tracking Receiver (Spacec ra f t ) . . . . . . . . . . . . . . . . . . . F-25

F.5 The Spacecraft Turnaround Ranging Channel

. . . . .F-29

F.5.1 Equ ivale nt Noise . . . . . . . . . . . . . . F-32 F.5.2 Equiva lent Si gn al . . . . . . . . . . . . . F-35

APPENDIX G.PHASE MODULATED SIGNAL DESIGN . . . . . . . . . . G-1 G . l Solu t ion fo r Modulation Ind ice s . . . . . . . . . . G-2 G.2 Maximization of Su bc ar ri er Channel Si gn al -t o

no i se R a t i o s . . . . . . . . . . . . . . . . . . . G-7 G . 3 Boundary Condit ion on Residual Carrier . . . . . . G-7 G . 4 Si gna l E f f i c i ency . . . . . . . . . . . . . . . . . G-9

APPENDIX H.SUPPLEMETYTARY THEORY . . . . . . . . . . . . . . . H-1 H . l The Equivalent Noise Bandwidth of Linear

Networks . . . . . . . . . . . . . . . . . . . . . H-1 H.2 Equ ivale nt Noise Temperature of Linear

Systems . . . . . . . . . . . . . . . . . . . . . . H -5 H.2.1 Si ng le Networks . . . . . . . . . . . . . . H-5 H.2.2 Cascaded Networks . . . . . . . . . . . . . H -7

H . 3 The Band-pass Amplitude Limiter . . . . . . . . . . H - 1 0 H .4 The Range Equation . . . . . . . . . . . . . . . . H - 1 3 H .3 Antenna Polar izat ion Loss . . . . . . . . . . . . . H-17 H.6 In t e l l i g i b i l i t y o f C li pped Voice . . . . . . . . . H - 1 9

i v

.__-.. ...-.....- - 1....... 1.11 . I I, . I . I II I I.I. I 111 I 11- ~ I I I



Section Page

REFERENCES . . . R-1



TABLES

T a b l e Page C.2.2-I INPUT FUNCTIONS . . . . . . . . . . . . . . . . . . . C - 1 0 C .2 . 4.1 CONFIDENCE VALUES VERSUS LOSS-LOCK PROBABILITIES . . . C-17 C - 4 - 1 INPUT SNR VERSUS LOSS-LOCK PROBABILITIES . . . . . . . C-22 c .5.1 I"JT SNR VERSUS LOSS-LOCK PROBABILITIES . . . . . . . C-26 F.1.1-I PROGRAM STATE VERSUS CORRELATION . . . . . . . . . . . F-4

vi



FTGURES

Figure Page

2.2-1 Up-l ink channels . . . . . . . . . . . . . . . . . . . 9

3.3-1 The ranging channel . . . . . . . . . . . . . . . . . 17

3.4-1 PCM te le m e tr y cha nn el . . . . . . . . . . . . . . . . 20

3.5-1 The voice channel . . . . . . . . . . . . . . . . . . 23

3.6-1 The biomedic al da ta ch annel . . . . . . . . . . . . . 26

3 -7-1 The emergency voice channel . . . . . . . . . . . . . 29

3.8-1 The emergency key channel . . . . . . . . . . . . . . 32

4.1-1 The FM c a r r i e r c h a n n e l . . . . . . . . . . . . . . . . 34

4.2-1 The t e l ev i s io n channel . . . . . . . . . . . . . . . . 36

4.3-1 PCM t e l emet ry channel . . . . . . . . . . . . . . . . 38

4.4-1 The voice channel . . . . . . . . . . . . . . . . . . 41

4.5-1 The biomedical data channel . . . . . . . . . . . . . 43

B.1-1 In pu t noise spect rum . . . . . . . . . . . . . . . . . B-2

B.3-1 Lim iter model . . . . . . . . . . . . . . . . . . . . B-5

B.4.1-1 Nonpre l imi t ed product 'de t ec tor . . . . . . . . . . . B-6

B.4.2-1 Pre l imi t ed product de te e to r . . . . . . . . . . . . . B-8

B.4.3-1 Nonprel imited product mixer . . . . . . . . . . . . . B-9

c.1-1 Phy sica l loop model . . . . . . . . . . . . . . . . . c-1

c.2-1 Linea r loop model . . . . . . . . . . . . . . . . . . c-5

c.2.I-1 Asymptotic Bode p l o t s o f t r a n s f e r f u n ct io n s . . . . . C - 8

c.3-1 Pre l imi t ed phase- locked loop . . . . . . . . . . . . c-18

D.1-1 Demodulator configurat ion . . . . . . . . . . . . . . D-1

D .2-1 Demodulator configurat ion . . . . . . . . . . . . . . D-9

v ii



Figure Page

E-1 Demodulator configuration . . . . . . . . . . . . . . E-1

E- 2 Asymptotic Bode pl o t . . . . . . . . . . . . . . . . E-2

F.l-1 Range clo ck rec ei ve r . . . . . . . . . . . . . . . . F-1

F.L 5-1 E r r or p r o b a b i l i t y v e r s u s s i g n a l -t o - n o i s e d e n s i t yr a t i o . . . . . . . . . . . . . . . . . . . . . . . . F-17

F .2-1 PCMtelemetry subcarr ier demodulator . . . . . . . . F-18

F.3-1 C a r r i e r t r a c k in g r e c e iv e r . . . . . . . . . . . . . . F-22

F.4-1 C a r r i e r t r a c k i ng r e c e i v e r . . . . . . . . . . . . . . F-25

F .5-1 Spacecraft turnaround channe1 . . . . . . . . . . . . F-29

H.1-1 Linear network model . . . . . . . . . . . . . . . . H-1

H . 1 - 2 Contour of i n t e g r a t i o n . . . . . . . . . . . . . . . H -4

H .2.1-1 Equivalent noise temperature of a n o i s y l i n e a r n e t -

work . . . . . . . . . . . . . . . . . . . . . . . . H-6

H .2.2-1 Cascaded l inear noisy networks . . . . . . . . . . . H-8

H .2.2-2 Cascaded passive and noisy networks . . . . . . . . . H - 1 0

H . 3 - 1 Band-pass l i m it e r model . . . . . . . . . . . . . . . H-11

H . 3 - 2 Limi te r s ig na l and no ise suppress ion versus

in p u t SNR . . . . . . . . . . . . . . . . . . . . . . H-12

H * 3-3 Exact and approximate s ignal suppress ion . . . . . . H - 1 3

H.4-1 Communication l i n k model . . . . . . . . . . . . . . H-14

H.4-2 Antenna geometry . . . . . . . . . . . . . . . . . . H-14

H.5-1 E l l i p t i c p o l a r i z a t i o n . . . . . . . . . . . . . . . . H-18

viii



UNIFIED S-BAND TELECOMMUNICATION TECHNIQUES FOR APOLLO

VOLUME I1

MATHEMATICAL MODELS AND A-NALYSIS

B y John H. Painter and George HondrosManned Spacecraft Center

SUMMARY

This i s t h e second volume i n a s e r i e s done a t Manned SpacecraftCenter, documenting communication techniques used i n the Apol lo Uni f ied

S-band Telecommunications and Tracking System. A s s t a t e d i n t h e f i r s tvolume, NASA TN D-2208, the present document i s concerned with deta i ledmathematical modeling o f c e r t a in c h an n el s i n t h e sy st em . S p e c i f i c a l l y ,t h i s volume provides s imple mathemat ical too ls usable f o r pr ed ic t i ng ,approxim ately, th e performance of vario us communications and tr ac ki ngchannels i n th e sys tem.



SYMBOLSS i g n a l s t r u c t u r e :

A c ( t >

nwi

Acpreff

peak amplitude of a s i n u s o id a l c a r r i e r

a square-waveform, having values +1 and -1, which

m a y b e s u b s c ri p te d f o r i d e n t i f i c a t i o n

baseband modulation function of a frequency-modulatedcw r i e r

a r b i t r a r y s i g n a l f u nc ti on

outpu t fun c t ion o f an i d e a l b an dp as s l im i t e r

o u tp u t f u n c t i o n o f a n i d e a l m u l t i p l i e r

a de si re d sig n al time f un ct io n which may appear

w i t h i d e n t i f y i n g s u b s c r i p t s

peak f requency dev i a t ion i n rad ians / sec o f a s inus o id a l s u b c a r r i e r o n a down-link frequency

modulated carr ier

peak phase dev ia t ion , i n rad ia ns , o f a s in u s o id a l

s u b c a r r i e r o f a down-link phase modulatedc a r r i e r

peak phase de v ia t i on , i n rad ian s , of a s in u s o id a lsu bc ar rie r of an up- lin k phase modulated

c a r r i e r

peak phase d ev ia t ion i n rad ia ns o f a pseudo-random

ran gin g code on an up- l ink phase modula ted ca r r ie r

e f fe c t iv e peak phase dev ia t ion o f a pseudo-random

ranging code on down-link phase modulated carrier

equiv alent phase modulat ion fun ct io n on a s u b c a r r i e r

on a down-link frequency modulated c a r r i e r

equi valen t phase modulat ion fun ct i on on a down-linksubcarr i e r

equivalent phase modulat ion funct ion on an up- l ink

s u b c a r r i e r

equiv alent phase modulat ion f un ct io n of an angle

modulated carr ier

2



U t ) a modulat ing func t io n of an ang le modulated s inu soi dal

c a r r i e r

iu m o d u l a t e d r a d i a n f re qu en cy o f a s i n us o i da l c a r r i e rC

iui

radian f requency of a down-link subcazrier

e. ra d i an frequency of an up- l ink sub car r ie rJ

Phase-locked loops :

ampli tude of input s inusoid

amplitude of VCO s in u s o id

eq uiv ale nt one-sided closed-loop no is e bandwidth

Laplace trans form of e ( t )

maximum va lu e of e ( t )

loop modula tion t rack ing e r ro r fun c t ion

Laplace t ra nsfo rm of h ( t )

e q u iv a l e n t c l o s e d lo o p i n p ut o u tp u t t r a n s f e r f u n c t i o n

lo o p f i l t e r im pu lse re sp on se f u n c t i o n

open loop gain constant

VCO c o n s t a n t

p o l e f re qu en cy o f t h e l o o p f i l t e r

Laplace t ransform of v d ( t )

VCO d r iv in g f u n c t i o n

o u t pu t f u n c t i o n of v o l ta g e c o n t r o l l e d o s c i l l a t o r (VCO)

p e a k f a c to r f o r VCO p h a s e j i t t e r

z e r o f re qu en cy of t h e l o o p f i l t e r

loop damping ra t i o

v a r i a n c e o f VCO p ha se j . i t t e r p r o c e s s

Laplace t ransform of t p i ( t ) i n complex var ia bl e s

3



Laplace t ransform of output phase fun ct io n

input phase funct ion

VCO output phase funct ion

loop na tu ra l r esonan t f requency

a sample function of a narrow-band-limited whiteGaussian noise process

sample fu nc ti on s of independen t low frequen cy

white Gaussian noi se processes der ived f rom n ( t )

v a r i a n c e s o f t h e v a r i a b l e s n , x, y

absolute , nonzero, values of the f l a t s p e c t r a ld e n s i t i e s

I2:

(u)),@Y

(iu),@11

(u)) n oi se s p e c t r a l d e n s i t i e s o f t h e f u n c ti o ns x ( t ) ,

Y ( t > , n ( t >

I absolute , nonzero, value of t h e f l a t s p e ct r a ld e n s i t y

sp ec tr a l densi t y of an equivalent low frequency

white Gaussian phase process der ived from n ( t )

Miscellaneous:

B f i l t e r t r a n s m i ss io n ba nd wid th

BLequivale nt square t ransm iss io n bandwidth of an

i d e a l ba nd pa ss l im i t e r

BObandwidth of an o u t p u t f i l t e r

f mmidfrequency of an output bandpass f i l t e r

Kff i l t e r t r an sm is si on c o ns t an t

KP

peak t o r m s f a c t o r

K' m u l t i p l i e r c o n s t a nt o f an i d e a l m u l t i p l i e rcp

M f re qu en cy m u l t i p l i c a t i o n r a t i o NO

an output noise power pL

output power of a n i d e a l b an dp as s l im i t e r 4

I



-S

SNR”

vL

s igna l - to -no ise power ra t io

v o l t a g e l i m i t i n g l e v e l of a n i d e a l b an dp ass l i m i t e r

s i gn a l suppress ion fac to r of an ide a l bandpass l i m i t e r

1 . 0 INTRODUCTION

1.1 Background and Pur pose of t h e Document

Work s i m i l a r t o t h i s volume has been performed prev iousl y , ex te rn al

t o Manned Spa cecra ft Center. I n gene ral , such work w a s fragmentary andw a s performed t o meet c e r t a in immediate needs such as re sp on se s t o NASAre qu es ts f o r prop osa l. When work began on t h i s volume i n mid-1963, t h ea uth or s f e l t t h a t a n ee d e x i s t e d f o r a comprehensive t u t o r i a l documents e t t i n g down gene ral ana lyses of t h e types of channels employed i n t h e

Apollo system. I t w a s f e l t t h a t for such a document t o b e u s e f u l t o NASAengineers it shou ld con ta in , i n appendix fo rm, su f f ic ie n t bas ic exp lana

t i o n t o complete ly and independent ly support t he body o f the document.This work i s th e authors ’ answer t o t h a t need.

1 . 2 Theoretical Approach

1 . 2 . 1 Analyt ical ScopeThe a na ly s i s presented i n t h i s document has been performed with th e

a i m of o b t ain in g t r a c t a b l e e q u a ti o n s w i t h which the ou tpu t da ta qua l i tycan be predic ted for each channel for a va r i e t y of t ransm iss ion modesand conditions.

The approach has been t o der iv e output da ta s ignal- to-noise powerr a t i o s which a r e r e l a t e d t o t h e i np u t c a r r i e r -t o - n o i se r a t i o f o r eachcommunication channel, by an expression containing generalized signalmodulation parameters and channel trans miss ion parameters. The req uir ement t h a t th e channe l equat ions be t ra c t ab le w a s t ak en t o & pl y t h a t t h eexpress ions be r e l a t iv e ly s imple and amenable t o hand ca lcu la t io n wi tht h e a i d of mathemat ical ta bl es . Addi t ional ly , it w a s d e s i r e d t h a t t h eform of th e channel equat ions should give some in tu i t i v e in s i gh t in toth e operat ion of th e channel .

I n pu t -o u tp u t s i g n a l - to - n o i s e r a t i o r e l a t i o n s were d e ri v e d s e p a r a t e l yfo r each type demodulator and each type si gn al . Where simp lifyin g as

sumpt ions were made, they were s t a t ed e x p l ic i t ly i n the de r iva t io ns .Add i t iona l ly , th e most important of th e an al yt ic al assumptions have beenl i s t e d i n t h i s s ec ti on .

This ana lys i s has t r ea te d only des i red s ig na l s and the rmal sys temnoise . No a t tempt has been made t o t r e a t in termodulat ion ef f e c t s or

equ ipment no n l in ea r i t i e s . Sys tem no nl in ea r i t i e s , such as l i m i t e r e f f e c t sor t h e e f f e c t s of m od ulat io n r e s t r i c t i v e d e t e c t i o n , have b ee n t r e a t e d .

5



It i s not expec ted th a t thes e channe l equa t ions w i l l y i e l d r e s u l t sof abso lute accuracy. Rather , eas e of handl ing channel pr edi c t i on s has

been ob ta ined wi th t o l e r ab le accuracy th rough the u se of s im p l i f y in g a s

sumptions. The philosophy has been adopted t h a t t h e performance of th e

system analy zed he re may be measured i n t h e lab ora tor y . The accuracy of

th e p red ic t ion a l equa t ions hav ing been determined , a requ i red channe l pe r

formance margin may be employed f o r t h e purpo ses of p re di ct in g f o r o thertran smi ssio n modes and con dit ion s than th os e measured i n t h e l a b o r a t o r y .

1.2.2 Method of P r e s e n t a t i o nThis document i s , i n a c e r t a i n s e n se , t u t o r i a l , and i n an o th e r se n se ,

Much ma te ri al , which has been b a si c al ly d eriv eda working document.elsewhere, has been extended or mod ified and reprodu ced he re . Enough

ma ter ia l has been included t o make t h e document almost se lf -s u ff ic ie n tfo r the purpose of making performance calculations on th e system. The

scheme employed i n th e wr it in g of t h i s document has been t o p r e s e n t a l l

ba si c d er iv at io ns i n appendix form. The main body of th e paper w a s r e seyved for assembling the individual channel equations from componentequat ions appear ing in th e appendices . I n t h i s manner, th e m a i n body

of t h e t e x t i s us ef ul fo r working computations, and th e appendices pro

v i de t h e r e q u i r e d a n a l y t i c a l s u pp o rt .

1.2.3' AssumptionsThe most important of th e s imp lify ing assumptions which appear

throughout th e document ar e tab ula ted below as an a id t o t h e r e a d er .

a . Modulat ion re s t r i c t i v e phase-locked loops ope rate wi th no mod

u l a t i o n e r r o r , e xc ep t f o r Doppler e f f e c t s .

b . Modula tion t rack ing phase- locked loops opera te l in ea r l y wi th

r e s p e c t t o p h as e.

e . Thresholds fo r phase-locked devices may be defi ned on a l i n e a r

ba s i s a f te r t he method of Mar tin ( re f . 7) .

d. All p r ed e te c ti o n a nd p o st d et e ct io n f i l t e r s a r e i d e a l w it h f l a t

ampli tude t ransmiss ion ch ar ac te r i s t i c s and square cu t -o f f f requencyc h a r a c t e r i s t i c s .

e . I n pu t n o i s e t o a l l channels i s charac te r ized as be ing band-l imited, whi te , and Gaussian.

f . All a m pl it ud e l im i t e r s a r e i d e a l s n a p- a ct i on w i th i d e a l p re

l i m i t i n g and p o s t l im i t i n g f i l t e r s of e q u a l b an dw id th .

g . A l l d i g i t a l waveforms have an i d ea l square shape.

h . A l l o u tp u t d a t a s i g n a l- t o -n o i s e r a t i o s a r e d e r i v e d f o r c h an ne l

demodulators above threshold .

i. Proper s ig na l des ign insu res no in-channel in termodulat ionproducts .6



1 . 3 System Desc ript ion

A ph ys ica l de sc r ip t io n of th e syst em concept, space craf t and groundequipment, s ignal design, and system opera t ion of t h e u ni f i ed S-bandsystem has been discus sed i n volume I of t h i s s e r i e s , NASA TN D-2208.

Although th e pre se nt volume con tain s block diagrams of the system channels ,

it i s recommended th a t t h e re ade r fa m i l ia r i ze himself wi th volume I p r i o rt o r ead i ng t h i s volum e.

2 -0 GROUND-TO-SPACECRAFT CHCWNEL ANALYSES

The s i gn a l t r ans m i t t e d from t he g round t o t he spacec ra f t i s t aken as

a s i nu so i d a l ca r r i e r phase m odu la ted by a sum of rangin g code, up-datas u b c a r r i e r , a nd v o ic e s u b c a r r i e r . T h i s composi te s ignal i s demodulateda t t h e s p a c e c r a f t , a nd t h e ba se ba nd s i g n a l i s rou t ed t o t h e p remodul at ionp ro c es so r f o r r ec o ve ry o f t h e s u b c a r r i e r s , and a l s o t o t h e PM modulatorfo r down- link t ransmiss ion . In t h i s se c t io n , we w i l l presen t t he ana l y seso f t h e tr ansponde r ca r r i e r t r a ck i n g channe l, and a l so t h e vo ice and up-da ta channel s .

2 . 1 Carrier Tracking Channel

The per formance c r i t e r io n of the spacec raf t ca r r i e r - t ra ck ing channeli s t h e th resho ld of th e ca r r i e r - t r ac k i ng phase lock- loop . The per formanceof such a loop has been analyzed i n appendix C . 4 .

The input s ig n a l power a t t h e s p a c e c r a f t i s obta ined from equat ion

A . 3 . (7 ) , page A-8 , as

n L

where

A = s igna l ampl i tude

A + r = phase deviat ion of t h e up -l ink ca r r i e r by t he r ange code

Adj = phase deviat ion of t h e u p- li nk c a r r i e r by t h e j t h s u b c a r r i e r

The inp ut no is e power, computed i n a bandw id th equa l t o t he c a r r i e rt rack ing loop no i se bandwidth , B N , i s given by

7



where

= t h e magnitude of th e f l a t i n p u t n o i s e s p e c t r a l d e n s i t y .

From equations (1)and ( 2 ) , w e o b t a i n

where

The thr es ho ld val ue of may be determined acco rding t o t h e de si r ed

s p e c i f i c a t i o n of t h e p r o b a b i l i t y o f l o s s of c a r r i e r phase lock . The

reader may r e f e r t o a pp en dix C -4 where v e r s u s t h e p r o b a b i l i t y o fB

l o s s o f c a r r i e r p ha se l o c k i s t r e a t e d .

2 . 2 The Voice Channel

The sp ac ec ra ft voice channel i s shown i n f i gu re 2.2-1 a long wi thth e up-da ta c hannel . Le t t he nar rowest bandwidth p r i o r t o t he wide-band de tec tor be denoted as B

LP *T hen t he i npu t s i gna l - t o -no i se r a t i o

computed i n BLP

i s

where


I‘nil

= th e magni tude of t h e f l a t i n p u t n o i s e s p e c t r a l d e n s i t y

8

. . - .. . .-. . . ..... . . . 1111 I I, I



BVLOW - PASS

LIM ITER LOOP FILTER " FILTER b

rJB w= B l v - B v

NsVJ~,,, BAND-PASS - LOW - PASS BVLIM ITER LOOP FILTER " FILTER b

rJ-

B vBw= B lv -

BAND-PASS

LIMITER

B w B l p

BAND- PASSREF

LIMITER LOOP FIUER

a11- BW= BLUD

BLUD

Figure 2.2-1.- Up-link channels

FILTER



From equatlon D.1.4.1 (61, page D-7, w e o b t a i n t h e v o i c e s u b c a r r i e r

s ignal- to-no ise r a t i o computed i nBLV'

t h e bandwidth of t h e band-pass

l i m i t e r of th e vo ice channel . Thus, from t h i s appendix , and tak ing i n to

c o n s i d e r a ti o n t h e s p a c e c r a ft l i m i t e r suppress ion ( a s t r e a t e d i n s e c t i on

H . 3 ) , w e o b t a i n

where

aL = l i m i t e r s u p p r e s s i o n f a c t o r

A$r = p ha se d e v i a t i o n o f t h e c a r r i e r b y t h e r an ge code

A$v = p ha se d e v i a t i o n of t h e c a r r i e r by t h e v o i c e s u b c a r r i e r

A$j = p ha se d e v i a t i o n o f t h e c a r r i e r by t h e j t h s u b c a r r i e r

BLV= voice sub car r ie r demodulator p re de t ec t ion bandwid th

The qua l i t y of th e vo ice channel may be determined by computing the

voic e informat ion peak-squared s i gn al t o mean-squared no ise r a t i o . T h i s

r a t i o w a s chosen because much work has been performed (r ef er e n ce 2 3 ) re

l a t i n g t h e p eak -squared s i g n a l t o mean-squared n o i s e r a t i o t o i n t e l l i g i b i l i t y w it h cl i p p i n g de pt h as a parameter. Thus, from equat io n E. 3 . 2( 6) ,

page E-9, w e have

Using now equations ( 2 ) and ( 3 ) , we o b ta in

10

-...



Since

equa t ion ( 4 ) becomes

where

= peak f requency dev ia t io n of t he vo ice sub car r ie r by i t s

A f rpeak informat ion

BV= bandwid th o f vo ice channe l pos tde t ec t io n f i l t e r

Equation ( 6 ) g i v e s t h e v o i c e s i gn a l -t o - no i s e r a t i o i n t er m s of t h e r a n g e

code , th e vo ice channe l pa ramete rs , and L s u b c a r r i e r s t ra n s m i tt e d t ot h e s p a c e c r a f t .

2.3 The Up-Data ChannelRe l’er ring a ga in t o S igure 2.2-1, che i n p u t s i g n al - to - n o is e r a t i o

(sm) computed i n ELP

i s

-‘i

‘i

F r o m equa t ion D.1.4.1 (6), page D-7,we ob ta in th e up-da ta subc ar r ie r

SNR computed i n BLUD’ t h e bandwidth of th e band-pass l i m i t e r of t h e

up-data channel .

Thus

11



IXlS

A s in t h e c a s e oi" t h e vo ice channel, t h e peLk-squared s i gn a l t o mean-

s qu ar ed n o i s e r a t i o w i l l be used to determine th e up-data channel

qu a l i ty . Thus, f rom equat ion E . 3 . 2 ( 6 ) , page E - 1 1 , we have

where

- of th e up-da ta s ig na l .Kp = peak

Using now equations ( 2 ) , (3), and t h e f a c t t h a t

w e o b t a i n

where

aL

= l i m i t e r s u p r e s s i o n f a c t o r

A f peak= peak frequency dev ia t io n of t h e up-data su bc ar r i e r by

~i t s i n format ion

BUD = bandwia th of th e up-da ta channel pos tde t ec t io n f i l t e r

A%= phase d ev i a t i on o f t h e ca r r i e r by t h e r ange code

A%l= phase dev i a t i on of t h e ca r r i e r by t h e up-da ta subca r r i e r

APj= p ha se d e v i a t i o n of t h e c a r r i e r by t h e j t h s u b c a r r i e r

12



s i n c e t h e u p- da ta r e c e i v e r i s a subsystem separate from t h e s p a c e c r a f tS-band subsystem, i t i s assumed t h a t t h e ou tpu t da ta qu a l i ty o f t h e ,upd a t a r e c e i v e r may b e u n iq u e ly r e l a t e d t o t h e o u tp u t s i gn a l- t o- n o is e r a t i o

of t h e up-da ta su bc ar r i e r demodula tor, b] . Given a s p e c i f i c a t i o nB,,

f o r %] , equat ion ( 4 ) may be used t o i n f e r t h e c h a nn el q u a l i t y .N

UD BUD

3.0 SPACECRCIFT-TO-GBOUND PHASE MODULATED CHANNEL ANALYSES

3.1 Carrier Tracking Channel

T he re a re two per fo rm ance c r i t e r i a fo r t he c a r r i e r t r ack i ng channe l.One c r i t e r io n i s t h e i n p u t si g na l -t o -n o is e r a t i o a t which the channelt h re sho l ds . The o t he r c r i t e r i o n i s t he i npu t s i gna l- t o -noi se r a t i o a t

which the VCO p h a s e j i t t e r i s acceptab le f or Doppler t ra ck ing .

I n s e c t io n F.3, it w a s shown t h a t t h e ground c a r r i e r t rack ing loopmay be t r e a t ed fo r t h re sho l d as i n s e c ti o n C . 4 , given a knowledge of theloop's equiva len t th r es ho ld no i se bandwidth B

N '

The ground-rece ived s i gn a l power i n the r ec e iv er c losed loop no i sebandwidth i s obtained from equat ion F.5.2 ( 7 ) , page F-36, as

where

2-0

e O s = s i g na l suppre s si on f ac t o r due t o phase modulated noi se i nthe spacecraf t t u rnaround channel

A = peak value of t h e r e c e iv e d s i n u s o i d a l c a r r i e rg

13



and

h f j

The r eade r shou l d r e fe r t o s e c t i on F.3 f o r t h e d i s c u s s i o n of t h e

spacec ra f t t u rna round channel and t he de r i v a t i on o f equa t i ons ( 2 ) and

( 3 ) . Other t e rms a re def ined as :

= spa cecr af t tu rnaround channel phase ga in , neg lec t ing l i m i t e r

suppress ion

Acpreff = e f f ec t i ve phase dev i a t i on of t he ca r r i e r by t he t u rna roundrange code

A q j , A(ph = subc a r r i e r phase dev i a t i on on t h e up - li nk ca r r i e r

A q= sub car r i e r phase dev ia t ion on th e down- link ca r r i e r

ctLS

= s p a c e c r a f t l i m i t e r s i g n a l s u p p r e s s i o n f a c t o r

Ap.eff = e f f ec t i ve phase dev i a t i on o f t h e c a r r i e r by t h e tu rnaroundJ s-Jbca r r ie r s

The ground rec e iv er no i se sp ec t ra l den s i ty i s a t t r ib u t e d t o t h enormal syst em noi se sp ec t r a l dens i ty p l us t h e phase no i se

l'ni It rans mi t t ed f rom th e spacecra f t dur ing the tu rnaround process . Thet o t a l re c e iv e r n o is e s p e c t r a l d e n s i ty i s def ined here as

l% land i t i s t r e a te d i n d e t a i l i n s e c ti on F.5 .1 . Thus, t h e n o i se i n t h e

ground rece iver c losed loop noise bandwidth BN i s

N = 2 +

c 1 nTIBN(4)

The s igna l - to -no i se r a t i o computed i n the c losed loop no i se band

width may be obtained from (1)and ( 4 ) . Thus, we have

SC-

Nc-

II1



Since

equat ion ( 5 ) may be simpliyied. Thus

The th reshold va lue of may be sp ec if ie d as i n s e ct io n C . 4 .

3 . 2 Angle Tracking Channel

The closed l o o p noi se bandwidth of t he angle channel i s cons ider a b l y s m a l l e r t h a n t h a t of t h e c a r r i e r t r a c k i n g c h an n el . S in c e t h e a ng l echannel depends on the car r i e r channel VCO fo r phase r e fe renc e , t heangle channel performance i s d i r e c t l y t i e d t o t h e c a r r i e r channel per formance. I n pa r t i c u l a r , t h e angle channel does no t per form when theca r r i e r channe l t h re sh o l ds . T he re fo re , equa t ion 3 .1 (5) may a l s o beu s e d t o d e f i n e a n g le c h a nn e l t h r e s h o l d .

3.3 Ranging Channel

The ba si c model re qu i re d t o analyze the performance of th e rang ingchannel i s shown i n f i gu re 3.3-1. Si nce t he r ang i ng channe l i nc l udest he spacec ra f t t r ansponde r as w e l l as t he ground range c lock rec e iv er ,

t h e r e ad e r i s u r ge d t o r e a d s e c t io n s F . l and F.5 .

Various t e rms a r e def ine d be low:

'is= i n p u t s ig n a1 ,p o we r t o t h e s p a c e c r a f t r e c e i v e r

'n

i s

( w ) = s p a c e c r a f t i n p u t n o i s e s p e c t r a l d e n s i t y

= s p a c e c r a f t s i g n a l- t o - no i s e r a t i o a t l i m i t e r i n p u t ,

computed i n l i m i t e r bandwidth



aLS

= spacecraft band-pass l i m i t e r s i g n a l s u p p r e s s i o n f a c t o r

%S= spacecraft band-pass l i m i t e r bandwidth

vLs= spacecraft band-pass l i m i t e r v o l t a g e l i m i t i n g l e v e l

K b= spa cec raf t wideband de te c to r ga in cons tan t

= sp ace cr af t tu rnaround channel phase ga i n cons tan t+m

= spacec ra f t ou t pu t ca r r i e r powersos

S = i npu t s i g na l power t o t h e g round r ece i ve ri g

'n( w ) = ground t he rm a l inpu t no i se sp ec t r a l dens i t y

i g@nT(W ) = ground t o t a l " equ i va l en t " i n p u t n o i s e s p e c t r a l d e n s it y ,

i nc l ud i ng t u rned a round no i se

= ground s igna l - to -no i se r a t i o a t r ange c l ock r ec e i ve r i np u t ,

computed u s i ng t o t a l equ i va l en t no i se sp ec t r a l dens i t y

N= c l ock s i gnal - to -no i se r a t i o a t l i m i t e r input , computed in

l i m i t e r bandwidth

BLR= range c lock rece ive r l i m i t e r bandwidth

BN= range clock loop noise bandwidth

There a r e two per formance c r i t e r i a f o r the ran ging channel . One

concerns t h e s i gna l - t o -no i se r a t i o (SNR), r e q u i r e d a t t he i npu t of t h e

r an ge c l o c k r e c e i v e r t o i n s u r e t h a t t h e c l o c k l o o p ( s e e f i g u r e F.1-1)i s above th re sho ld . The second c r i t e r i o n concerns th e input S N R r equ i r ed

f o r a g iv en ra ng e code a c q u i s i t i o n t h e .

3.3.1 Clock Loop ThresholdGiven a s p e c i f i c a t i o n f o r t h e p r o b a b i l i t y o f loss of l ock , t he

t h re sho l d p ro pe r t i e s o f t he r ange c l ock l oop a re i m p l ied by t he c l ock

SNR a t t h e l i m i t e r i n p u t, , computed i n B t h e c l o c k lo o p n o i s e"

bandwidth. Combining equation F.1.4 (l), page F-12, and F.5.2 (4),

page F-36, we o b ta i n

16



C SS. \7BAND-mss PHASE

LIMITER I b. MODULATOR

A + m

REF:SIGNAL

LIMITER b CLOCK LOOP .

, K L R ~ B L R B N I

I 1 IRECEIVER

RE FSIGNAL

CODECODE I-'CLOCK

GENERAT0R

Figure 3.3-1.- The ranging channel



(1) where

aTreff = The equiv a len t tu rnaround phase dev ia t io n of t h e range code

on the down c a r r i e r as given by eq uat ion F. 5.2 ( 5 ) , page

F-36.

aV.eff = The equiv alen t turned-around phase dev iat ion of t h e up-J

sub ca r r i e r s on t he down ca r r i e r as given by eq uat ion F.5.2

(6), page F-36.

LD = a s i g n a l d e t e ct i o n l o s s as d e fi n ed i n s e c t i o n F.1.3 , pageF-12.

LKA c o r r e l a t i o n l o s s as d e f in e d i n s e c t i o n F.1.3, page F-11.

The input SNR computed i n BN i s given as

where

A = t h e pea k v a l u e of t h e s i n u s o i d a l c a r r i e r r e c e i ve d a t t h eg ground

CT2 = t h e mean-squared va lue of t h e turned-around phase no i se as'ps given by eq uat ion F.5.1 ( 5 ) , page F-34.

The value of may be obtained from equation F .5 .1 ( 1 2 ) , page F-33.

Using equation (1)and ( 2 ) , t h e r ange c l ock l oop may be t r e a t ed fo r

t h r e s h o l d as i n s e c ti o n C . 4 .

3.3.2 Range Code Acquisition T i m e

From f i g u r e F.1.5-1, page F-17, t h e a c q u i s i t i o n t i m e f o r t h e

18

...1--.1-.-.-.-.--.I.-.. ..-1.11.-.1. I.. 1 11 I I. 1 1 1 1 1 - 1 1 1 1 1 I111111111 I I I I I111111111IIIIIIIIII11111111I I I I II I I1111111111111111111111 11



pseudo-random ranging code may be d i r ec t l y re l a t e d t o t he r a t i o of

e f f ec t i v e ou t pu t s i gna l- t o -no ise sp ec t r a l dens i t y - , us ing equa t ion

P o lF.1.5 (lb), page F-16.

Combining equations F.1.5 (lo), page F-15, and F.5 (4), page F-30,we o b ta in

CY L K n

where

a n d t h e q u a n t i t i e s Ag’

CT

‘PS”

Acpreff, IQn,l)and Acp.eff a r e th e sameJ

a s i n s e ct io n 3.3.1.

3 . 4 The PCM Telemetry Channel

The PCM t e l emet ry channel i s shown i n f ig ur e 3.4-1. “he f igure

inc lude s only tho se components o f t he channel neces sa ry fo r t he ana l y s i s

t o fol low.

L e t us aga i n de f i ne t he channel inpu t s i gna l - t o -no i se r a t i o as

2- 0

e vsA13

I ‘I

where as before

A = am pl it ude o f t h e s i nuso i d a l ca r r i e r r ece i ved a t t h eg ground

BLP= band-pass f i l t e r b andw idth



Iu0

BAND- PASS BAND- PASSa-

FILTER LIMITER - ( I* LOOP FILTER

Bw = B L P Bw = B L T

x 2Ref

A

1- vco

I I

Low-PASSI b FILTER -

Bw = B T

Figure 3.4-1.- PCM t e l em e t r y channel



I 111 II I I I1 I 1111 I I I I 1 II1 I I1 I I 1111 I 1 I 11.11 1111111111.1111111111

Qn ( f ) = t o t a l e f f e c t i v e t t in p u tt tn o is e s p e c t r a l d e n s i t yT

A t t h i s p o in t the r e a d e r s ho ul d n o te c a r e f u l l y t h a t

only when th e t ransp onde r rang ing channel i s closed. When t h i s channeli s open, Q

"T( f ) incl ude s th e t ransponder turned-around noise as w e l l

as t h e t t n 6 m 1 t tground system inp ut noi se. The term Gn ( f ) i s given

T

i n appendi x F.6 .1 .

The recovery o f a s u b c a r r i e r u si ng a band-pass f i l t e r has beent r e a t e d i n a pp en dix D. Thus, f r m equat ion D . 1 . 4 . 1 (61, page D-7, andappendix F, equat ion F.5.2 (4), we ob ta in

K L

= 2 cos2(&reff) J:(%) J:(Acpi)i=1 j =1

i # T

where AcPreff and AcP.eff a r e as def ined i n s e c t i o n 3.1 andJ

2-0

e " = s i gn a l suppress ion fa c t or due t o phase modulat ion of t h edown-l ink carr ier by t ransponder turned-around noi se .

Now ( 3 )

Combining now equations ( 2 ) and ( 3 ) , we ob ta in

K L = 2 cos2(AcPreff

1=1 j=1

?]BT i # T LP

21



Considering now that

equa t i on ( 4 ) becomes

T= 2 cos2(A'P,eff) J:(AVT) J:(Acpi) ','(A,.'f)[>

The t e lemet ry demodulator i n th i s rep or t has been cons idered as as p e c i a l i z e d d e t e c t o r . A s such, it h a s b ee n t r e a t e d i n s e c t i o n F.3.The reader may r e f e r t o t h i s s ec t i on f o r discussion of the demodulatort h r e s h o l d .

3.5 The Voice Channel

The voice information i s t r ansm i t t ed from t he spa cec ra f t on a sub-

c a r r i e r which i s a l s o used fo r t ransmiss ion of b iomedical da ta . Thus,

th e ground su bc ar r i er demodulator i s common t o bo th vo ice and biomedic al

da ta channel s . The voice channel i s shown i n f ig ur e 3.5-1. The channel

i n p u t s i g n al - to - n o is e r a t i o i s de f i ned as

where

A = ampli tude of the s ig na lg;

BLP

= band-gass f i l t e r bandwidth

= i n p u t n o is e s p e c t r a l d e n s i t y

I@"Tl

A s i n t h e c as e o f PCM te l em etr y only when th e t ranspo nder

ranging channel i s closed. Otherwise; i t ' i s d e fi ne d i n s e c t i o n F.6.

22



Re fI

Lp b FILTER LOOP FILTER -B W= BLP

Figure 3.5-1.- The voice channel

Iuw



The s u b c a r r i e r s i g n a l - t o - n o is e r a t i o may be ob ta ined f r o m appeh

d i x D.1.4.1, e q u a t i o n ( 6 ) . Thus

K L

J i # v

A s shown i n f ig ur e 3.5-1, t h e v o i c e i n f o r m a tio n i s recovered wi th a low-passf i l t e r a t th e ou tpu t o f th e modula tion t r ac k in g loop . The voice channel

qu al i t y may be determined by computing th e peak-squared si g n a l t o mean-

s qu ar ed n o i s e r a t i o a t t h e output of th e voice channel low-pass f i l t e r .

T h i s r a t i o w a s chosen by the au thors , s o t h a t it may be used d i r ec t l y t o

e va lu at e v o ic e i n t e l l i g i b i l i t y . -

Now, from s e c t i o n E . 3 .2 ( 6 ) , page E -1 1 , w e have

Using now equations (2) and (3), we o b t a i n

("' =

6p rj2K

cos2 ( A v ~ e f f ) J t ( A T v )Nv

i f v

Now s i n c e

24



i f v

K = 6yqaj'cos2(Ayreff) J:(Aq) i=1

where

= peak f requency dev ia t ion of the vo ice subc ar r i e r by i t sM~ peak information

Bv = pos t de t ec t ion f i l t e r ba.ndwidth

= phase dev i a t i on o f t he c a r r i e r by t h e vo ice subc a r r i e r

Acpreff = phase dev ia t ion of the c ar r i e r by the range code as de

f ined i n appendix E. 6

Acp.eff = phase de v ia t io n of the ca r r i e r by the j th s u b c a r r i e rJ t u rned a round in the s pace cra f t as de f i ned i n

appendix E . 6

wi = phase de v i a t i on o f t he ca r r i e r by t he ith s u b c a r r i e r

o r i g i n a t i n g i n t h e s p a c ec r a f t

The voice channel threshold may be treated as in appendix C.

3.6 Biomedical Data Channels

The seven biomedical d at a channels are i de n t i ca l . Therefore , on lyone of them w i l l be analyz ed her e. One of th es e channels i s shown i nf i g u r e 3.6-1.

Examination of figure 3.5-1, sec t i on 3 . 5 , r e v e a l s t h a t e q u a t io n s (1)

and (2 ) of th e voice channel ar e th e same for t he b iomedica l da tachannels. W e may then proceed wi th th e s ignal - to-noise r a t i o of onebiomedica l da ta subcar r i e r a t t he ou t pu t of the voice and biomedical

da ta su bcar r i e r modula t ion t rac k ing loop . S ince the band-pass f i l t e r i s



Re f[ucT\

BAND-PASSFILTER LIMITER

B =BLp Bw = BL"

LOOP FILTER

v c o

I

BAND-PASS LOW- PASSLIMITER DETECTOR FILTER

Bw =BLB Bw= BEA

F i g u r e 3 .6 -1 . - The biomedical data c h a n n e l

0



u se d f o r t h e r e c o ve r y o f t h e s u b c a r r i e r , w e may r e f e r t o appendix E ,

s e c t i o n E. 3 . 1 , equat ion (81, page E-10 . Thus,

where

A fb

= f requency dev ia t ion of t h e voice and biomedical data sub-c a r r i e r due t o t h e b io me dic al d a t a s u b c a r r i e r i n q u e s t io n

f b= f requency o f t h e b iom ed ica l da t a sub ca r r i e r i n ques t i on

The d et ec to r shown i n biomedical da t a channel may be e i t h e r a modulationt r a c k i n g l o o p or an iDl d i s c r i m i n a t o r . Exclud ing th reshold cons idera t ions ,

t h e a n a l y s i s o f t h i s s e c t i o n h o ld s f o r e i t h e r ty p e o f d e t e c t o r .

The recov ery of da ta from a s u b c a r r i e r u s i ng a modulat ion t rackingl oop and a low-pass f i l t e r i s c ove red i n d e t a i l i n a p pe nd ix E. Thus,from se ct io n E.3.2, equ at io n ( 8 ) , w e f i n d t h a t

Using now eq ua ti on s (1)and (2) of se c t i on 3.6 and (1)and ( 2 ) o f t h i s

s e c t i o n , w e ob t a i n

cos2 (Avreff) J12(A%) fii=1

where

KP

= f a c t o r r e l a t i n g t h e peak t o rms valu e of t h e basebandmodula tion of the b iomedical da t a su bc ar r i e r i nques t i on

27



af, peak= t he peak f requency devia t ion of the b iomedica l data sub-

c a r r i e r by i t s modulat ion

BB '

= bandwidth of biomedical da ta po std ete ct i on f i l t e r

The res t o f the terms of equat ion (3) are as de f i ned i n s e c t i o n 3.6.Now s i nce

equat ion ( 3 ) becomes

'nT

The thre sho ld of th e biomedical data channels i s e s s e n t i a l l y t h a tof th e voice demodulator s in ce voice and th e biomedical data s u b c a r r i e r s

ar e f requency mul t ip l exed . The voice demodulator threshold treatment

may be found i n appendix D.

3.7 The me rg en cy Voice Channel

The emergency voice channel i s s h m i n f i g u r e 3.7-1. The readeri s reminded a t t h i s po in t t h a t t h e emergency vo ice s ig na l does no t con

t a in the " tu rn-around" no i se . Thi s i s because t h e vo i ce s i g na l i s modu l a te d d i r e c t l y on th e VCO output whi le th e turn-around rangin g channel

i s ina ct i ve d uring t h i s mode of t ransmission .

The vo ice channel input s ign a l - to -no i s e r a t i o may be defined as:

28



t

b

BAND-PASS LOW -PASS

FILTER FILTER

Bw= BL P B E

Ref

Figure 3.7-1.- The emergency voice channel

L



where

A = ampli tude of inpu t s ig na lg

BLP = bandwidth of band-pass l imiter

= i n p u t n o is e s p e c t r a l d e n s i t y

Again th e peak-squared si gn al t o mean-squared noise will be used for

eva l ua t i on of t he channel qua l i ty . Thus, from s e c t i o n D.1.4.2 ( 5 ) ,page D - 8

2= sin ("V peak)[2IB

LP

where

@%peak= peak phase dev ia t ion of th e c a r r i e r by the vo ice in f or

mation

Using now equation ( 2 ) , and t ak ing in to cons idera t ion the bandwidth

r a t i o , we obt ain

where

BE = bandwidth of low-pass f i l t e r used t o recover the voice in fo r mation

Now s i nce

equat ion (3) becomes



The threshold of the emergency voice demodulator i s e s s e n t i a l l y t h a t of

t h e c a r r i e r t r ac k in g loop. The c a r r i e r t r a c k in g loop th reshold has been

d i sc u ss ed i n s e c t i o n 3.1.

3.8 The knergency Key Channel

The emergency key channel i s shown i n f ig ur e 3.8-1. A s t h e f i g u r ei n d i c a t e s , a b e a t - f r e q u e n c y o s c i l l a t o r i s used for demodulation. Thereade r i s reminded t h a t no turn-around n o i se e x i s t s i n t h e c ha nn el s i n ce

the transponder ranging channel i s no t ac t i ve du r ing t ransmiss ion of

emergency key.

Again , th e channel input s ignal- to-noise r a t i o i s def ined as

When p re se nt , t h e emergency key si g n a l appears as a s u b c a r ri e r i nth e incoming si gn al . Thus, from equa tion D.1.4.1 ( 5 ) , page D-7, wef i n d t h a t t h e s i gn a l- t o -n o i se r a t i o of a subcarr ier , recovered by ab a n d - p a s s f i l t e r , a t the ou tpu t of a m o d u la t i o n r e s t r i c t i v e loop i s

where

ATK = phase dev ia t ion o f the ca r r ie r by the key subca r r i e r

However, since

( 3 ) Equation ( 2 ) becomes

No w s i n c e



w[u

31-BAND- W SS BAND-PASSd B L P

b F1LTER FILTER b qBw = B L P B w = B s K NK BK

Ref. BFO

Figure 3.8-1.- The emergency key channel



equat ion ( 4 ) becomes

where

BK = k ey p o s t d e t e c t i o n f i l t e r b and wid th

The emergency key chann el thr es ho ld i s e s s e n t i a l l y t h e c a r r i e r lo op

t h re sho l d . Thus, t he reader may r e fe r t o s ec ti on 3.1 f o r t h r e s h o l dre l a t i ons o f t he key channe l .

4.0 SPACECRAFT-TO- GROUND FREQUENCY MODULATED CHANNEL ANALYSES

The sp ac ec raf t f requency modulated c a r r ie r i s assumed t o con taint e l e v i s i o n a t baseband and two su bc ar r ie rs . One su bc ar r i er i s i d e n t i f i e das t h e PCM t e l em et ry sub ca r r i e r , and may conta in e i t h e r h igh or low b i t

r a t e real-t ime t e l em e t ry or appa ren t h i gh b i t ra t e recorded te lemetry.The o ther sub car r i e r i s i d e n t i f i e d as th e voice s ub ca rr ie r, and may cont a i n e i t he r r ea l - ti m e c l i pped vo i ce p l u s bi om ed i ca l da t a , or recordedvoice.

The fol lowing se ct io ns w i l l t r e a t s e p a r at e ly t h e o u tp u t d a ta s i g n a l t o- no is e r a t i o s f o r t e l e v i s i o n , v o ic e, PCM te lemetry, and biomedicalda ta . The rea der should no te th a t t he mathemat ica l re l a t i on sh ips w i l linvolve K s u b c a r r i e r s , r a t h e r t h an two, i n o r d e r t h a t t h e s e r e l a t i o n

sh ips remain genera l .

4 . 1 Carrier Demodulation Channel

The performance c r i t e r i o n f o r the c a r r i e r demodulator , which i s amodulat ion t racking phase-locked loop, i s i t s th re sh ol d. The demodu

l a t o r t h re sho l d may be t r e a t ed as i n appendix C given a knowledge ofthe equivalent c losed-loop noise bandwidth B

N'The FM c a r r ie r demcdu

l a t o r i s shown i n f i gu r e 4.1-1. The input s igna l - to -n o i se r a t i o i s

t aken as

33



BAND-PASSLIMTER b LOOP FILTER

BW' BLP

Figure 4.1-1.- The FM c a r r ie r c ha nne l



. . . . .. . ,., .. , .-..

where

s i n u s o i d a l c a r r i e r a mp li tu de

in p u t n o i s e s p e c t r a l d e n s i t y m agnitude

bandwidth of t h e band-pass l imi ter

A t t h i s p oi nt we w i l l assume t h a t t h e c a r r i e r channel l oop no ise bandwidth i s much larger than the bandwidth of th e band-pass l i m i t e r p re ceding it. There fore , the th resho ld of t h i s channel can be t rea te d as

i n s e c t i o n C . 5 .

4 . 2 Television Channel

The p erf orm an ce c r i t e r i o n f o r t h e t e l e v i s i o n c h an n e l is i t s outpu t

d a t a s i g n a l -t o - n o i s e r a t i o . S in ce t h e t e l e v i s i o n d em od ula to r i s t h e

ca r r ie r demodula to r, i t s t h r e s h o ld i s t r e a t e d as s t a t e d i n s e c t io n 4 .1 .Figure 4 . 2 - 1 shows t he te le vi s i on demodulator.

The inpu t s igna l - to -no ise r a t i o i s taken as

where

A = s in u s o id a l c a r r i e r a m p l i t u d e

= i n p ut n o i s e s p e c t r a l d e n s i t yI@ n iI= width of th e band-pass l i m i t e r which feeds th e c a r r i e r de-

BLPmodulator

The r a t i o of peak-squared si gn al t o mean-squared no is e out of the

output low-pass f i l t e r i s taken from equation E . 3 . 2 ( 6 ) , page E-11.

Thus

where AFTIT i s the peak f requency dev i a t io n in c yc les per second of

t h e t e l e v i s i o n s i g n a l on t h e c a r r i e r .

35



wcn

S 3

B ~p BAND- PASSLIMITER - LOOF

8, = B L P I

Figure 4.2-1.- The television channel



U t i l i z i n g a peak t o rms fac tor Kp, for t he t e l ev i s ion waveform,

as i n s e ct io n D.3 .2 , th e mean-squared ou tput s ignal - to -noise r a t i o

i s given as

F r o m equat ions ( 2 ) and (3 ) t h e da t a ou t put s i gna l - t o -no i se r a t i o i s

7 = 3 K 2[y MTvL9 [-12[3NTv BTV BTV BTv i Bw

However, since

then

(41

Tv

4 . 3 PCM Telemetry Channel

The performance cr i ter ia for t he t e l emet ry channel a re i t s outputdata s igna l - to -no i se r a t i o and the t e l emet ry demodulator th reshold .

Figure 4.3-1 shows the telemetry channel .

The i n p u t s i g n a l n o i s e r a t i o i s t aken a s

where

A = s i n u s o i d a l c a r r i e r a m p lit ud e

r n i I = i n p u t n o i s e s p e c t r a l d e n s i t y

BLp= width of th e band-pass l i m i t e r w hich f eeds t he ca r r i e r

demodulator

37



w03

BAND-PASS

LIMITER LOOP FILTER

STl

L O W - PASSFILTER

-b LOOP FILTER-SQUARE LAW

DETECTOR

FR EQ MULT.L-lIi

Figure 4.3-1.- PCM t e l e m e t ry channel



The t e l e m e t r y s u b c a r r i e r s i g n al - t o- n o is e r a t i o a t t h e c a r r i e r

demodulator output, computed in BLS, the bandwid th o f the su bca r r i e r

band-pass l i m i t e r , i s taken from equation E. 3.1 ( 8 ) , page E-10. Thus,

where

fT = subcar r ie r f requency

Ai?T

= peak f requency devia t ion of t h e s u b c a r r i e r o n t h e c a r r i e r

Equation (2 ) employs the assumption that

12 f TLp]2<<I

Now the outpu t dat a sign al- to- no ise r a t i o may be obtained. Thus,

(41

where

BT

= bandwidth o f t h e p o st d et e ct io n f i l t e r

Using now equations ( 2 ) and ( b ) , we obta in :

However, since



- -

The threshold of the te lem etry demodulator has been discussed i n

se ct io n F.3 where th e demodulator an al ys is i s given.

4 .4 Voice Channel

The performance c r i t e r i a fo r the vo ice channel are t h e d a t a o u tp u t

s igna l - to -no ise r a t i o and the vo ice su bc ar r ie r demodulator th resho ld .

The voice channel i s s h m i n f i g u r e 4.4-1.

The inpu t s igna l - to -no ise ra t i o , cmputed i nBLP’

the bandwidth

of t h e c a r r i e r ba nd -p as s l i m i t e r i s t aken as

-A2 2

where

A = s in u s o id a l c a r r i e r a m pl it ud e

= i n pu t n o i s e s p e c t r a l d e n s i t y

The vo ice sub car r i e r signa l - to -no ise r a t i o a t the carr ier demod

ul a t or output , computed i n BLs, the sub car r ie r band-pass l i m i t e r band

width, i s t aken fYm equa t ion E . 3 . 1 ( 8 ) , page E-10, as

where

fSV = voice s ubc ar r ie r f requency

Afm = peak f re qu en cy d e v i a t i o n o f t h e s u b c a r r i e r on t h e c a r r i e r .

Equation (2 ) uses the assumpt ion tha t

L p j <<IfSV

The peak-squared si gn al t o mean;squared no ise a t th e low-pass

f i l t e r o u tp u t , computed i n BV Y

the low-pass f i l t e r bandwidth i s t aken

from equation E.3.2 (61, page E-11 , as

40



BAND- PASS LOW-PASS- -LIMITER

B W - B L P

LOOP FILTER b LOOP FILTER FILTER

B W ' B v

Figure 4.4-1.- The voice channel

0



where

= peak f requency devia t ion of the vo ice s i gn a l on thepeak v o ic e s u b c a r r i e r

From equa t ions (2 ) and ( 4 ) t h e o u tpu t d a t a s i gn a l -t o -n o i se r a t i oi s given a s

('V peak

BV

= 2 [ . - [ f v B ; e a j 2 p ] :IBLp ( 5 )Af sv

f S VN

However, since

('V peak)1BV = 3rfS V I 2 rfVB;eaK] [>]NV * fSV i BV

The threshold of th e voi ce demodulator may be tr e a t e d as i n appendix C .

4 . 5 Biomedical Data Channel

The performance cr i ter ia for t h e b i o m e d i c a l ch a n n e l s a r e t h e d a t a

o u tp u t s i g n al - t o - n oi s e r a t i o , t h e i n d iv id u a l b iom e dic al s u b c a r r i e r d e

modulator th re sh ol ds , and th e voice demodulator thr esh old , s ince the

b iomedical s ubc ar r i e r s a re f requency mul t ip lexed wi th t he normal vo ice .

Only one biomedical channel i s t r e a te d i n t h i s s ec t i on , s ince the gene r a l equat ions a r e the same for a l l biomedical channels .

The inpu t s igna l - to -no ise r a t i o , computed i n BLp, t h e c a r r i e r

l imiter bandwidth i s t aken as

42



- bLIMIT ER LOOP FILTER LIMITER -B W - B L SB W - B L P

BAND- PASS LOW- PASS 'y'B-LOOP FIUERd LIMITER b FM DETECTOR b FILTER

B W - B L E Bw = B EL

Figure 4.5-1.- The biomedical data channel-Fw

0



- -

where

A = s i n u s o i d a l c a r r i e r a m p li tu d e

I 'nil

= i n p u t n o i s e s p e c t r a l d e n s i t y

The v o ic e s u b c a r r i e r s ig n al -t o -n o is e r a t i o a t the carr ier demod

u la to r output , computed i n BLs, t he subca r r i e r l i m i t e r bandwid th , i s

taken from equat ion E. 3.1 ( 8 ) , page E-10, as

where

fSV = voice subcar r i e r f requency

A fw = peak frequency deviat ion o f t h e s u b c a r r i e r on t h e c a r r i e r

Equation ( 2 ) uses the assumpt ion tha t

2

1 fSV12 [k] (3 1

The b iomedica l sub car r i e r s igna l -to -no i se r a t i o a t t he ou t pu t ofth e voice s ub ca rr ie r demodulator, computed i n

B~~~the biomedical

su bca rr i er band-pass l im i t er bandwidth, i s ta.ken from equat ion E.3.1 ( 8 ) ,page E-10, as

%IN2

- BLs

where

fSB = biomedical subcarr ier f ' requency

MSB = peak frequency de via t io n of the b iomedica l subcar r i e r on

t he vo ice subc a r r i e r .

Equation (4 ) uses t he a ssumpti ons t h a t t he no i se sp ec t r a l dens i t y ou t

of t he vo ice subcarr ier demodulator may be considered f l a t a c r o s s t h e

b i a n e d i c a l s u b c a r r i e r l im i te r bandwidth, and t h a t

44



2

1[k] << 1 ( 5 )

l2 fS B

The peak-squared s ig n a l t o mean-squared no ise r a t i o a t t he ou t pu tof t h e law-pass f i l t e r , computed i n BB, the low-pass f i l t e r bandwidth,

i s taken f’rom equat ion E. 3.2 ( 6 ) , page E-11, as

Using a peak-to-rms factor, KP’

for t he b iomedica l data s i gna l ,

th e mean-squared da ta s igna l - to -no i se r a t i o a t t he ou tpu t i s t aken

from equat ion E. 3.2 (7),page E-11, as

Cmbining equat ions (21, (41, ( 6 ) , and (71, the output biomedicald a t a s i g na l - to - n o is e r a t i o i s given as

However, since

equat ion (8) becomes

The threshold of the biomedical data channels i s e s s e n t i a l l y t h a tof th e voice demodulator s in ce voice and th e biomedical data s u b c a r r i e r s

are frequency mult iplexed.



APPFNDIX A

ANGL;E MODULATION

A s explained i n volume 1 of t h i s s e r i e s , t he A po ll o lunar communica t ion system employs s in us oid al angle modulat ion. I n pa rt ic ul ar , bothphase modulation (PM) and freq uency modulation (FM) are employed. Iti s th e purpose of t h i s appendix t o deriv e usable mathemat ical modelsfor two ang le modulated sig na ls. The f i r s t s i n u s o i d a l s i g n a l i s modul a t e d b y K s inu so id a l sub car r i e rs . The second s ig na l i s modulated byt h e sum of K s i nuso i da l subca r r i e r s p l u s a re cta ng ula r wave, generatedfrom a pseudo-random ranging code.

A.l Basic Considerat ions

A sin us oid al angle modulated s i gn al may be s imply represe nted as

s ( t ) = A CO S $ ( t ) (1)

where


$(t)= t im e va r i a t i on o f t h e s i nuso id

I f a s i gna l func t i on , f ( t ) , i s t o be i ncorpo ra ted i n t he s i gn a l ,two simple methods may be used. For phase modulation, l e t

$(t)= w c t f f ( t ) (m ( 2 )

then

sPM( t) = A cos kc t + f ( t ) ] ( 3 )

I f w i s t aken as t h e m o d u l a t e d f’requency o f t he s i nu so i da l s i gna lC

(or c a r r i e r ) , t h e n it i s s ee n t h a t t h e s i g n a l f u n c ti o n f ( t ) appearsd i r e c t l y i n t h e s i g n a l phase . For frequency modulation, l e t

then s,(t) = A COS w e t + f ( t ) d t

c 4 IA -1



Now, the instantaneous f ' requency of t he s i nus o i da l s i g na l may be def ined

as t h e time de r i v a t i ve o f t he ang le $(t)

It i s s ee n t h a t i n FM, t he s i gn a l func t i on f ( t ) a pp ea rs d i r e c t l y i nt he s i g na l f requency .

Based on th e preceding equation s, one no ta t i on may be used t or e p r e s e n t e i t h e r PM or modulated signals.

as ( t ) = A cos ECt+ ys(t)l (7)

v s ( t ) i s the equivalent phase modulat ion of t h e s i g n a l . I n t erm s o f

a s i g n a l f u n c ti o n f(t),f o r PM: c p s ( t ) = f ( t )

f o r FM: c p s ( t ) = J f ( t ) d t (9)t

Most usefu l per iod ic s igna l func t ions can be represen ted as terminated

F o ur ie r s e r i e s i n t h e form.

Kf(t)= a0 + c pi cos wi t + b i s i n W i t

1i=1

K

f ( t ) = c0

+ic=1

ci cos W i t + ei do r

K. 7 1

f ( t ) = d0

I-

i=1di s i n w . t + viI. _I

A-2



For FM, we may de fin e th e frequency fun ct io n

K6 (t) = c AWi COSS

i=1

where

AWi = peak rad ian f requency devia t io n due t o the ithcomponent

It i s s ee n t h a t t h e d e f i n i t i o n of e q u a ti on (16) p l a c e s n o r e s t r i c t i o n

on e . . For the case where b s ( t ) r e p r e s e n t s K s u b c a r r i e r s , t h e1

su bc ar ri er s themselves may be angle modulated, i n which case 8i i s

a funct ion of t ime. It i s d i f f i c u l t t o d ete rm in e, a n a l y t i c a l l y , t h e

equivalent phase modulat ion cp,(t) when 8i i s a func t i on o f t i m e ,

due t o the d i f f i cu l t y of in t eg ra t ing th e fYequency f’unction . However,t h i s d i f f i cu l t y does no t i nva l i da t e t h e concept o f r ep re sen t i ng FM by

an equivalent phase modulat ion.

A.2 The Ca rr ie r wit h K S u b c a r r i e r s

This s ig na l may be rep rese nted as

We w i l l examine f i r s t , t h e PM case. L e t

K r 1cp , ( t ) = c s i n pit +- cpd

i=1

Then

Equation ( 3 ) has previously been t reated by G i a c o l l e t o ( . r e f . l ) . Ther e s u l t o b t a i n e d i s

A-4



The Bess el f un ct io n expansions and summing pro ces ses lea din g t o equat i o n ( 4 ) a r e unaff ec ted by the t i m e behavior o f

(pi.Therefore ,

equa t ion ( 4 ) i s va l i d fo r ang l e m odula ted sub ca r r i e r s .

The r e s i d u a l c a r r i e r i s de f i ned as t h a t t e rm rem ai ni ng a f t e r modu

l a t i o n a t t he f r equency w of t h e unmodula ted ca r r i e r . Observa tionC

of equat ion (4) shows t h a t t h e r e s i d ua l ca r r i e r te rm o f f r equency wC

i s t h a t t e r m f o r which a l l ni

a r e i d e n t i c a l l y z e r o .

ni

E O ( 5 )

ThenK

s c ( t ) E A

The FM ca se may be tr ea te l exac t l y on l y f o r unmodu-ated subcarriers,

where B i i s con stan t . The cas e may be t r e a te d approximately i f t h e

gr ea te st f requency component i n'i

i s much l e s s t h a n t h e s u b c a r r i e r

frequency wi '

The e quiv alen t ph ase modulat ion i s

A-5



The equivalent peak phase deviat ion i s s ee n t o be

Aw,

ATi= r (9)i

Then, the FM si gn al may be repres ented as

Equation (10) h o ld s o n l y w i t h t h e r e s t r i c t i o n on Bi, prev ious ly

mentioned. The G ia co le t to ( r e f . 1) expansion i s t hen

A s i n e q u at io n (6) t h e r e s i d u a l c a r r i e r i s obtained as

A . 3 The Ca rr ie r wi th K Su b ca rr ie rs and Range Code

T h is s e c t i on ex tends t he G i aco l e t t o ( r e f . 1) expansion for aca r r i e r w hich i s phase modulated by K s i n u s o i d a l s u b c a r r i e r s plus a

square waveform, rep re se nti ng a pseudo-random ranging code.

The modulated signal i s r ep re sen t ed as

whereACPr

i s th e peak phase de via t io n of the range code on th e c ar r i er ,

and c t ( t ) i s a code waveform, hav ing on ly th e val ue s +1. I t - i s as

sumed th a t t h e code makes instan taneou s t ra n s i t io n s between the + and - 1s t a t e s .

A-6



Equation (1)may be expanded as

NOW

s i n Acp, ; c t ( t ) = +10

i-)( 3 )

- s i n0Acp, ; c t ( t ) = -1= c t ( t ) s i n (a ~ p

and

Combining equations ( 2 ) , ( 3 ) , and (4), we obtain

-K

w c t + C

mis i n

pit+ cpi

i=1

cos (“Pi-) K+

-ct(t) s i n (LY i-)

1 sin WC-t + C cCpi s i n pit + cpi$ ( 5 )

i=1

A-7



Equat ion ( 5 ) co nta ins two exp ress ions which may b e t r e a t e d w i t h t h eG i o c o l e t t o ( r e f . 1) expansion to give

r K l\

Equation (6) w i l l be used subsequent ly i n appendices D and G to o b t a i n

e x p l i c i t e x pr e ss i on s for de te c ted s ub car r ie r s and range code . A s i n

equa t ion A . 2 (12), page A-6, t h e res idual c a r r ie r t e rm may be ob ta ined

as

K

A- 8



APPENDIX B

NOISE

This appendix sets down a l l t h e r e l a t i o n s h i p s f o r n oi s e which a r e

used i n th e rema inder of th e document. The governing asswapt ion sh a l l

be t ha t a l l noise processes encountered a t t h e i n p u t s of t he va r i ous

systems and subsystems sh a l l be considered as charac te r i zed by Gauss i ans t a t i s t i c s . T h a t i s , a l l in pu t no is e wave forms w i l l be taken as sta

t io na ry , random, Gaussian proce sses . Since th e t re atm en ts of Gaussian

p r oc e s se s and c m b i n a t i o n s o f d e t e r m i n i s t i c s i g n a l s s m e d w i t h G au ss ia nn o i s e a r e w e l l documented, on ly the pe r t in en t re s u l t s w i l l be s e t down

h e r e, a lo n g w i th r e f e r e n c e s t o t h e o r i g i n a l t r e a tm e n ts .

B . l The Narruw-Band Ga us si an Random Proc es s

A Gaussian noise process which has a s p e c t r a l w id th , Af, much l e s s

than i t s center f requency , fc , can be expressed in a very meaningful ,

useful form. A sample func t ion of the process w i l l be represen ted as

n ( t ) . The sample fun ct io n n ( t ) may be expr ess ed as t h e d i f f e r e n c e o f

two components i n phase q uad ratu re as i n r e f er e n c e 2.

where w i s t he r ad i an cen t e r f requency o f t he spec rum o f n ( t ) .C

B enne t t ( r e f . 2 ) shows t h a t x ( t ) and y ( t ) are sample funct ions ofindependent Gaussian pro ces ses . Davenport and Root ( r e f . 3) show t h a tt he poss i b l e va l ues o f x ( t ) and y ( t ) a r e de te rm ined by G aussi an

va ri ab le s x and y, which have expected valu es, o r means, of zero,and whose var i ances a re r e l a t ed t o the var i ance of the or ig in a l samplef u n c t i o n n ( t ) b y

Moreover, i f th e narrow-band pro ces s i s c h a r a c t e r i z e d b y a n o i s e s p e c t r a l

dens i t y , ' P n ( w ) w a t t s pe r cyc l e o f bandw id th , t he sp ec t r a l de ns i t i e s o f

th e x and y components are r e l a t e d as

where equation ( 3 ) holds regard less of l i mi t a t io ns of bandwia th of th eo r i g i n a l p r oc e ss ( r e f . 2 ) .

B-1



I l l 1 I

A t ransformat ion f r o m r ec t angu l a r t o po l a r coo rd i na t e s y i e l d s

where p(t) i s i d e n t i f i e d as th e "envelope", and cp(t) the "phase" ofn( t ) . B o t h p ( t ) and t p ( t ) are sample functions of random processes

which ar e not independent (r ef . 3 ) . The d e n s i t y o f the random variablep i s Rayleigh, whi le th e den si ty of the random va ria bl e (p i s uniform

i n t h e r an ge (0, 27~).

It should be noted t h a t s i nce spec t r a l de ns i t i e s o f ti me func t i ons

a re e i t he r Fou r i e r t r ans fo rm s o f a u t o c o r r e l a t i o n f u n c ti o n s or products

of Fou rier t ransform s wi th conjugate Fourier t ransform s, and s ince the

o r i g i n a l time f lmct ions are real , t h en t h e s p e c t r a l d e n s i t i e s are two-s ided; t h a t i s , t h e s p e c t r a l d e n s i t i e s e x i s t f o r p o s i t iv e and n eg at iv e

r e a l f requenc ies .

A spec i a l c a se of th e Gaussian narrow-band pro ce ss occurs when thes p e c t r a l d e n s i t y , @,(W), may be considere d, wi th in th e narrow frequency

r e gi on o f i n t e r e s t , t o be a constant . That i s , f o r a center frequency,

W it may be assumed thatC'

2 0; a l l o t he r W ( 51

where i s a co ns tan t. The spectrum of mn(w) c o n s i s t s o f two

square b locks of in t e ns i t y I Q n I and width AW, centered on +wC

and

- W r e spec t i ve l y , as shown i n F igure B . l - 1 .C'

Figure B.l-1.- Input noise spectrum

Then t h e l o w frequency components have s pe ct ra l de ns i t ie s

= 0 ; a l l o t h e r w

B-2



Th i s s p e c i a l c a s e i s usefu l i n t h e t r ea t me n t of thermal ly genera ted

noise processes .

B.2 Angle Modulated Carrier Plus Noise

T hi s s e c t i o n t r e a t s t h e sum of an angle modulated (PM or FM) c a r

r i e r pl us Gaussian narrow-band no ise . It i s a s s u m e d th a t t h e c a r r i e ri s c en te re d i n t h e s p e c t r a l d e n s i ty of th e no ise . The s i gn a l i s r e p r e

sen ted as

s ( t ) = A cos f.t + v,(t)]

where

A = c a r r i e r a m p l i t u d e

w = c a r r i e r r a d i a n f re qu en cyC

cps(t) = carr ier "equivalent" phase modula t ion as t r e a t e d i n

appendix A

The sum of s i g n a l p l u s n o i se i s w r i t t e n as

where

n ( t ) = a sample function of a narruw-band Gaussian process

s(t) + n ( t ) = A cos Ect + c p , ( t )1 + x ( t ) cos wCt - y ( t ) s i n u)Ct

s ( t ) + n ( t ) = cos q s ( t ) + x ( t q cos w Ct - [A s i n ps(t) + y ( t g sin w Ct

A t r ans fo rma t ion t o po la r coo rd ina te s g ives

s ( t ) + n ( t ) = A ( t ) CO S ( 5 )

B- 3



where

A cos V s ( t ) + x ( t )cos $ ( t )=

A( t)

+(t)= a r c t a n

Equation ( 5 ) show s t ha t t he e f f ec t of summing Gaussian noise with anangle modula ted ca r r i e r can be in t e r pr e t e d as t ha t o f p roducing a s i g nal which i s simultan eously a mpli tude and ang le modulated. It i s seen

that the ampl i tude or envelope funct ion A ( t ) i s not nega t ive .

For t h e s p e c i a l c a se of an an gle modulated si g n a l embedded i n

narrow-band white Gaussian noise wi th a r e l a t i v e l y h ig h r a t i o o f ca r r i e r to-noise , Bennet t ( r e f . 2) has shown that equat ion ( 3 ) and ( 5 ) may bewell approximated by

s ( t ) + n ( t ) S A COS w c t + ' p s ( t ) + A A2 >> y 2 ( t ) (10)1Thus, i n th e low-noise case, ad di t ive whi te band-l imi ted Gaussian noise

may be c ons ide red t o add a s e p a r at e " ph as e" j i t t e r t o an angle modulated

sig na l . Where th e approximation holds, th e phase no ise may be consid ered

t o be a sm@e func t i on o f a Gaussian process. Thus, for t h e s p e c i a l

case , t he sp ec t r a l dens i t y , @ (W), of the phase noise may be expressedas

'p

o r

where S i s th e ang le modulated c a r r ie r puwer.



--.- . . -- . .

B . 3 Transmis sion of Sig nal Plu s Noise Througha Pe rf ec t Band-pass Li mite r

The treatment of pass ing a car r ie r p lus Gauss ian no ise th rough a

l i m i t e r i s complex and has been performed by Davenport ( r e f . 4), Mid

d l e t o n ( r e f . 5 ) , and others. Sec t ion H . 3 summarizes some resu l t s ofDavenport s a n a ly s i s .

The treatment here w i l l be more in t u i t iv e , drawing on Davenp ort’sr e s u l t s as needed. Figure B . 3 - 1 shows the block model.

s ( t ) Band-pass z ( t ) I d e a l ~ Band-pass - l(t),

f i l t e r l i m i t e r f i l t e r

Figure B.3-1 . - Limiter model.

The sum o f i n p u t s i g n a l p lu s n o i se i s taken i n polar form from

equat ion B.2 ( 5 ) , page B-3, as

s ( t ) + n ( t ) = A ( t ) co s I.Ct + $(tqThe i d e a l c h a r a c t e r i s t i c s o f t h e l im i t e r ou tp ut V , ( t ) are represented

i as

v1( t ) = +vL; Z ( t ) > 0

= -vL; z ( t ) < o ; z ( t ) = s ( t ) + n ( t )

= 0 ; Z ( t ) = 0 Jwhere

VL = v o lt ag e l i m i t i n g l e v e l

The o ut pu t f i l t e r o f t h e l i m i t e r i s assumed -0 be a p e r f e c - b an d-p assf i l t e r h aving a f l a t ampli tude t ransmiss ion ch ar ac te r i s t i c , square f re quency cu t -o f f ch ar ac te r i s t i c , and f l a t phase t ransmiss ion charac te r i s t i c

acr oss th e pass-band. The trans miss ion cons tant i s a r b i t r a r i l y t a k e n asuni ty . The f i l t e r pass-band i s assumed wide enough t o pa ss a l l zonale ne rg y a s s o c i a t e d w i th t h e c a r r i e r f re qu en cy u) and narrow enough t o

C

r e j e c t a l l other zones. From Davenport (ref. 4), t h e t o t a l powerPL

o u t o f t h e ba nd -p ass f i l t e r i s taken as

B- 5



2

PL =8pJ ( 3

The output waveform l(t) i s t aken as

Jcl(t)= 42 cos pCt + +(t) l ( 4 )

A s i n equa t i on B. 2 (lo), page B-4, for t h e s p e c i a l c a se of narrow-band,

whit e Gauss ian no i se and h igh car r i e r - to -n o i se ra t i o , equa t ion (4)maybe approximated as

vLl(t) 4-7T cos + c p , ( t ) 4- A ; A2 >> y 2 ( t ) ( 5 )

For t h i s s p e c i a l c a se , t h e ph ase n o is e s p e c t r a l d e n s i t y i s given as

where

1 'n 1 = cons t an t value o f t h e wh it e n o is e s p e c t r a l d e n s i t y i n t o t h el i m i t e r

S = l i m i t e r i npu t s i g na l power

Aw = bandwidth of t h e i n p u t no i se s p e c t r a l d e n s i ty

B .4 Transmission of Si gna l Plus Noise Through

a Perfect Product Device

B.4.1 A Nonprel imited Product Detector

The product det ec to r of f ig ur e B.4.1-1 i s fe d an ang le modulated

s i g n a l p l u s narrow-band w hite Gaussian no is e. The re fe re nc e s ig n a l has

nega t i ve s i ne phase w it h r e spec t t o t he ang l e m odula ted ca r r i e r .

- s i n w tC

Figure B.4.1-1.- Nonprelimi ted product d ete cto r

B- 6



I

The product detector i s assumed t o have some ga in cons tan t , K and t ocp y

r e j e c t a l l exce pt t h e "d. c. '' or d i f f e r e n c e terms of the product . Then

m ( t ) = -Ki s i n w e t [s(t) + n ( t q

cp

(2 1The dif f er en ce terms a t t h e o u tpu t a r e

It i s s ee n t h a t t h e o u tp u t f u n c t i o n i s e a s i l y s e pa ra b le i n t o a signalcomponent, s

m(t),and a noise component, nm(t), as

n m ( t ) = K' yo( 5 )r p 2

The ou tpu t no i se s pe c t ra l den s i ty i s given as

2

= o ; a l l o t h e r w (7)B.4.2 A Pre l i mi ted Produc t D e tec to r

The band-pass l i m i t e r of f i g u r e B.4.2-1 i s fe d an an gle modulatedsi g na l pl us narrow-band Guassian noi se and dri ve s a p r o d u c t d e t e c to r ,

hav ing ga in cons tan t K'cp.

B-7



The re fe rence s i gn a l has negat ive s in e phase .

- s i n W tC

Figure B. 4.2-1. - P r e l im i t e d p r o d u c t d e t e c to r

From equa t ion B .3 ( 4 ) , page R-6,

JIl(t)= 4 5 cos

pet + c c t qwhere VL i s v o l ta g e l im i t i n g l e v e l . Equation B.2 ( g ) , page B-4, de

s c r i b e s $(t). Then

m ( t ) = -K'' s i n w e t l(t) ( 2 )

V K'm ( t ) = -4-

fl's i n w

Ct cos pet + $( t i ] (3)

The "d.c." or d i f f e r e n c e term i s given as

2V K'm ( t ) =-' s i n $(t) ( 4 )

rl

Prom equa t ion B. 2 (7) page B- 5

For t h e s - pc i a l c as e o f r e l a t i v e l y hi gh s i gn a l -t o - n oi s e r a t i o i n t o t h e

l im i te r , the approximation ho lds th a t

A ( t ) Z A (6)

B-8



and

Where t he approx imation hold s, th e product de tec tor ou tpu t i s sepa rab l ein to s i gn al and noise components

2V K’

sm(t) =-’

fl9 s i n q s ( t )

2v P’n

m( t )= -L -

Y ( t ) ( 9 ).rrA

and t he ou t pu t no i se spec t r a l dens i t y i s given as

= o ; a l l o t h e r w

B.4.3 A Nonprelimited Product MixerFor th e device shown i n f i gu re �3.4.3-1, s e v e r a l c o n d i t i o n s a r e s t a t e d .

The input t o th e product dev ice , hav ing ga in cons tan t K I P , i s an angle

modulated s ig n a l and narrow-band Gaussian no is e, ce nte re d on a r ad i anfrequency w . The r e fe rence s i g na l ha s nega t i ve s i n e phase , a rb i t r a r i l y ,

C

and i s of a frequency wd s uc h t h a t t h e o u tp u t s p e c t r a o f t h e sum and

di f fe ren ce t erms do no t over l ap . The id ea l band-pass f i l t e r t ran smi t sa l l ene rg y a s s o c i a te d wi th t h e d i f f e r e n c e t e r m s, a r b i t r a r i l y , and r e j e c t st h e sum terms.

6- s i n w t

C

Figure B.4.3-1.- Nonprelimited product mixer

B-9



The band-pass f i l t e r has a n a r b i t r a r y g a i n c o n st a ntKf .

vo(t) = Kfktjd i f f e r e n c e

terms

+ cp,(t) + x ( t ) i n (".- wd)t1+ y ( t ) cos

("c - "d)t)

It i s seen t h a t t he ou t pu t func t i on con s i s t s o f a signal component and

a noise component.

B-10



The ou tput s ig n al a.nd nois e fun cti on s have t he same form as t he i npu tfunc t ions , be ing merely mul t ip l i ed by cons tan t s and t ran s l a t ed i n f r e

quency. Equation (6) can be t rans fo rm ed t o t he usual recognizable form

of equat ion B.l (l), page B-1, by a s u i t a b l e r e d e f i n i t i o n o f v a r i a h k s .

For th e s pe ci al case of white Gaussian noise a t t h e input , t he ou tpu t

n o is e s p e c t r a l d e n s i ty i s given as

c d ( 7 )0

= o ; a l l o t he r w

It i s i n t e r e s t i n g t o c ompute t h e r a t i o o f o u t p u t c a r r i e r p w e r So t o

tne magriitude of t he output noise sp ec t r al den si t y f o r th e white Gaus

s i an ca se

where

S = i npu t c a r r i e r power

I$( = magnitude of t he i npu t no i se spe c t r a l dens i t y

It i s s ee n t h a t th e r a t i o i s constant through a perfect product mixerand i d ea l band-pass f i l t e r , r ega rd l e s s o f t ransmiss ion ga in cons tan t s .

B-11



APPENDIX c

PHASE-LOCKED LOOP THEORY

C . l A Phy sica l Approach t o t he Phase- locked Loop

Figure C.l-1.- Phy sica l loop model

Figure C . l - 1 shows a model of th e phase-locked loop which i s essent i a l l y a c losed loop feedback mu l t i p l i e r . The dev ice con s i s t s o f a

m u l t i p l i e r , h av in g m u l t i p l i c a t i o n c o n s ta n t K 'cp' a l o o p f i l t e r , of t h elaw-pass type, and a v o l ta g e c o n t ro l l e d o s c i l l a t o r ( V CO ) . The physicalope ratio n of t he loo p may be st be explore d by assuming a n o i s e l e s s i n p u ts i g n a l , s i ( t ) .

The input s i gn al i s t aken as an angle modulated sinu soid of ampli

tude A, frequency W having "effect ive" phase modulat ion c p i ( t ) .C'

1 s . ( t ) = A CO S bCt+ ( P i ( t q

Without l o s s o f g e n e r a l i t y , t h e VCO may be considered t o producean angle modulated s inu so id of am plitude Av, f requency W

C,having an

effect ive phase modulat ion cp0

(t),and having neg ativ e si ne phase with

resp ec t t o the in pu t s igna l . Assuming nega t ive s in e phase and a

c-1



frequency W e x a c t l y eq u a l t o t h e i n p u t s i g n a l f r e qu e nc y i s n o t a re-C

s t r i c t i o n on g e n e r a l it y , s i n c e a term l i n e a r i n t and a cons tan t may

be t aken i n the o therwise unspec i f ied ou tput phase c p o ( t ) .

The output of the mu l t ip l i er m ( t ) i s

It may now be assumed t h a t th e m u l t i p l i e r i s f u l l y b a la nc ed sothat the double f ' requency term i s r e j e c t e d . A ls o, t h e lo w-p ass f i l t e r

w i l l not pass double flrequency components. I n any even t, the des i redlow-f requency mu l t i p l i e r ou tpu t i s given as

The VCO d r i v i n g f u n ct i on v d ( t ) i s given as t h e time convolut ionof the mul t i p l i e r s i gn a l wi th the impulse response func t ion h ( t ) ofth e low-pass f i l t e r .

A ss mi ng z er o i n i t i a l c on d it io n s a t t i m e zero

W

c-2



The output phase c p o ( t ) o f t h e VCO i s p r op o rt io n a l t o t h e time

i n t e g r a l of t he d r i v i ng s i gna l . The p ro po r t i ona l i t y cons tan t i sKv

t

=%Jv d ( t ) d t

0

Thus

Equation (9) i s t he exac t non l i ne a r . i n t eg r a1 equa ti on g i v ing t he

VCO phase response t o an inp ut phase funct io n. Equat ion (9) shows thatthe output phase c p o ( t ) responds t o the input phase cpi(t), b u t t h e

equat ion does no t g ive one an i n t u i t i ve " f ee l " for th e manner i n whichthe loop responds.

Suppose, through some uns pec ified means, over some i n t e r v a l o ft ime, th e outp ut phase approaches and remains ne ar th e inp ut phase, saywi th in 30". Then the s ine func t ion of the input -ou tpu t phase d i f fe rence

i s ve ry nea r l y t he phase d i f f e rence i t s e l f . Equation (9) may then bere wr i t te n, wi th good approximat ion, as

Equation (10) i s l i ne a r and ea s i l y L ap lace t r ans fo rm ab l e. First , assum

i ng t he i n t eg rand i s w e l l behaved, the order of i nt eg ra t io n may bechanged t o g ive

D ef in i ng t he k p l a c e t r ansfo rm o f cp (t) t o be @ ( s ) , equat ion (11)maybe transformed t o

c- 3



No w t h e i n t e g r a l o ve r 7 o f t he l oop f i l t e r impulse response i s simply

the Laplace t ransform or t h e t r a n s f e r f u n c t i on o f t h e l oo p f i l t e r which

w i l l be labeled FL(s). Then

Q o ( S ) = (14)

Next, a ga i n cons t an t for t he l oop i s def ined as

K =aV&v2

and equat ion (13) i s rea. rranged a s

The form of equation (15) i s w e l l known f r o m t he t heo ry o f servo

mechanisms, representing a l i n e ar servo loop whose ou tput cp , ( t ) i ssub t r ac t ed f r o m i t s i npu t v i ( t ) , and having a l oop ga in func t ionF,(4

KSL. The t ra ck ing pr op ert ies of servomechanisms are we l l known and

well documented ( re f . 6). There fore, once th e VCO phase of a phase-

locked loop i s brough t s u f f i c i en t l y nea r t he i npu t phase s o that t h e

loop ope ra t e s l i ne a r l y , t hen t he l oop ope ra t e s as a l inear servomechan

i s m for phase. The loop output phase w i l l "t rack" the input phase

wi th in the dynamic capabi l i t i es of t he s e rvo as determined by the loop

f i l t e r t r ans fer func t i on FL( s ) ; hence, th e name "phas e-locke d loop.

C . 2 The Lin ear ized Model o f th e Phase-locked Loop

From t h e work i n the preced ing sec t ion , i t i s e v i d e n t t h a t as long

as the output phase t r ac ks the inpu t phase c lo se ly enough, say wi th in

c-4



30°, th e loop may be d esc ribe d for phase by l i ne a r t r a ns fe r func t i ons.

Equation C . l (l?), page C - 4 , may be w ri t t e n i n an expanded form f r o mwhich the equ i va l en t l i n ea r model of the loop, for phase, may be drawn

by inspec t ion .

f igure C. 2-1. - Lin ear loop model

Inspec t ion of equa t ion (1)and f igure C . 2 - 1 shows that the mul t i p l i e r ha s been r ep l aced by a phase sub t rac tor wi th ga in co ns tan t K

cp'which i s p r o p o rt i o na l t o t h e m u l t i p l i e r g a i n c o ns t an t K '

cp'and the

ampl i tudes of t h e i n pu t s i g n a l A and VCO s ignal %.

C . 2 . 1 The Closed Loop Transfer FunctionsFrom t he l i n ea r m odel o f t h e l oop t h r ee t r a ns fe r func t i ons of i n t e re ' s tmay be der ived . These th re e func t ions re l a t e th e input s ign a l phase

Q. ( s ) t o t h e o u tp u t s i g n a l ph as e QO( s ), t h e VCO d r i vi n g s i g n a l V D ( s ) ,1

and t h e phase e r ro r s i g na l E(s).

c- 5



KP[l - G(s]

1 +y r c p F L

S

Equat ions (l), (2), and ( 3 ) hold f o r any loop f i l t e r . They may be

s p e c i a l i z e d f o r t h e usual loop f i l t e r t r an sf er func t ion which has one

each, re a l , f i n i t e t ransmiss ion zero , and po le .

where

Kf = f i l t e r cons tan t , a dimensionless number

The closed loop t r a n s f e r f u n c t i o n may now be w r i t t e n as

s - zG ( s ) = Iv(tpKf 2i. + s(!%%Kf - 4 - KJCciKfZ

( 5 )

A t o t a l g a in c o n st an t K may nuw be def ined as

The three t r an sf er func t ions may be r ew r i t t e n as

Is - z

G ( s ) = K ( 7 )+ s ( K - p ) - Kz

C-6



V D W s(s - 2) m,o = K6f[

s2 -f- s ( K - p ) - KzI ( 9 ) 1

It i s seen t h a t the d en om in ato rs o f t h e t r a n s f e r f u n c ti o n s a r e of- the form

D(s) = s2 + 2 5 W n s + wn2 E s2 + s ( K - p) - Kz (10)

where

A va l id approximation f o r most second orde r loops i s tha t

K>> - p (121

t h e n

and

n

2 1 - 2 5

It i s i n form a t ive t o d r a w the asymptotic Bode diagrams of t h esteady s t a t e t r a n s f e r f u n c t i o n s . It should be understood t h a t these

diagrams are v a l i d only so long as t h e loop i s locked and i s opera t ingl i n e a r l y . The diagrams apply on ly t o per iod ic input phase func t ions .There are ce r t a i n aper iod ic phase func t ions which w i l l cause t h e loopt o become n onl in ear and to unlock. These func t i ons w i l l be examinedi n a fo l lowing sec t ion .



I

ab I I

a.

(

' IPII III

b.

c

C.

C

a.C

Figure C. 2.1-1. - Asymptot ic Bode plots of t ransfer funct ions



C.2.2 Modulat ion Tracking Error

I t has been s t a t ed t h a t t h e l i n ea r tr ea t m en t o f t h e phase- lock l oop

i s v a l i d o n l y so long as t he i n s t an t aneous e r ro r between input phase

p i ( t ) and ou tpu t phase p0( t ) remains smal l ; say l e s s than 30°. T h i s

sec t i on exam ines t h e e f f ec t s o f ce r t a i n i npu t s i gn a l phase func t i ons o n

t h e e r r o r f u n c ti o n.

e ( t ) = vi( t ) - cpo(t) (1)

A ltho ug h t h i s e r r o r a n a l y s i s i s l i n e a r , w hi le t h e v e ry e r r o r it a t t em p t sto ana lyze causes the loop t o become n onl inear , s t i l l , t h e r e s u l t s areuse f u l fo r i n fe rence o f t he l oop ope ra ti on . The an a l y s i s w i l l be ca r r i edthrough by eva lua t ing th e inverse Laplace t ransform of t h e e r r o r f u n c ti o n

f o r e a ch of fou r inpu t phase s igna l s . The input s ig na l s and corresponding Laplace t ransforms are given i n t a b l e C .2 .2-I .

~ .

Case

1 2

K2-2

S

3 2 53S

K K K w4 U ( t ) C K~ s i n w n t c n n

n = l n = l s2 + w2

n...

TABLE C.2.2-I I N P U T F UNC TI ONS

U ( t ) i s t h e u n i t s t e p f un c t io n , a s de f i ned as

u ( t ) = 1 ; t 2 0

= O ; t < O

Case 1 i s a phase s te p inpu t of ampli tude K1 r ad i ans . Case 2 i s a

phase ramp inp ut wit h slope K2 radians/second. Case 3 i s a phase

a c c e l e r a t i o n w i t h a c c e l e r a t i o n 5 radians/second2. Case 4 i s a sm

mation of si ne waves having amp li tudes Kn and f requencies u)n'

c-9



The e r r or funct ion t o be t r ea te d i s given as

E(s) = mi(s) - B0(s) E

K S(S - P )

G i b ) ( 3 )

Case 1:

K1=-

S

Icp( s - PI

(3)E(s) =

K K

s2 + s ( k - p) - Kz

where 1

4 = t a n-1

- 4 - h K z - ( K - I( 5 )

It i s seen tha t t he re i s no steady-state error , only a t ransient . Thepeak tr an si e nt e rr or may be found by se t t i ng the f i r s t der ivat ive ofe ( t ) t o zero.

Case 2:

K2=2s

K2KP(s - P )E(s) =

[s s 2 i-s(K - p ) - Kz] ( 3 )

c-10



where

I t i s s ee n t h a t t h e r e i s a t r a n s i e n t e r r o r and also a s t e a d y - s t a t e e r r o r

which i s dependent on t h e po l e f r equency p of t h e lo op f i l t e r . T h i s

s t e a d y - s t a t e e r r o r i s g e n e r a l l y s m a l l enough t o be neglec ted.

Case 3:

2 K KE ( s ) = 3 @

( s - p )

( 3 )s 2 L 2 + s ( K - p ) - Kz]

c-11



where

JI = 2 tan-'

-t a n

-1

K - p K + P

I t i s s ee n t h a t t h e r e i s a t r a n s i e n t e r r o r , a s t eady - s t a t e e r ro r w hich i san i n c r e a s i n g f u n c t io n of t ime , dependent on th e pole frequency, and

two cons tant s te ady -sta te e r r o r s , one dependent on th e pol e frequency.

Case 4:

K

c p . ( t ) = U ( t ) c Kn s i n WritI n = l

K K K (U S ( S - p )

E ( s ) = C n c p n

n = l [I.' + on'] [s2 + S (K - p) - ~4

K

e ( t ) = Cn = l

c-12



where

-

- t a n4

- (K - p ) 2 + w + Kzn -

It i s s ee n t h a t t h e r e i s a t r a n s i e n t e r r o r and a l s o a s t eady - s t a t eerror which i s a sum of s inu so id s hav ing f requencies t he same as t h einpu t si nu so id s, and ampli tudes and phases which a r e dependent onsin uso id frequency and loop parameters . The s tead y-sta te peak er ro rin rad ia ns may be seen t o be

K AVi e c . ( 7 )pk i=l

wi

wi

K AY,

2wi

C.2.3 Loop Phase NoiseI n t h e f o ll ow i ng s e c t i o n s it w i l l b e n ec es sa ry t o b e a b l e t o r e l a t e

t he phase no i se or " j i t t e r " of t h e VC O s i gn a l t o th e no i se accompanyingt h e i n p u t s i g n a l . If t he i npu t phase no i se i s cha rac t e r i zed as being

Gaussian and has a f l a t s p e c t r a l d e n s i t y w i th v a l u e I O , p I , t h e n t h e

var i anceIS:

or mean squared value of t h e VCO pha se no is e may be

w r i t t e n as t h e p roduc t of t h e i npu t spec t r a l dens i t y t i m e s t he "c losedloop equi vale nt no is e bandwidth" %, For tw o-sided sp ec t r a l de ns i t i e sw e may w r i t e

C-13



The closed-loop equ iva len t noi se bandwidth may be c m p u te d i n t er m s of

gene ral loop parameters , us ing th e l oo p t r a n s f e r f u n ct i on of equat ion

C.2.1 ( 7 ) , page C-6, and the method of appendix H-1.

mob) G(s) = K (s - Z )"=1 s2 + s ( K - p ) - Kz

where

Then,

I t i s seen that t h e o n ly p o l e s i n t h e l e f' t half p la ne a r e a t s = Aand s = B, respec t ive ly . Def ine

The resid ue a t s = A i s given as

The residue at s = B i s given as

C - 1 4



The sum of t h e r e s i d u es i n t h e l e f t ha l f plane i s given as

Su bs t i t u t in g and reduc ing , w e have

F o r t h i s l oop t h e l a w f requency gain i s taken a s re fe rence

Then

2A% = TCK[s]o r

X K 2% = 2[e]For t h e a l t e r n a t e n o t at i o n,

u)n

K = 2 5 wn ' = - Z

2BN = 2 + 4 5 7

C.2.4 Thresho ld Pred ic t ionA phase-lock loop i s u s e f u l only when it i s locked. A phase-lockedloop which i s opera t ing a t a high inpu t s igna l -to -no ise r a t i o w i l lremain locked most of t h e t ime. A s t h e i n p ut s i g na l - to - n oi s e r a t i o i s

lowered the l o o p w i l l break lock more frequ ent ly, but w i l l rega in locki f t h e s i gn a l- to -n o is e r a t i o i s not t o o low.

c-15



Perhaps t he s i m p l e s t way t o t r e a t t h re sh o l d i s t o d e f i ne t h e loop

as operat ing above threshold i f it i s i n lo ck a cer t a in average percen tof the time and define it as below threshold i f i t i s i n l oc k l e ss than

the requi red percen t of time. I n t h i s manner l oop t h re sho ld i s r a t h e r

subjec t ive and i s dependent on the loop's use, which defin es the thresh

o ld i n lock t ime percen tage .

The an a ly t i ca l methods and assumptions by which th e sig na l , nois e,

and l oop param e te rs a r e r e l a t e d t o t he pe rcen t inlock t ime have provided

a f e r t i l e f i e l d f o r a n a ly s i s. M ar tin ( r e f . 7) def ined a "p rac t i ca l "

"abso lu te" th reshold , p red ic t ab le f rom a l i n e a r loop model , which w a sreasonably subs tan t i a t ed by l abora to ry t e s t da t a . Later work by Develet( r e f . 8) t r ea t e d th res hold wi th non l inear loop models .

It i s t he purpose o f t h i s s ec t i on t o se t down the simplest method

of loop treatment which w i l l y i e l d r e su l t s o f t o l e rab l e accuracy. The

simplest method i s t o de f i ne t he cond i t i ons unde r which t he l i n ea r l oop

model i s va l i d and t hen use t he l i ne a r model t o i n f e r t he non l i nea r

threshold proper t i es o f the loop.

A n assumpt ion which h igh ly s impl i f i es the ana lys i s i s t ha t t he phase

component of the input signal i s sepa rab l e i n t o a d i s t i n c t s i g n a l termand a d i s t i n c t n o i s e term. A second simplifying assumption i s t h a t t h e

phase noise term represents a Gaussian noise process having a f l a t spect r a l den si ty . With the se two assumptions the modulat ion t ra cki ng er ro r

o f t h e loop i n response t o the inpu t s ign a l phase t e rm and the VCO phase

j i t t e r i n re sp on se t o t h e i n p ut phase n o i s e term may be e as i l y de te r

mined by the methods s e t f o r t h i n s e c t i o n s C.2.2 and C.2.3.

Wi'ih a knowledge of t he modula tion t rack in g e r ro r , and espe c ia l ly

t he peak t r ack i ng e r ro r em i n r ad i a ns , and a knowledge of the standard

devia t ion 0 of t he VCO phase no i se i n r ad i ans , M ar t i n ' s ( r e f . 7)cp

thr es hol d c r i t e r i o n may be employed. It i s given a n

where x i s a peak fac tor or conf idence fac tor for t h e VCO phase noise.

The s ig ni fic an ce of t he number E may be s een by obse rv atio n of2

equat ion C . l (91, page C - 3 , t h e nonl inear equa t ion g iv ing the loop response t o i nput phase . Suppose i n i t i a l l y bo th cp.(t) and cpo(t) a r e

1 id en t i ca l l y zero. Suppose 'pi (t) i n c r e a s e s p o s i t i v e l y from zero. Then,

acco rdi ng t o equa ti on (9) c p o ( t ) w i l l i n c r e a s e p o s i t i v e l y t o t r a c k

cpi(t). This t ra ck in g i s caused by t he e r r o r func ti on s i n k i ( t ) - To(%],

i nc reas i ng as Y i ( t ) s e p a r a t e s i n v al ue from cp,(t ) . However, i f

T i ( t ) sep ara te s from cp,(t) rapidly enough s o t ha t t he i n s t an t aneous

c-16



JIvalue of c p i ( t ) - cpo(t) exceeds -

2rad i ans , t hen t he e r ro r func t i on

s i n F i ( t ) - cpo(t] w i l l decrease wi th increas ing phase e r r o r and

T o ( t ) w i l l no t t r ack c p i ( t ) . I n o ther words , i f t he ins t an taneous va lueJI

of T i ( t ) - c p o ( t ) exceeds -2

f o r a loop which i s i n i t i a l l y locked, t h e

loop w i l l break lock.

Given a peak t r ack i ng e r r o r emy

t he n t h e s t a t i s t i c a l p ro b ab a bi li ty

o f - t he loop break i ng l ock i s impl ied by equat ion (1)above. For Gaussian

phase j i t t e r t h e p r o b a b i li t y of t h e loop break ing lock a t t he t ime of

peak t r ack i ng e r r o r em may be dete rmined . I n pr ob ab i l i s t i c no ta t ion ,

t h e p r o b a b i l i t y t h a t t h e l oc ke d l oo p loses lock i s given by

where

Hx(x) = norm al d i s t r i b u t i o n func t i on .

I n te rm s o f t h e e r r o r function, which i s ta bu la te d, we have

L

Equations (3) and ( 4 ) show t h a t given a t r a ck i n g e r r o r em’

t h e

s t anda rd dev i a t i on u of t h e VC O phase j i t t e r un ique ly de te rm ines t hecp

p r o b a b i l i t y of l o s s of lock. Given a r equ i r ed p robab i l i t y o f l o s s oflock, the confidence value x may be ob ta ined f r m t a b l e s of t h e e r r o rfunc t ion .

Table C.2.4-I g i v e s t h e v a l u es of x corresponding t o l o s s of lock

p r o b a b i l i t i e s f o r f i v e ca s es .

TABLE C .2.4-1. - CONFIDENCE VALUES VERSUS LOSS-LOCK PROBABILITIES

c-17



I 1 I l l I 1

I n s e ct io n s C . 4 and C . 3 t o f ol lu w, d e t a i l e d t h r e s h ol d r e l a t i o n sw i l l be de r i ved fo r two s p e c i a l ca s es o f p ha se -lo ck ed lo op s, r e l a t i e

t h e p r o b a b i l i t y o f l o s s of lock t o t h e i n p u t s i gna l - t o -no i se r a t i o .

C .3 Signal and Noise C ha rac te r i s t i cs ofPr el im ite d Phase-Locked Loops

The remainder of appendix C w i l l be r e s t r i c t e d t o phase-locked

loops preceeded by an id e al band-pass l im i te r . The model i s g i ven i nf i gu re C . 3 - 1 .

CPI

S & t ) I d e a l5 band-pass

m ( t )* Loop

n i ( t l o l i m i t e rf i l t e r

-~

vco <

Figure C. 3-1. - Pre l imi te d phase-locked loop

The inp ut s ig na l s i ( t ) i s t aken a s an id e a l angle modula ted&

s i g n a l from equat ion A.l (7),page A-2.

The noise i s t ak en i n t h e form of equat ion B . l (l), page B-1, as a

sample function of a Gaussian proce ss, w ith a f l a t s p e c t r a l d e n s i t yband-l imi ted t o the input bandwidth of the l imi ter .

n i ( t ) = x ( t ) cos wCt - y ( t ) s i n w e t (2 1

The l im i t e r ou tpu t func t ion i s taken from equation B.3 (4), page �3-6, asT T

c-18



where, from equat ion B .2 ( 9 ) , page B-4,

For h ig h s i g n a l -t o - n o i s e r a t i o (SNR) i n t o t h e l i m i t e r t h e o ut pu tfunc t ion i s approximated, from equat ion B . 3 ( 5 ) , page B-6, as

l(t) = 4-vL

cos W C t + c p p + a1 ( 5 )fl 1 A

For high inpu t SNR i n t o t h e l i m i t er , i t . i s seen from equat ion ( 5 ) t h a tth e inpu t s ig na l phase func t ion t o the phase- locked loop i s cps(t).

A l s o , frm equat ion B. 2 (ll), page B-5, t h e p ha se n o i s e s p e c t r a l d e n s i t ya t t h e i np ut t o t h e l oo p i s

where

AW = 27tBi

(7)

i s the co ns tan t value o f the f l a t n o i se s p e c t r a l d e n s i t y i n t o t he

l i m i t e r and Bi

i s the l imi te r inpu t bandwid th .

C . 3 . 1 Limi te r Ef fec t s on Loop ParametersSect ion H . 3 , page H-9, which i s based on Davenport ' s ( ref .4) work,

d i s c lo s e s a proper ty of band-pass l i m it er s which a f f e c t s t h e p a r a m e te r sof a l im it e r dr iven phase-locked loop. A t l o w l i m i t e r SNR the ampli tude

of th e s in usoi d feeding th e phase-locked loop i s suppressed by a f a c t o r

Lf r o m i t s v a lu e a t h ig h l i m i t e r SNR. M a r t i n ' s ( r e f . 7 ) approximation

t o ciL i s reproduced here from equation H.3(3), page H-11

where

121= l i m i t e r i np u t n o i se - to - s ig n a l r a t i o

c-19

ci



Equation C . 2 (2), page C - 3 , shows t h a t th e phase s ubt rac tor cons t a n t K i s propor t iona l t o the ampl i tude of t h e s i n u s o id fe e d in g t h e

cpphase-locked loop. When th e l im it e r supp resse s th e sinus oid, it a l s o

suppresses K by th e same fa ct or y. The value of K which i s sup'p cp

pressed by the l imiter a c t i o n w i l l .be denoted by K .'a

then

K = a K'pa L c P

The loop parameters der ived i n s ec t io n C . 2 . 1 and C . 2 . 3 may be

modified for l i m i t e r suppress ion . S ince th e phase sub t ra c to r ga in i s

reduced, s o i s the loop ga in K. If K i s understood t o be the max

imum loop ga in for no l imiter suppress ion, then the modif ied gain i s

Ka = % K ( 3 )

and

It should be noted t h a t th e t rac k ing e r r o r of the loop changes i n al i k e manner. The e r r o r s f o r p a r t i c u l a r m o du la ti on s and p a r t i c u l a ralp has may be e valu ated through use of the express ions of s e c t i o n C . 2 . 2 .

C . 4 Modulation Res t r ic t ive Loop

This sec t ion w i l l cons ider the th resho ld t rea tment of a s p e c i a l

t y pe of loop known as "modula tion re s t r i c t iv e" . Th is type loop i s usedt o t r a c k an m o d u l a t e d s in u so i da l c a r r i e r or t h e r e s id u a l c a r r i e r com

ponent of a narrow-phase modulated signal. It i s assumed th a t by su it

a b l e i n p u t f i l t e r i n g and p ro p er signal design, a m o d u la t i o n r e s t r i c t i v e

loop w i l l see l i t t l e or no s i gn al modulat ion and w i l l operate wi thneg l ig ib le modula t ion e r ro r e

mexcept for t h a t error caused by Doppler

e f f e c t .

c-20



The following treatment w i l l be fo r the spe c ia l case o f ze ro Dopplere f f e c t . I f Doppler e f f e c t cannot be neglected , loop thresho ld may bet r e a t e d eas i l y using the r e su l t s of sec t ion C . 2 . 4 .

Several assumptions are made. First, the l imi te r bandwid th i s

taken t o be much wider than th e loop equ ival ent noise bandwidth . Second,t h e s t a t i s t i c s o f t h e p hase n o i s e pr o c es s p a s s in g from th e l i m i t e r i n t ot h e l o o p are assumed t o be approximately Gaussian f o r any l i m it er SNR.

The e f fe c t o f l i m i t e r suppression i s inc luded i n the de te rmina t ion ofth e c losed loop noise bandwidth BN.

For t h i s s p e c i a l c a se , t h e t h r e s h o ld d e f in in g e q u a ti o n C . 2 . 4 (l),page c-16, s p e c i a li z e s t o

where the e qu a l i t y de f i nes the th resho ld VCO phase j i t t e r . Dividingequa t ion (1)by x and squaring,

From equa t ion C . 2 . 3 (11, page C-14,

f i o m equa t ion c.3 (61, page C-19 ,

and

But the qu an t i ty on th e r ight -hand s id e o f equa t ion ( 5 ) i s i d e n t i c a l l yt h e no i se - to - si g na l r a t i o i n t o t h e l imi te r , computed i n th e loop noi sebandwidth BN.

c-21



t h e n

and

Thus, t h e l imi ter i n p u t SNR, t aken i n the loop no ise bandwid th , hasbeen re l a t ed t o the conf idence va lue x. F'rom t h e resul ts o f t a b l eC. 2.4-1, page C-171, t h e r e q u i r e d SNR may be tab u la ted f o r va r ious

p r o b a b i l i t i e s o f loss of phase-lock . These are as fol lows;

7.47db 8.62db

T A B U C. 4-1. - INPUT SNR VERSUS LOSS-LOCKPROBABILITIES

It should be note d t h a t when making computations in vol vin g %,th e va lue o f BN used shou ld be th a t va lue a c t ua l l y p roduced by t h el imi ter inpu t SNR. The cal cu la t io n of BN i s t r e a t e d i n t h e f ol lo w in g

sec t ion.

C.4.1 Loop Noise Bandwidth Above Threshold

For a prekimited modulat ion r e s t r i c t i v e phase- locked loop which

has been opt imized ( f o r thr esh old ) a t some par t icular loop s ignal- to-noise

r a t i o , it i s n e c es s a ry t o be a b l e t o d e te r mine t h e e f f e c t i v e l oo p n o i s ebandwidth fo r s ignal- to-noise r a t io s above thr esh old . The usu al assumptions

ar e made t h a t th e loop i s l o ck e d , f e d by a n i d e a l l im i t e r , an unmodulateds inuso id , and whit e band-limited Gaussian no is e. From equ ati on C.2.3 (15),page C-15, t h e t w o s ided c losed loop noise bandwidth i s t aken as

where

K = open loop gain

c-22



- -

z = loop f i l t e r zero frequency

p = loop f i l t e r pole f requency

R e s u l t s w i l l be obta ined for a s p e c i a l c a se w hich r e p r e s e n t s a widec l a s s of loops . The loop f i l t e r parameters w i l l be se t such that

] P I <<K

5 = 0.707

where

5 = loop damping factor

t h e n

Az = z -=

KO

0 2

= ao%

where

a = l imi te r signal v o l t a g e s u p p re s s ion f a c t o r

% = maximum o r h ig h s i g n a l v a lu e o f l o o p g a in

0= c o n d i t i o n s a t t h e l o o p d e s ig n t h r e s h o ld

Then

Now the th res ho ld loop no ise bandwidth occurs f o r a E a so that0’

2BN = 3 a~ %0

The r a t i o of bandwidth for any a t o t h r e s h o ld b an dw id th i s

This result i s given by Mart in ( re f . 7).

( 3 )

( 5 )

C-23



U sing Ma r t i n ' s ( r e f . 7) approximat ion t o a as a func t ion of

l imi t e r i n p u t si g n a l- t o -n o i s e r a t i o , w e have

1 a s

or

t hen

2BN 1 + 1- =

2%0

L I( 7 )

( 9 )

A r e l a t e d p roblem i s t h a t of de te rm in ing t he l oop s i gna l -t o -no i se r a t i o

a t which a l oop w a s optimized, given a p lo t of loop bandwidth versus inpu t

s i g n a l . It i s e a s i l y d e te rm in ab le t h a t

where

= s t rong s i gn a l va l ue of loop noise bandwidthBNH

C . 5 Pr ef i l t e re d Modula tion Track ing Loops

Thi s sec t i on will cons ide r t he t h re sho l d t r ea t m ent o f a s p e c i a l

type of loop known as "p re f i l t e re d modula tion t rac k ing ." This type of

loop i s used to demodula te angle modulated s inus o ida l c ar r i e r s . Forsome typ es o f si gn al modulat ions i t i s po ss ib le t o reduce peak modula

t i o n t r a c k i n g e r r o r em

t o a very s m a l l value by making the loop

C-24



na t u r a l r e sonan t f requency wn

ve ry l a r ge compared t o t h e h i gh es t

modulating fkequency. Simu ltaneou sly, t o reduce the t ransmiss ion ofphase noise or j i t t e r t h rough th e loop, th e input s ign a l p lus no i se

may be processed through a sharp cut -off band-pass f i l t e r having a

bandwidth j u s t wide enough t o pass t he modulated s ig na l . For high in

pu t s i gna l- t o -no i se r a t i o s and l i n ea r loop opera t ion, th e bandwidth ofthe equiva len t phase no i se w i l l be h a l f t h e in pu t bandwidth. For mod

u la t io n ind ices no t to o l a rge , t he bandwidth of t he phase no i se w i l lform, es se nt ia l l y , th e closed loop noise bandwidth. Such a modulation

t r ack i ng l oop i s c a l l e d a p r e f i l t e r e d l o o p .

Assuming the loop natural fkequency wn

i s much la r g e r than the

hig he st modulat ion f’requency, th e peak modulat ion trac ki ng er ro P be

comes negligibly small. The t h re sho l d d e f i n i ng equa t i on f o r t he p re

f i l t e r e d modula tion t rac k ing loop then becomes t he same as f o r t hem odul ati on r e s t r i c t i v e loop, s i nce aga i n e

mapproaches zero.

3Txu =

c p 2

then

2

.2=k] cp

Due t o the p re f i l t e r in g and the assumption th a t t h e in put bandwidth i s

much l es s t han t he l oop Wn’

the e f fec t ive no i se bandwidth of the phase

no i se i s hal f t he input bandwidth o r2’

Then, from equation C.2.3 (l),

page C - 1 4 ,

rum equat ion c.3 (61,page ~ - 1 9 . ~

and

C-2 5



then

where

= n o i s e -t o - s ig n a l r a t i o i n t o t h e l i m i t e r , computed i n t h el i m i t e r bandwidth Bi

t hen

From t he r e su l t s o f t ab l e C .2 .4 - I, page C - 1 9 , t h e r e q u i r e d SNk i n t o

t h e l imi te r may be t abu l a t ed f o r va r i ous p r ob ab i l i t i e s o f l o s s of

phase-lock. These a r e given as fol lows:

.- .. . .

10-3 I O - ~ 10-7... -. - . - - . - - ... . .

2.91ab 4.47db 5.62db

TABLE c.5-I.- INPUT SNR VERSUS LOSS-LOCK PROBABILITIES

I t i s seen t h a t fo r t he l ow er p r ob ab i l i t i e s t he l i m i t e r i npu t SNR

i s l o w enough t o vio la t e the assumpt ion of high S m which w a s used t o

obta in the phase no i se sp ec t r a l dens i ty . Therefore , ca re should be

exerc i sed i n a p p l y i n g t h e r e s u l t s o f t a b l e I fo r t he l ower SNR.

C-26

~~

I, I, I I I , I 1111 I I I I I I I I 1111



APPEN3IX D

PRODUCT DEMODULA!IT ON

D . l Linear Product Demodulator

F igure D.1-1 shows the configurat ion of a prod1 z t demodulator, use't o coheren t ly de t ec t phase modulat ion . This i s t h e p roduc t de t ec t o r o fs e c t i o n B.4.1, page 13-6, fol low ed by an i de a l ou tpu t f i l t e r .

I s o woutput

1 * f i l t e r -n o w

BO

- s i n w tC

Figure D. 1-1.- Demodulator configurat ion

The ou t put f i l t e r i s d e fi ne d t o be e i t he r l aw -pass or band-pass, withf l a t un i ty ampl itude and phase t ransmiss ion cha ra c t e r i s t i c s , squaref requency cu t -of f ch ar ac te r i s t i cs , and t ransmiss ion bandwidth of B

0cps.

The inp ut s ig na l and noise are t aken i n t he u sua l forms as

where

s i ( t ) = ang le modulated s i gn al wi th equiv alen t phase modulation V s ( t )

n i ( t ) = sample function of a random Gaussian process

D-1



n . ( t ) i s f u r t h e r d e f i n e d as white and band-l imi ted t o R cps.1

The noi se power sp ec t r a l de ns i ty at t h e m u l t i p l i e r o u tp u t i s g l1ven

from equat ion B .4 .1 (7),page B-7, as

2

= o ; a l l o t he r va l ues o f W ( 3 )

where

I@-- 1 = cons tan t va lue o f t h e i n p u t n o i s e s p e c t r a l d e n s i t y @-- ( w )LIi

The s i g n a l component, f'rom equation B. 4 . 1 ( 4 ) , page B-7, i s

K' 2s

m(t)= .-Y A s i n cps(t) (4)

Equation ( 4 ) will be subsequently t re a t e d for s p e c i f i c signal types.

D . 1 . 1 Detec t ion of S inuso id a l Subc ar r i e rsThis s ec t i on t r e a t s demodulation of a c a r r i e r which i s phase modu

l a t e d by t h e sum of a pseudo-random range code plus K s u b c a r r i e r s .

The model of f i g u r e D.l-1 a p p l i e s .

The input s ignal i s t aken In usua l form as

K

s . ( t ) = A cos + Acprct(t) + c ncpi s i n ( W i t + cpi1

i=1 t h e m u l t i p l i e r o u tp u t i s

2 K

m ( t ) SK'c p 2A s i n + c .AV, s i n pit + cpi

i=l

Expanding equation ( 2 ) and app l y i ng t he i den t i t i e s of' equat ions A . 3 ( 3 )and A . 3 (41, page A- 7 , w e o b t a i n

D- 2



J

The f i r s t term of equation ( 3 ) i s a f u n c t i o n o f t h e s u b c a r r i e r s m u l t i pl i ed by the range code and i n t e r fe re s wi th the des i red s ig na l whichi s the second term of equa t i on (3).

Then, the des i r ed mu l t i p l i e r te rm i s

Using t he G i aco l e t t o expans ion ( r e f . 1)

I m m K

L

-K

s i n

Frm equat ion ( 5 ) t h e j th de t ec t ed subca r r i e r te rm s , t he f i r s t orde rterms having frequency w

3'may be obtained by se t t in g

ni

= O ; i # j

D-3



then, for n i = +1

NOW

and

J ( x ) = ( - l l n Jn(x)-n

t hen

K m . ( t ) = K'A cos (ATr) J1 (ATj) [.o(ACPd s i n (mjt + CPj) (11)

J CP i=1

i # j

A s i n appendi x A, equa t i on (11)holds whether or no t t he i nd i v i dua l

s u b c a r r i e r s a re , thems elves, an gle modulated.

The o u t p u t s i g n a l f'rom t he ou t pu t f i l t e r i s

K

D . 1 . 2 Detec t ion of Arbitrary Baseband ModulationThi s se c t i on t r e a t s baseband modulat ion which i s not a s i nuso i d o r

sum of si nu so id s. The most workable method i s t o d e s c ri b e t h e a r b i t r a r y

func t ion by i t s peak phase de via t io n and by an e mp i r i ca l ly de te rmined

peak t o r m s r a t i o , o r form f a c t o r .

D - 4



The m o d e l of f i g u r e D . l - 1 app l i e s . The i n p u t s i g n a l i s t aken as

s i ( t ) = A

where now c p s ( t ) i s a baseband signal, having peak phase deviat ion Acp, -The output s ignal f'rm t h e ou tp ut f i l t e r i s

K's o w = 2 A s i n c p s ( t ) ( 2 )

The peak squared output signal i s

It i s noted t h a t fo r t he ou tpu t s i gna l t o be a l i n e a r r e p l i c a o f

the phase modulat ion, AT, must be l e s s than about 30". I f l i n e a r i t y

i s not of grea t cons i de ra t i on , as for cl ipped speech, ATp may be

increas ed. I n no case may ATp be g re a t e r t han 90". R e s i d u a l c a r r i e r

suppress ion cons idera t ions w i l l gene ra l l y l i m i t ATp t o less t han 90".

It i s des i r ab l e t o p lace a bound on t he r e s i du a l ca r r i e r r em ai ni ng

a f t e r modula tion s ince g ene ra l ly the demodulator re fe rence s ig na l i s de

r i ved from t he r e s i du a l c a r r i e r . For square wave modulation th e r e m i n i ng c a r r i e r i s given by th e l im i t in g case of equa t ion A . 3 (7),page A - 8 ,for ATi i d e n t i c a l l y z e r o , as

sC(t) = A cos (ATp) cos (4)

Likewise, f o r si nu so id al modulation, where ATr i s i d e n t i c a l l y z er o,

t h e r e si d u a l c a r r i e r i s

s , ( t ) = A Jo (ATp) cos u) tC

For the sake of s imple ana lys i s , it i s assumed t h a t for a r b i t r a r y b aseband narrow de vi at io n modulation of peak de vi at io n &JP, t h e r e s i d u a l

c a r r i e r t e r m i s bounded by equations ( 4 ) and ( 5 ) .

D-5



D.l.3 N oi se C ha rac t e r i s t i c sThe nois e spect rum out of th e mul t ip l ie r i s f l a t w i t h i n t h e l i m i t s

s e t by the inpu t bandwidth B. Therefore, th e bandwidths Bo of the

low-pass or band-pass f i l t e r s a re the e quiva len t no i se bandwidths a tt he ou t pu t .

The outpu t no is e powers fo r both th e band-pass and low-pass c ase s

are g iven by

N =0

where

= f l a t ampl itude of th e in put whi te noise spect rum

jmnil

D.1.4 Output S igna l - to -no i se Rat ios

The resu l t s of t he p r i o r t h re e s ec t i ons may now b e i n t e g r a t e d t og ive ou tpu t s ign a l - to -no i se r a t i o s fo r coheren t demodula tion of bo th

subc ar r i e rs and a rb i t ra ry baseband modula t ion .

D.1.4.1 S u b c a rr i e r and b an d- pa ss f i l t e r . - The s i g na l - to - n o is e r a t i o

o u t o f t h e i d e a l b an d- pa ss f i l t e r Bo cyc les wide, fo r the j t h sub-

c a r r i e r si g n a l may now be determined.

&om equat ion D.l.l (12), page D-4, t he ou t pu t s i gna l for t h e j t h

s u b c a r r i e r i s

i f j

The output signal power i s then

h r r K

i # j

D-6



From equa t ion D. 1.3 ( 2 ) , page D-6, the output noise power i s

2-- 1

N = ( 3 )

The ou tpu t s igna l - to -no ise r a t i o i s then

The q u a n t i t y i n b r a c k e t s i s s e e n t o b e th e t o t a l i n p u t s i g na l - to - n o is er a t i o computed i n a physical bandwidth B

0'Then

BOi # j

A s i n a pp en dix A, equa t ion (6) holds, whether or n o t t h e s u b c a r r i e r s a r e ,themselv es, an gle modulated.

D. 1 . 4 . 2 Baseband modulat ion and low-pass f i l t e r . - Proceeding a si n t he previous sec t i on , the peak squared ou tpu t s ig na l i s t aken fYomequa t ion D . 1 . 2 ( 3 ) , page D-5, as

D-7



I I I 1 I I l l 1 I I 1

Again, from eq ua tio n D. 1.3 ( 2 ) , page D-6, the output noise power i s

2KI-2

No - 2

I % /2Bo

The peak squared signal to mean-squared noise r a t i o i s then

2= sin (*'p) (4)

NO

The bracketed quanti ty i s s ee n t o be t h e t o t a l i n p u t s i g na l - to - n o is e

r a t i o computed i n a physical bandwidth Bo. Then

0

I f t h e r e e x i s t s a f a c t o rKp

r e l a t i n g t h e p e a k t o rms s igna l vo l tage

such tha t

So r . m . s

= Kp

so peak

(6)

then the r a t i o of mean squared signal So to mean sq uare d no is e No

may be written

D-8



.._. - .. . ....-.....__.. .. . . . _..... ...-. _.._..-___._

D.2 Prelimited Product Demodulators Results similar to those in the preceding section are obtained for

a product demodulator preceded by a hard band-pass limiter. The con

figuration is shown in figure D.2-I. This is the detector of sec

tion �3.4.2, page B-7, followed by an ideal output filter

Figure D.2-1.- Demodulator configuration The ideal output filter has the same characteristics as in D . 1 ,

page D- 1 .

The input signal and noise are taken in the usual form as s.(t) = A cos [wct + cps(t)11

where the signal and noise characteristics are the same as in D.l,

page D - 1 .

For a high input signal-to-noise ratio into the limiter of, say,

10 db, the multiplier signal may be taken from equation B . 4 . 2 (7),page B-9, as approximately

For high input signal-to-noise ratios, equation (3) gives output signal-to-noise ratios identical to those for no prelimiting. However, for decreasing input signal-to-noise ratios, limiter effects become pronounced.

Lacking a useful rigorous treatment, the following rough approximation will be made, which highly simplifies the analysis. The

D-9



m u l t i p l i e r o u tp u t m ( t ) will be approximated f o r a l l inpu t s igna l - to noise r a t i o s b y

where

aS

= l i mi te r s ig na l suppress ion fac -; ;o r.

M a r t i n ' s ( r e f . 7) approximat ion for aS w i l l be used.

where

= l i m i t e r i n p ut s i g na l- to - no is e r a t i o i n t h e l i m i t e r band

width BL.

It remains t o determine th e n ature of an'

Th is may be de term ined

by no t in g t h a t t h e l i m i t e r i s a con sta nt power out pu t devi ce. That i s ,

regard less o f t h e l i m i t e r o u tp ut s i gn a l -t o -n o i se r a t i o , t h e t o t a l ou tp uts ignal p lus noise power i s constant . This means th at reg ard les s o f the

l i m i t output sp ec tr a l composi tion, the t o t a l power acr oss the spectrum

i s constant. Next, it i s noted tha t a product detector i s simply as p e c t r a l t r a n s l a t o r . It does not change th e nature of the l im it er out

put spectrum, bu t merely t r an sl a t es i t i n f 'requency. Therefore, th e

t o t a l power ou t o f the mu l t ip l i e r i s cons tant. This constancy of m u l t i p l i e r output power Fm w i l l be used to s o lv e f o r a

n'

D-10



Using equation ( 8 ) , i t i s s ee n t h a t t h e s i g n al - t o - n oi s e r a t i o a tt h e m u l t i p l i e r o u tp u t i s given as

2

[# ] (9)

N j BL

It i s apparent that the rough approximation of equat ions ( 4 ) and(8) has given a p e s s i m i st i c r e s u l t for o u tp u t s i g n a l -t o - n o i se r a t i owhich i s analogous t o Davenport 's ( r e f . 4) l i m i t e r r e s u l t a t low input

s i g n a l - t o - n o i s e r a t i o s .

Equation ( 4 ) i s r e w r i t t e n a s

The s ig na l component and noise s pe ct ra l dens i ty ar e given sep ara te l ya s

It should be emphasized t h a t th e ma te r ia l presented above i n sec

t i o n D.2 i s t h e re su l t of physical reasoning and approximation and i s

not mathemat ical ly r igorous . This ma te r i a l should be app l ied withcare .

D-11



APPENDIX E DEMODULATION WITH MODULATION TRACKCNG LOOPS

Figure E-1 shows the conf igurat ion of a modulat ion t racking phase-locked loop used to de te c t f requency modula t im.

OUtPUtI d e a l l(t)

* f i l t e r

vd(t)e f i l t e r 00 -band-pass .

Loop

n i ( t ) - l i m i t e r BO n o ( % )-

vco

-The input s ignal i s t a k en i n u s u a l form as

s i ( t ) = A cos pet + q s ( t g

where

c p s ( t ) = "equivalent" phase modulation of t h e s i g n a l

The inp ut noi se i s t aken as

n i ( t ) = x ( t ) cos w Ct - y ( t ) s i n (uCt ( 2 )

The l im i t e r ou tpu t s ig na l i s t a k e n f'rom e q u a t i m B . 3 (4), page B-6, as

with c p i ( t ) i d e n t i c al t o $(t) of th e re fe renced equa t ion . For suf

f i c i e n t l y hi gh l i m i t e r s ig n al -t o -n o is e r a t i o , V i ( t ) may be separated

in to s i g n a l p ha se m c d u la t io n , c p s ( t ) , and noise phase modulation, ( p p p ( t ) .

E-1



Since a modulat ion t racking loop i s use fu l on ly when opera t ing re l a t i ve ly

l i ne a r l y , t he a s sm p t i o ns a re made t h a t t he s t ea dy - s t a t e m odu la ti on

t r a c k i n g e r r o r i s l e s s t han , s ay , 30" and t h a t t h e s e t of l i n e a r t r an s

f e r func t i ons de r i ved i n append ix C adequate ly descr ibe loop opera t ion .

The t r a ns fe r func t i on of i n t e re s t i s t h a t r e l a t i n g t he VCO dr i v i ngs i gn a l v d ( t ) t o i np u t s i g n a l v i ( t ) . I n t ransform nota t ion , f i m

equat ion c.2.1 ( 9 ) , page C-7,

(4) where

H i ( " ) = t ransform of t h e i n p u t signal "equivalent" phase modulat ion

For s ig na l s which ar e f requency modula ted, t he t ransform re l a t i on between VCO dr i v i ng si gn al and in pu t frequency modulation i s given as

vD(K K

s - z

= ' [g2 f s ( K - p ) - Kz1where

sai(s) = t h e i n t e g r a l of t h e " e q u iv a l en t " ph ase m od ula tio n orfrequency modulat ion

The as ym pto tic Bode p l o t of t h e s t e a d y - s t a te t r a n s f e r f u n c ti o n d e r iv e df'rm equat ion (3 ) w a s g i v e n i n f i g u r e C.2.1-1, page C-8 , and i s repro

duced here as f igure E-2.

db

Figure E-2.- Asymptotic Bode plot

E-2

0



The f i gu re shows t h a t t he s t eady - s t a t e t r a ns fe r func t i on i s asymp

t o t i c a l l y f l a t for frequency modulat ion f r m ze ro f'requency ou t to t h ereg ion of w = 1.1, t h e l oo p f i l t e r z e r o f re qu en cy . For a n a l y t i c simp l i c i t y , it may be assumed th a t most o f th e si g na l modulat ion energy

w i l l be of frequency l e s s t han 1.1. This assumption i s n ot s t r i c t l y

necessary, s ince equa l i za t ion i n the ou tpu t f i l t e r may be employed i fthe modulat ion frequencies do extend beyond 121. For modulat ion sat i s f y i n g t h i s f re qu en cy r e s t r i c t i o n , t h e VCO d r i v i n g s i g n a l i s given as

For the assumpt ion o f r e l a t i v e l y h ig h i n p u t s i gn a l- t o- n o is e r a t i o i n t o

th e l imi t e r and l i ne a r l oop ope ra t ion , t he VC O d r i v i n g s i g n a l i s sep

ara b le i n to ind iv idu a l s ig na l and no i se components. Then

The demodulator may be treated f o r s i gn a l and f o r no i se , s epa ra t e l y .

E.l Detec t ion of S inuso id a l Subc ar r i e rsand Ar bi tr ar y Baseband Modulat ion

T h i s s ec t i on t r ea t s the demodulat ion of a carr ier f requency modu

l a t e d by a c m p o s i t e f u n c t i o n c o n s i s t i n g of a summation of K sinuso i da l sub ca r r i e r s p l u s some a r b i t r a r y baseband func t i on . F igu re E-1

a p p l i e s.

The input s ignal i s t aken as

1s . ( t ) = A cos bet + V s ( t y

where

c p , ( t ) = c a r r i e r phase modula tion due t o s ig na l a lone

The func t ion descr ib ing t he ins t an taneous c a r r i e r f 'requency dev ia t io n i s

def ined as

E- 3



where

f b ( t ) = ar b i t ra ry baseband fun c t ion and

AW.1

= peak rad ian f requency devia t ion of t h es u b c a r r i e r

The s ig na l por t ion of t h e VCO dr i v i ng func t i on

K

-t- C AW, cos Fit +i=1

c a r r i e r b y t he it h

i s given by

7( 3 )

The output s ignal f r o m t he ou t pu t f i l t e r (band-pass) f o r t h e jth sub-c a r r i e r i s given as

Equation ( 4 ) holds whether or no t t h e sub ca r r i e r s t hem sel ves are angle

modulated.

The peak output signal f rm t he ou tpu t f i l t e r (low-pass) for base

band modulation i s given as

where

hbpeak= peak radian frequency de vi at io n due to the baseband

modulat ion

E. 2 N oi se C ha rac t e r i s t i c s

Observat ion of f igure C.2.1-1, page C-8, shows that the modulat ion

t r ack i ng loop has no f in i t e ou tpu t no i se bandwidth f o r f l a t input phase

noise. Output f i l t e r s of the law-pass o r band-pass type are used t or e s t r i c t t he ou t put no ise . The f i l t e r s are assumed to have i d ea l p rope r

t i e s , t h a t i s , square f requency cu t -of f ch ar ac te r i s t i c s and f l a t t r a n s

m i ss i on c ha rac t e r i s t i c s i n t he pass -band.

For t he a s sm p t i on t h a t t he m odu la ti on f r equenc i e s are l e ss than

12 1, t hen t he ou t pu t no i se sp e c t rm i s pa rabo l i c , or propor t i ona l t o

2w as shown by figure C.2.1-1, page C-8 . Theref ore, the bandwidths Bo

of t he ou tpu t f i l t e r s are not the equivalent noise bandwidths a t t h eoutput. These w i l l now be computed.

E- 4



The noise spectrum a t t h e l o o p f i l t e r o u tp u t i s t a k e n as

where

CP ( j w ) = f l a t input phase noise spectrum.cp

Equation E.2 (1) fo l lows from e q u a t i o n E ( 6 ) , page E-3.

E . 2 . 1 Low-pass Output F i l t e r

The low-pass ou tpu t f i l t e r i s t a k en t o h av e an ampli tude t ransmiss i o n c o e f f i c i e n t o f u n i t y a n d p h y s i c a l b an dw id th B cps whichcor responds t o rad ian bandwid th

0

The equ iv al en t no ise bandwidth Be i s d e f in e d as that bandwidth having

a t r a n s m i s s io n c o n s t a n t of un i ty which passes th e same noise power from

f l a t i n p u t s p e c t r a l d e n s i t y iP ( j w ) a s i s a c t u a l l y p r es e nt i n th e o ut cp

uut. Equating noise powers, w e have

where

e = 2nBe

th e n

[ jA w ( j w ) ' d ( j w )0

'="'e = .J

- j a w0

Awe =(n'U0)

3

( 3 )

(4)

( 5 )

E-5



The output noise power i s seen t o be

E . 2 . 2 Band-pass Output F i l t e r

The band-pass f i l t e r i s t aken t o have an ampl i tude t ransmiss ion

co ef f i c i en t of u ni ty and ph ysi cal one-sided bandwidth of B cps which0

i s symmetric t o a frequency of f cps .m

I n r a d i a n n o t a t i o n ,

Aw0 = 2fiBO

w = 2nfmm

Equating noi se powers give

E-6



Where

aW = equivalent noise bandwidth a t t h e o u tp u te

2( j u , ) d( jw) = j a w e

-2 Ij ( w m - 9 )

ALIIe = nu, ( 5 )

The ou tpu t no is e power due t o a two-s ided equivalent phase noise input

s p e c t r a l d e n s i t y ipCp

( j w ) i s given as

or

E- 7



It i s seen tha t f rom equat ion ( 5 ) an approximation may be made.

2 2Be = (2n) fm Bo ; fm

2 >>-1 B 212 0

This approximation i s a c c u r at e t o w i th i n a bo u t 5 pe rcen t fo r Aw0

approaching 3 om. This approximation i s e s s e n t i a l l y t h e same as as

suming f l a t n o i s e w i t h s p e c t r a l d e n s i t y

P ( j w ) = w 2 @ v ( j w )

n m0

acr oss th e bandwidth Bo when Bo i s much l e s s t h a n t h e c e n t er f r e

quency'm

. Approximately

o r

2

No 2 ($) lPvl 2(2fl)*BOfm2

E. 3 Output Signal - to-noise Ra t io s

The r e s u l t s o f t h e p r i o r s e c t i o n s a r e now u sed t o o b t a in o u t pu t

s ig n a l- to - n oi se r a t i o s f o r two cases: a r b i t r a r y baseband modulat ion

wi th a lo w-p ass f i l t e r , an d a n i n d i v i d u a l s u b c a r r i e r w i t h a band-passf i l t e r .

E . 3 . 1 Sub car r ier and Band-pass F i l t e rWe may now det erm ine th e sign al- to- no ise r a t i o out of th e assumed

square band-pass f i l t e r of bandwidthBo

f o r t h e j t h s u b c a r r i e r ,

w here t he i npu t s i gn a l i s assumed t o be fre quen cy modulated by th e sumof K su bc ar r i e r s p lu s a r b i t r a r y baseband modula tion . The input

no i se i s white, band-l imited, and Gaussian.

E- 8



From equa t ion E.l (4), page E-&, t h e o u tp u t s i g n a l for t h e jt h

s u b c a r r i e r i s

The output signal power i s then

From equa t ion E. 2.2 ( g ) , page E-7, the ou tpu t no i se

-No - - I @ v 1 2 ( 2 ~ ) 2

kj2+ q B o

where

f . = center f requency of the band-pass f i l t e rJ

The output s ignal- to-no ise r a t i o i s then

AW2

L A

SA = . Kv2 _ 2

r

where

power i s

(4)



The second bracke ted qu an t i ty i n equa t ion ( 5 ) i s s e en t o be t h e i n pu t

s igna l - to -no i se r a t i o computed i n a physical bandwidth Bo. Then

S

A =1.2NO

where

t hAf2 = peak c y c l i c e e q u e n c y d e v ia t i o n o f t h e c a r r i e r by t h e j

J s u b c a r r i er

Approximately

SA&

2NO

f o r

E . 3.2 Baseband Modu lation and Low-pass F i l t e r

Proceeding as i n E.3 .1 , th e peak output s i gn al from equat ion E . l

( 5 1 , page E-4, i s

1S

ob peak =7@'%peak (1)

The peak-squared signal i s

From equat ion E . 2 . 1 (9),page E-6, the output noise power i s

E-10



The ra t io of peak-squared signal t o mean-squared noise i s then

where

Llfb = peak cy cl ic f 'requency dev ia ti on of t h e c a r r i e r by t h epeak baseband modulation.

The second brac kete d qu an ti ty i s seen t o be th e inpu t s igna l - to -no iser a t i o computed i n a physical bandwidth B Then

0'

I f a s p e c i f i c a t i o n f a c t o r KP

e x i s t s , r e l a t i n g t h e peak t o rms value

of th e baseband modulation, such th a t

fbr.m. s.

= K f iPeak

(7)P

the n the mean-squared out put sig nal- to-n oise r a t i o may be formed as

- = 3KP2 [mpeak l 2[dB0Bo NO

E- 11



APPENDIX F

SPECIALIZED DETECTORS

F . l Range Clock Recei ver and Code C or re la to r

Re ce i v e r

Figure F.l-1.- Range clock receiver

The above figure shows the block diagram of t h e c i r c u i t r y whichrecovers th e received range c lock s ig na l and generates t he range code

c o r r e l a t i o n s i g n a l . The p h y s i c a l o p e r at i o n o f t h i s c i r c u i t r y h as b ee nt r ea te d i n volume I of t h i s s e r ie s .

S i ( t ) + n i ( t ) = t h e I F i n p u t s i g n a l p l u s n o is e

cr (t1 = re ce iv er code having only valu es of fl

s,(t)= product of c

r(t)an d t h e I F r e f e r e n c e s i g n a l

6 (t)+ n m ( t ) = s ig n a l p lu s no i s e o u t of t h e m u l t i p l i e rm

F-1



= c o r r e l a t o r d r i v i n g s i g n a l

Sk ( t = c o r r e l a to r o u tp u t s i g n a l

= c lo c k l o o p d r i v in g s i g n a l

The o p e ra t io n o f t h i s d ev i ce w i l l b e t r e a t e d for an i n p u t signal, phase

modulated by a t ransmit ted ranging code C , ( t ) , p lu s K subcar r ie r s .

The input noise i s assumed Gaussian, f l a t , and band-limited to Bi

cps.

F.l.l Signal Treatment

where s . (t) has th e same proper t i es t r ea te d i n appendix A. 2.1

s r ( t ) = - c r ( t ) s i n (2)

(4) It i s as sm ed th a t th e mul t ip l i e r p roduces on ly the d i f fe rence f'requency

term.

then

As m ( t )= 5 c r ( t ) s i n ( 5 )

Eguatri.on ( 5 ) may be ex-panded a s i n se ct io n D.l.l.

F-2



K co

* cos n i p i t +

O3

i=1nl= nk= OD

(6) The f i r s t t erm i n equa t ion (6) i s t h e d e s i r e d term. The second term i st he in t e r f e rence . Equation (6) holds whether or no t the subca r r i e r s

themselves are ang le modulated. For t he d i g i t a l l og ic employed by t h i ssystem the product o f th e two analog code wavefoms c r( t) and c t ( t )

corresponds t o the Boolean modulo two add it io n, or "exclu sive or",*of the two codes C and Ct . The receiver code output i s programable

ri n seven s t eps . A t each step, th e co rr el at io n between the two codeschanges. This i s shown i n t ab le I. It can be s h m , e i t h e r al ge br a

i c a l l y or wi th a t r u t h t a b l e , that the "exclus ive or" of the two codesi n program s ta te P7 i s i d e n t i c a l l y c l o c k , C 1 . This means th a t

( t ) c t ( t ) = c l ( t ) (7)r7

where

c l ( t ) = square wave of u n i t ampli tude, having the clock f re quency w

cl

*For a de ta i l ed phys ica l exp lana t ion of the ranging equipment,

see volume I of t h i s series .

F-3



Transmitter code cT = CI @ X(abvbcvac)

B i tProgram Receiver Component I n i t i a l Final length

s t a t e code acqu ir ed co r r e l a t i on co r r e l a t i on

P10 I c1

P2 ,

P3-X a

IX

-P4 I

X a II

a

p5 RI !b

-P6 xc C

-p7 X(abvbcvac) check a ,

cmponent length

,x11 a 31

b 63

C 127

,

0 50% 0

25% 341

50% 75% 341

50% 75% 693

j 50% 75% 1 397

b, c 75%



I n t e rm s o f Fou r i e r s e r i e s ,

c ( t ) c t ( t ) =: C $F - c os ( p n j s i n pwc lt

r7 p = l

where

TTw = c l R

R = c l o c k b i t p e r i o d ( 9 )

Then t h e c o r r e l a t i o n d r i v i n g s i g n a l f o r s t a t e P7 i s

where it has been assumed th a t t h e band-pass f i l t e r having bandwidth Bpasses only th e fundamental s i nu so id al component of th e clock square

wave.

F.1.2 Noise TreatmentThe input noise i s t aken i n t h e u sua l form as

havi ng an i npu t no i se s pe c t r a l den s i t y Hni(w), band- limi ted t o Bi.

The noise term from t h e m u l t i p l i e r i s

2 r(t)y ( t ) ( 3 )nm( t)5 -1

c

We are now i n t e re s t e d i n ob t a i n i ng "(w), t h e s p e c t r a l d e n s i t y of t h e

noi se term o u t of t h e m u l t i p l i e r . We make t h e assump tion t h a t c r ( t )

a n d y ( t ) may be r ep re sen t ed as sample funct ions of independent randm

processes and tha t

F- 5



whereBi A AWi

; I w [ < 23-r -=- 22 ( 5 )

= 0 ; a l l o th e r w

Frm Titsworth and Welch (ref. 1 0 ) we approximate th e sp e c tr a l

dens i ty o f c r ( t ) i n i t s program s t a t e s g r e a t e r t h a n P2, as t h a t o f a

Markov sequence.

Then

@ (w )c r

where

R = bit r a t e o f c ,( t)

" 9

L

Since

- y ) = 2

= o

t hen

m ( w - y ) = 2Y

= o

1 s i n 2 ( g )= -R

J

nw,2 < w - y < 2

AWi; ( w - Y J ' 2

Awi2 2

; a l l o t h e r y (9)

F-6



It fo l lows that

AW,I

w + -2 s in2(k)

aW;1

w - 2

Let 2R

= x ; y = 2Rx ; dy = 2Rdx

then

w - 2

2R

Qm ( w ) = 121J nw2s i n 2 x

dx2RX2

w - -Q

2R



r 1

-X

2 sin 2x

a x + 2 \ x x

X1

X1

2 s in2 x ax = L {$ - 1+cos 2x

2c os 2x1

IXx2 2 1X

2X

2X1

X1

where

s i n x Si(4 = ("x

J 0

F-8



and

2.R 2R

2.

w - 2

2R

+ - 1R

Equat ion (21) i s t he gene ra l exp re s s i on f o r t h e n o is e s p e c t r a l d e n s i t ya t t he ou t pu t o f t he m u l t i p l i e r . W e a re i n t e r e s t e d i n ev a lu a ti n g t h i s

s p e c t r a l d e n s it y a t t he cen te r f requency of the narrow band-pass f i l t e r .W e w i l l then assume "(w) to be f l a t acro ss t h i s nar ruw bandwidth .

The frequency of i n t e r e s t i s



The assumption i s a l s o made t h a t t h e input bandwidth may be l i m it e d t o

Awi = l O c R ( 2 31

U s i n g t h e a s s u mp t io n s s t a t e d a bove, t h e n o i s e s p e c t r a l d e n s it y a t t h eoutput of the band-pass f i l t e r B may be wri t t en as

( 2 5 )

@c(")) $ n i l

(-10

+ -1

+ -1 + si(18.85) + Si(12.6.rr 4.rr

Now, the noise power a t t h e o u tp u t of the band-pass f i l t e r i n t he

bandwidth B i s

(29)

F . l . 3 S ignal - to-no ise Ra t iosFrom equations F . l . l ( 1 0 ) and F.1.2 ( 2 7 ) w e may obta in th e s ig na l

t o - n o i s e r a t i o a t th e ou tpu t o f th e band-pass f i l t e r o f bandwid th B

for program s ta te P7

as

F-10



K ("'p.> i=1 ~ o ~ ( * ~ i )

Equation (2) r e l a t e s t o t h e i n p u t s i g n a l - t o - n oi s e r a t i o computed i n a

bandwidth B as

For program s t a t e P7'

t h e product c r ( t ) c t ( t ) i s a square wave

clock s i g n a l 100 p e r c e n t o f t h e t i m e . For ot he r program s t a t e s

c , ( t ) c t ( t ) i s a sq ua re wave which rev er se s phase some per ce nt of t h e

time, on th e average, depending on th e perce nt c or re la t i on of Cr

w i th Ct. T h i s b eh av io r, c ou pl ed w i th t h e f i l t e r i n g a c t i o n of the band-

pass f i l t e r , i s i n t e r p r e t e d as caus ing t h e amplitude of s , ( t ) t o be

p r o p o r t i o n a l to the average amount of t i m e c r ( t ) c t ( t ) i s cons tan t

phase clock, or p r o p o r t io n a l t o th e c o r r e l a t i o n o f

p r o p o r ti o n a l i ty f a c t o r FK,normalized t o th e P7

o f s c ( t ) , i s employed t o account fo r th e v a r i a t io n .

v alue o f 1 . 0 i n s t a t e P and minimum value of 0.257

def ined a s a c o r r e l a t i o n l o s s.

Equation (3) may be generaliz,ed as

C with Ct. Ar

va lue of amplitude

$4( h a s maximum

i n s t a t e P3. $ i s

F-11



From equation A.3 ( 6 ) , page A-8, it may be determined that the receiver

input signal-to-noise ratio for the range code component only,

p2] is related to the input signal-to-noise ratio for the total

Nir B

carrier by

It is seen that there has been an effective signal loss due to the effects of the receiver code on the input noise, given by the factor 2*62 or 0.835.

Tr

Equation ( 4 ) may be generalized as

B where

LD= .835 is defined as detection loss.

F.1.4 Receiver Threshold

The threshold of the range clock receiver is the threshold of the

clock loop in the receiver. The threshold treatment of the clock loop

is that for a modulation restrictive loop as given in section C . 4 . Thesignal-to-noise ratio used for threshold computations is that given by

equation F.1.3 ( 6 ) above, except that it is computed in the clock loop

noise bandwidth. Thus,

YBNN

where [dBis the total carrier-to-noise ratio at the input to the N

range clock receiver, computed in the bandwidthBN

. The value ofLD

is 0.835. A worst case value ofLK

is -12 decibels.

F-12



F.1 .5 Range Code Acquisition TimeIt i s d e s i r a b l e t o r e l a t e t h e t im e f o r a c qu i ri n g t h e r an gi ng c o d e . t o

the s igna l and no ise pa ramete rs a t t h e i n pu t t o t h e r a ng e c l oc k r e c e i v e r ,

shown i n f ig ur e F.1-1, page F-1. Acquis i t ion t ime i s def ined as being

t h e t o t a l t im e t o o b t a i n i n d i c a t i o n s of c o r r e l a t i o n b etween t h e l o c a l l y

gen erate d code components and t h e rece ived code fo r a g iv e n p r o b a b i l i t yor e r r o r , g iv en p r i o r a c q u i s i t i o n o f t h e r an ge c l oc k s i g n a l . Rapid clock

a c q u i s i t i o n i s a s su r ed f o r t h e c l oc k l oop s i gn a l- to - no i se r a t i o s u f f i c i e n t l yhigh.

The code ac qu is i t io n process has been phys ica l ly desc r ibed e l sewhere( re f . Zl).. The process i s knuwn as "maxi" l i k e l i h o o d a c q u i s i t i o n ,"and has b ee n t r e a t e d by E a s t e r l i n g ( r e f . 12) . With a s l i g h t m o d i f i c at i o n

E a s t e r l i n g ' s t r e a tm e n t may be a p p l i e d d i r e c t l y to the range c lock recei ve r and code co rr e l a t or . The mo dif ic at ion i s t h a t h e r e t h e e ne rg yp er b i t i s g iv en by t h e d i f f e r e n c e i n c o r r e l a t i o n l e v e l s a t t h e o u tp u t

of t h e c o r r e l a to r an d n o t by t h e l e v e l s t he ms elve s.

F r o m e q u a t i o n s F.l.l (lo), page F-5, an d F.1.3 (6), page F-12, it

i s s ee n t h a t t h e s i g n a l i n t o t h e c o r r e l a t o r i s

K

The reference s i g n a l from the c lock loop VCO has si ne phase. The si g

n i f i c a n t c o r r e l a to r ou tpu t te rm , t h e d i f f e r e n c e te rm , i s then taken as

K

The desi red output s i gn al S ( t ) i s the change of s , ( t ) when a0

code component i s acquired. The change i s always posi t ive ( r e f . ll),

t h e r e f o r e , S (t) i s a lways pos i t ive . 6 has on ly va lues 0.25,0

0.50, 0 .75, or 1.00 ( r e f . 11). A c q u i s i t i o n o f a component i s s ig n a l e d

by J4( i n c r e a s in g from i t s i n i t i a l v alu e, s a y 0.50, t o the nex t

h igher va lue, say 0.75. Therefore , the e f f ec t iv e ou tpu t s ig na l may be

w r i t t e n as

As (t) = 0 Tr

F-13



where

p2= c o r r e l a t i o n v a lu e a f t e r a c q u i s i t i o n of a component

=

plv a lue b e fo r e a c q u i s i t i o n o f a component

Theref ore, K

The s i g n a l power a t t h e o u tp u t o f t h e c o r r e l a to r i s then

The magnitude of the assumed f l a t n o i se s p e c t r a l d e n s it y a t t h e

output of t h e c o r r e l a t o r i s

and i s r e l a t e d t o th e s p e c t r a l d e ns i ty a t t h e r e c e i v e r i n p u t throughequation F.1.2 (27),page F-10, as

The ra t i o o f ou tpu t s i gna l power t o no ise s pe c t ra l dens i ty i s next

obtained as

n K

K 1



- -

-

K' 0 s i n 2 ~ 'p , J:(A~~)- 'i

l @ o l -6.12 s ( ) i=l I'n i I

The t o t a l code a c q u i s i t i o n time for t h i s system, assuming pr io rc lock acqu is i t io qm ay be b roken i n t o two par t s : t h e i n t e g r a t i o n t i m e

requ i red t o make a d e c is i o n w i t h h t h e a ss ig ne d e r r o r p r o b a b i l i ty on amaximum li ke li ho od bas is , and t h e b u il t- in machine del ay time. Thef o l l o w in g d e f in i t i o n s are made:

Ta = t o t a l code a c q u i s i t i o n tb-3

T =m a c h in e d e l a y time between t r i a l c o r r e l a t i o n sm

T. = i n t e g r a t i o n time p e r t r i a l c o r r e l a t i o n1

Wi = t h e ith code component

p . =,p e r io d , i n e le m en ts of t h e it h

component Wi1

- log2 Pi = number of informat ion b i t s i n WNi i

1 = subscr ipt des ignat ing longest componentw1

T1

= i n t e g r a t i o n ti me p e r i n fo rm a ti on b i t i nw1

Tm'th e machine de lay t ime , i s a b u i l t - i n f i x e d pa ra m eter o f th e

r an gin g d i g i t a l c i r c u i t r y . T j i s a n i m p l i c i t f u n c ti o n o f th e r a t i o of

c o r r e l a t i o n s i g n a l power t o n o is e s p e c t r a l d e n s it y , p r o b a b i l l t y o f

error , and informat ion content Ni

of the code component W . . The1

d i g i t a l r an g in g s ys te m i s implemented such thatTi

i s f ixed dur ing

any s i ng le range code ac qu is i t io n. Therefore , Ti must be f ix e d t o

a c e m o d a t e the code component W1 of gr ea te s t l eng th , which has the

h ighes t in fo rmat ion con ten tN1

.

then

T. = N T1 1 1

F-15



The i n t e g r a t i o n i s performed s eq ue nt ia l ly on each element of a code

component, and s eq u en ti al ly on th e components themselves. Therefore,

t he t o t a l a cq u i s i t i on t i m e may be w r i t t e n as

T = T IC Pi - 11 + TINIC Pia m

Since i n u sab l e ca se s t he sum of the component periods i s much greater

than uni ty , th e approximat ion holds that

The in te gr at io n t ime p er informat ion b i t may be solved for, as

E a s t e r l i n g ’ s ( r e f . 1 2 ) f i g u r e i s reproduced here as f i g u r e F.1.5-1w i t h t h e a b s c i s s a units r e l abe l ed t o conform t o t h i s n o t a ti o n . For a

given e r r o r p rob ab i l i t y and in format ion conten t o f th e lon ges t codeS0

component, a value of T - may be read d i rec t ly f r o m t he curve .I @ o II ,

Given a requi rement for TI, stemming from a requirement for Ta , t h e r e -

S Aqui red ou tpu t va lue may be inferred.

I@,I

F-16



u-TJ_c1

1r), = 2015 -

I O 9

8

7 6

5

4

3 2- I

1.0 1 IO0

Figure F.1.5-1.- Error pro ba b i l i ty ve rsus s ignal -to-no-;e dens i -y ' r a t i o

F-17



F.2 PC M Telemetry S ub ca rri er Demodulator

This sec t ion w i l l develop an approximate an al ys is fo r th e demodul a t o r shown i n f ig ur e F.2-1.

Band- l ( t ) S qu ar e q ( t ) Band- c ( t ) Looppass *

l i m i t e rD l a w > pass

f i l t e r -device f i l t e r

iJL’ B L

Freq.

mult .

Figure F.2-1.- PC M te lemetry subcarr ier demodulator

The input s ignal i s t aken as purely phase modulated, of the form

1s . ( t ) = A cos kCt+ p s ( t 3

where th e modulation f un ctio n i s biphase

where

ct( t )= square waveform having only the values +1

The input noise i s t aken as a sample function of a narrow-band Gaussianp r o c e s s , b a n d -l im i te d t o t h e l im i t e r b a nd w id th , as

n . ( t ) = x ( t ) cos w e t - y ( t ) s i n wct ( 3 )

1

F-18



From equation B . 3 ( 4 ) , page B-6, t h e l i m i t e r outpu t l(t) i s t akenas

l( t) = 4 -JI

cos Iw e t + $(t)

1( 4 )vL

This t r ea tment w i l l be l im i t ed t o r e l a t i v e l y high s igna l - to -no iser a t i o s (SNR) i n t h e l i m i t e r . For h i g h l i m i t e r SNR,

l( t)2 4 -fl cos rw e t + c p s ( t ) + ddj( 5 )

A

VL c

from equation B . 2 (lo), page B-4.

F. 2 . 1 Output Data TreatmentThe l i m i t e r s i g n a l i s demodulated phase coh ere ntl y by t h e product

de t ec t o r hav ing ga in cons tan t K'p. It i s assumed th a t t he re fe ren ce s i gn a l

s r ( t ) from the VC O i s e s s e n t i a l l y n o i s e l e s s when t h e o u tp u t d a t a i s

usab le . This imp l ies th a t whenever th e output da ta i s usab le , the sub-c a r r i e r t r a c k in g l o op i s wel l above th res ho ld . The re fe r ence s ign a l i s

t aken as

sr( t) = - sin wct (1)

The o u tpu t s i g n a l and n o i s e s p e c t r a l d e n s i t y may be t a k e n d i r e c t l yf r m equa t ions B . 4 . 2 (8) and (lo), page B-9, as

U sin g t h e i d e n t i t y o f e q u a t i o n A .3 (3 ) , page A-7, equa t ion F.3 ( 2 )may be s ub st i t ut ed i n ( 2 ) t o g iv e

s (t)=;2

V K'c0 L c p t

(t) ( 4 )

F-19



- -

A qua n t i ty which i s u s e r u l for p r e d i ct i n g d a t a q u a l i t y i s- t h e-r a t i o of o u tpu t d a t a b i t en er gy -to- no is e s p e c t r a l d e n s i t yr;I @noI

This i s given as t h e b i t rate R t i m e s t h e r a t i o of outp ut power t o

n o i s e s p e c t r a l d e n s i t y .

4 2 12

R - V KE

= R71

2 L r p

I 'no1

where

n

b

i- r a t i o o f inpu t subcar r i e r power to inpu t no i se s p e c t r a lI@nil d e n s i t y

Equation (7) holds for reasonab ly h igh l imi te r SNR.

F.2 .2 Reference Loop Treatment

The square law device squares th e l im it er s i gn al l ( t ) and passesa l l z o na l e n er gy n e ar t h e second harmonic of t h e su bc ar ri er frequencythrough the band-pass f i l t e r t o t h e p ha se -loc ked l oo p . The output of thesquaring device i s

q ( t ) = 12( t ) = 8k]'{I + cos E w c t + 2$(t ] }

F-20



The ' d r i v in g s i g n a l f o r t h e l o o p i s t aken as th e double f requency term,

c ( t ) = 8 PI2cos [.met + 2$( t J1

For reasonab ly h igh l imi te r SNR, equa t ion (2 ) i s w e l l approximated by

c ( t ) = 8 B2cos [2wct + 2 q s ( t ) + 2 .i"i ( 3 )A

Due t o th e assumed square tel em et ry waveform, th e si g n a l term of equa

t i o n ( 3 ) i s i d e n t i c a l l y

2 c p ( t ) = 51 c t ( t ) = f 51 ( 4 )

then

2

c ( t ) = -8 7f COS [2m Ct + 291 ( 5 )

Comparison of equation ( 5 ) with equat ions B . 2 ( 1 0 ) an dB .2 (ll), both on page B-4, shows th a t th e phase no ise s pe c t ra ld e n s it y f o r t h e PCM t e lemet ry re fe rence loop i s given by

Since c ( t ) con ta ins no s ig na l modula tion , the loop w i l l not have modu l a t i o n t r a c k in g e r r o r , e x c e p t p o s s ib ly for Doppler ef fe ct s . Neglecting Doppler, th e loo p phase j i t t e r i s obtained from equa t ion (6) andequa t ion C . 2 . 3 (l), page C-14, a s

CT(P = 4 kIBNThe loop may be t r e a t e d f o r t hr e s h ol d as i n s e ct io n C.2.4, employ

ing equa t ion (7) above. Equation (7) i s v a l i d f o r r e a s on a b ly h ig hl i m i t e r SNR.

F-21



F.3 The Residual Carrier Tracking Receiver (Ground)

The ground c a r r i e r trac ki ng rec ei ve r, shown below i n f igure F.3-1,

i s a clos ed loop, phase track ing, double-superheterodyne rec eiv er. Thiss e c t i o n w i l l determine the equivalence between the receiver and a simple

phase-locked loop as t r e a t e d i n appendix C.

Freq.m u l t . 4 b

x3-

I- s i n

phaseIs i n wlot

I

Figure F. 3-1. - C a r r i e r t r a c k i n g r e c e i v e r

I n t h e f i g u r e , t h e v a ri o u s K ' s are ampli tude t ransmiss ion constants .

The numbered subscripts refer t o the nominal center f requencies of the

var ious s igna l s . T h a t i s , slO(t) r e p r e s e n t s a s ig n a l whose nominal

center f requency i s 10 megacycles . The frequency mu lt ip l ic a t io n fac to r

of the network between the VCO and the f i r s t mixer i s m. The inputs i g n a l i s taken as in appendix C.

r 1

F-22

. . . ,,, , ... .. - ., . , , ,11.11111 I 8 I I I I I I 111111 I I



The VCO output vol tage i s taken as i n appendix C, as

where i s n o t e q ua l t owC . The f i r s t m ix er i n j e c t i o n s i g n a l i s a

f requency mul t ip l i ed vers ion of t h e VCO s i g n a l .

s o ( t ) = - A ~s i n ( 3 )

The f i r s t in te rmedia te f requency s i gn a l i s one term of the p roduc t

I n p a r t i c u l a r,

(5)

The second inte rme diat e frequency sig n al i s one term of the product ,

K10K50AAosin- K l o ~ 5 0 ( t ) s i n w G O t = 2 + V i ( t ) - mcpTJ( t )

1

s i n u)

6ot

I n - p a r t i c u l a r ,

s l O ( t ) i s t h e l im i t e r i n p u t s i g n a l . The l im i t e r o u tp u t s i g n a l l( t)

has an ampli tude constan t dependent only on l im it in g le v e lvL'



T he m u l t i p l i e r s i gna l m ( t ) i s t aken as the low-frequency term, as i n

appendix C.

or

Equation (10) i s i d e n t i c a l i n form t o equat ion C.l (51, page C-2. There

fo re , t he VCO output phase funct ion C p v ( t ) may be w r i t t e n d i r e c t l y as

where

h ( 7 ) = impulse response function of t he l oop f i l t e r

% = VCO cons tan t

Mul t ip ly ing bo th s ides o f equat ion (11)by m, w e obta in

mqv(t) = 2K h( 7) s i n q i ( t - 7 ) - mqv(t - T] d7dtr0 0

Equat ion (12) i s i d e n t i c a l i n form t o e q ua ti o n C.l ( 9 ) , page C-3.

Therefore, by analogy, equat ion (1 2) des crib es a simple phase-locked

loop with inp ut Thase fun ction of ( P i ( t ) , output phase fu nct i on of

m qv( t) , and open l oop ga i n (neg l ec t i ng loop f i l t e r cons t an t ) of

It i s seen from equat ion (13) t h a t t he f r equency m u l t i p l i ca t i on cons tan t

m has been incorporated i n t o th e loop gain. The presence of t h e l imi te ri n the loop may be expec ted t o produce l im i t e r e f fe c t s , t r e a t e d i ns e c t i o n C.3.1, under condi t ions of low l i m i t e r SUR.

F-24



- -

The con clus ion t o be drawn from the above i s t h a t t h e ground r e s i d u al ca r r i e r t r ack i ng r ec e i ve r may be t r e a t e d f o r t h re sho l d as a normalphase-locked loop by the methods of s e c t i o n C . 4 .

F.4 The Residual Ca rr ie r Tracking Receiver ( Spa cec raft )

The spa cec raf t c a r r i e r t rack ing rece iv er , shown below in f ig ur e F.4-1,

i s a closed-loop, phase -t racking , double-superheterodyne rece iver . Thiss e c t i o n w i l l determine the equivalence between the receiver and a simplephase-locked loop as t r e a t e d i n append ix C.

mul;.qF' / ixm

i+ - .__ _ _

xa l

Figure F. 4-1. - C a r r i e r t r a c k i n g r e c e i v e r

I n t he f i gu re , t he va r i ous K's are ampl i tude t ransmiss ion cons tan t s .The numbered sub scr ip t s re f e r t o the nomina l cen te r f reque nc ies o f theva r i ous s i gna l s . That i s ,

s47 ( t ) r e p r e s e n t s a s ig n a l whose nominal

cen ter f requency i s 47 megacycles. m1

and m2

a r e f requency m u l t i p l i

c a t i o n f a c t o r s f o r the networks between the VCO and the mixers.

The input s i gn a l i s t aken as i n appendix C.

F-2 5



The VCO ou t pu t s i gna l i s t aken as

(2)

The VCO s i g n a l i s f requency mul t ip l i ed by a f a c t o r m2 to obtain

t he s econd m ixer i n j ec t i o n s i gn a l sm

( t ) .2

s (t)= cos t2%t+ m 2 v v ( t )1m2

The second in j e c t io n s ig na l i s f requency mul t ip l i ed by a f a c t o r "1 to

ob t a i n t he f i r s t m i x e r i n j e c t i o n s i g n a l s"I(t).

s (t) = A~ cos + m1 2 v ] ( 4 )m 'p (t)ml

The f i r s t in t e rm ed ia t e f requency s i gn a l47(t)

i s one term o f t h e

product

( 5 )

I n p a r t ic u l a r

The second interm edia-te frequenc y s igr .a l69(t) i s one term of the

product

L -1

F-26



I n p a r t i c u l a r ,

s9(t) i s t h e l im i t e r i n p u t s i g n a l . The l i m i t e r o u tp u t s i g n a l l(t) has

an amplitude constant dependent on ly on l im i t in g l e v e lvL

l(t) =-fl

cos w t + V i ( t ) (9)4vL r9The o th e r i n p u t t o t h e p ha se d e t e c to r i s S o ( t ) which i s t h e VC O

s ign a l , f requency d iv ided by 2, and phase sh if te d by 90".

s o ( t ) = -A 0 s i n

A s i n a pp en dix C, t h e m u l t i p l i e r s i g n a l m ( t ) i s t aken a s t h e low-

frequency term of the pr oduct of l ( t )and s o ( t ) .

m ( t ) =; 2 K'V A +>)C P L O s i n

{c p i ( t ) -

p1+jqt$

(11)

Equation (11)i s i d e n t i c a l i n form t o e q ua ti on C . l (51, page C - 2 . There

f o r e , t h e VCO output phase fun ct ion C p v ( t ) may be wr i t t e n d i re c t ly a s

h ( 7 ) = impulse response fu nct ion of t he loo p f i l t e r

% = VC O cons tan t .

F-27



W e nex t ob t a i n

L 2

h ( 7 ) s i n q i ( t - T ) \ J

Equation (13) i s i d e n t i c a l i n f orm to equa t i on C . l ( 9 ) , page C - 3 .

Therefore, by analogy, equa t ion (13) d e s c r i b e s a simple phase-lock.edloop wi th in put phase fun c t io n of q t ) , output phase funct ion of

yV( t ) , and open l oop ga i n (neg l ec t i ng loop f i l t e r[w]cons t an t ) o f

K = - 2 K'V A %71 ??Lo

I t 5-s seen from equat ion ( 1 4 ) t h a t t h e f re q ue n cy m u l t i p l i c a t i m

cons t an t s "1 and m2 have been incorporated i n t he l oop ga i n . A l s o ?

the presence of the l i m i t e r i n the loop may be expec ted t o produce

l i m i t e r e f f e c t s , t r e a t e d i n s e ct io n C. 3.1, under condi t ions of low

l i m i t e r SNR.

I t i s i n t e r e s t i n g t o n ote a di f fe r en ce be tween the ground c ar r i e r

t r a ck i n g r e c e iv e r , t r e a t e d i n s e c t io n F. 3 , and t he spacec ra f t r ece i ve r .I n the ground rec e ive r , t he input and ou tput phase func t ions of th eequiva le n t loop were the s i gn a l input phase fun c t ion c p . ( t ) and the

1 f i r s t m i xe r i n j ec t i on signal phase funct ion m cp , ( t ) . For cond i t i ons of

lock, t h e f i r s t m ixer i n j ec t i o n s i g na l t r acked t he i npu t s i gn a l phase

exac t l y , a ssum ing no s t a t i c phase e r r o r i n t he equ i va l en t l oop . Fort he spa cec raf t rece ive r , t he inp ut and ou tput phase func t ions of the

equ i va l en t loop a r e t h e s i g n a l i n p u t ph as e T i ( t ) , a n d t h e f h c t i o n

cpv(t). This equiva len t ou tpu t phase fun c t ion does no t

a c t u a l l y e x i s t anyw here i n t h e spacec ra f t r ece i ve r , as may be se en byexaminat ion of equat ions (17, ( 3 ) , ( 4 ) and (10). However, for t he casewhere the equivalent loop i s locked w i t h no loop e r r o r , t h a t i s , where

F-28



it may be easi ly seen b y s u b s t i t u t i o n o f e q u a ti o n (15) i n t o e q u at io n s

(31, (4), (61, (91, and (lo), t h a t t he i n t e rn a l phase- locked l oop i n

t h e s p a c e cr a f t r e c e i v e r t r a c k s e x a ct ly , as does the second mixer inj ec t i o n s ignal . The f i r s t m i x e r i n j e c t i o n s i g n a l t r a c k s t h e i n pu t s i g n a l

phase somewhat i n er ro r.

The conclusion t o be drawn from the above i s t h a t t h e s p a c e c r a f tr e s i d u a l c a r r i e r tr a c k in g r e c e i v e r may be t r e a t e d f o r t h r e s h ol d as anormal phase-locked loop by th e methods of se ct io n C . 4 .

F. 5 The Spacecraft Turnaround Ranging Channel

The phys ic a l de sc r ip t io n and opera t ion of t he spacec ra f t channel

which i s used f o r " tu rnaround ranging" has been t rea t e d prev io us ly( r e f . 13). T h is s e c t i o n w i l l develop an approximate t reatment for t h eA r igorous t reatment would be s o complex as t o be p r a c t i c a l lychannel.

unusable.

The turnar ound ran ging channel, shown i n fi gu re F.3-1, comprises

an i de a l ba.ndpass l im i t e r , a coherent product de tec tor , and a phase

modulator. The up -li nk range code i s demodulated, alon g w it h any sub-

ca r r i e r s and no ise i n the bandpass o f t he l i m i t e r , and i s remodulated

along wi th subcarr iers and noise onto the down-l ink.

'i(t) I I d e a l

1 I + \

References i g n a l s u b c a r r i e r s

Figure F. 5-1. - Spacecraft turnaround channel

The output s ignal of t h i s channel , s (t),d i f f e r s from t h a t0

S

assumed i n th e t rea tment o f de te c t ion , i n appendices D and F due tos e v e r a l e f f e c t s of t he channel . The t heory of appendices D and F mustbe s l i g h t l y m od if ie d t o a cc ou nt f o r t h e s e e f f e c t s . The e f f e c t s a r e a nadd i t j -on al noise spectrum on th e down-link due t o t in "-a ro un d noise ,

F-29



a suppression of th e des i red dawn-l ink s i gn a l due t o remodulated noise,

a suppression of t h e d e s i r e d s i g n a l due to t h e s p a c e c r a f i l imi te r , and

a suppression of th e des i re d dawn-link channels due to turned-around

subca r r i e r s .

"he i n p u t signal to t h e l i m i t e r ,si (t), i s t aken i n normal form

S

as a sinu soid , phase modulated by th e sum of range code plus an a r b i

t rary number, L, of subca r r i e r s .

si(t)= A s COS I.Ct + q S [ t j

S

where

The subscript j d en o te s up -l in k s u b c a r r i e r s e x p l i c i t l y .

The inp ut noi se i s assumed t o be a sample function of a Gaussian

noise process , whi te , and band-l imi ted t o BL , the physical bandwidthof t h e l i m i t e r . S

i (t) = x s ( t ) cos wCt - y , ( t ) s i n wCtS

The subscript s denotes a spacec ra f t quan t i t y .

From equat ion D. 2 (lo), page D - 1 1 , t h e m u l t i p l i e r s i g n a l d ri v i ng

the phase modulator i s approximated as

where

a

L

= l i m i t e r s igna l suppress ion cons tan t

S

F-30



from equa t ion D.2 ( 5 ) , -page D-10, % i s approximated a sS

2 - 1( 5 )-

"LS

S where

= s p a c e c r a f t l i m i t e r i n p u t n o i s e - t o - s ig n a l r a t i o

S The phase modulator sums the ranging channel ( s i g n a l ) w i t h t h e

normal dawn-l ink subcarr iers . A modulation gain Gr i s app l ied t .o

th e ranging channel . The normal su bc ar r i er s have phase de via t io ns &pi.

Then the output s ignal s0S

(t) i s given as

For s i m p l i f i c a t i o n , w e may d e f i n e a phase indexATm

for the turnaround

channel a s

2K'V

4'P Ls

Gr

A'Pm - 7((7)

then the phase modulator output i s

F-31



The i n p u t s i g n a l r e c e iv e d a t the ground i s t aken as

f

J

where A d i f f e r s f r o m A a c co r din g t o t h e s i g n a l g a in s and a t t e nu a g 0

S

t ions between spacecraf t and ground.

F . 5 . l Equivalent Noise

Observat ion of equat ion ( 9 ) shows t h a t t h e s i g n a l r e c e iv e d a t t h eground conta ins phase noise of the form

The e f fe c t s o f t h i s n o i s e a r e two -f old. F i r s t , t h e n o i s e a p pe a rs a tthe ground as noise and lowers t h e e f f ec t i ve ground inpu t s igna l - to no ise ra t io . Secondly, th e phase nois e removes power f rom t h e ca rr ie r

and eff ec t i ve ly suppre sses t h e remaining modulat ion. 'The suppress io n

ef fe c t has been t r ea te d by Middleton ( r e f . 1 4 ) .

The assumption i s made t h a t t h e r.m.s. phase dev i a t ion o f th e

s i g n a l due t o t h e p ha se n o i s e i s s m a l l enough s o t h a t t h e r e s u l t i n g

phase noise spectrum i s not spread o r broadened beyond t h a t of theor ig in a l spac ecra f t no i se . Then an equ iva len t ex te r na l " inc rementa l"

n o i s e f u n c t i o n NA( t ) may be po s tu l a t e d , which, when summed w i t h an

assumed "no isele ss" rece ived s ign al , g iv es t h e same phase noise as t h a t

of t h e a c t u a l r e ce i v ed s i g n a l . That i s , a n e q u a l i t y i s def ined as

F-32



where

and 2 2 02 = 4-

A ( p , a

L SYs2( t )

(4)

P I T 2S

AS

2 -avs

e i s t h e power su ppres s ion fa c t or o f Middle ton ( r e f . 14) due t ophase modulat ion by Gaussian noise having an r . m . s . va l ue o f a .

pS Assuming t h a t t h e modulation bandwidth encompasses a l l t h e n o i s e

pas sed t h rough t h e spac ec ra f t l i m i t e r , equa t ion ( 4 ) may be rewri t ten as

TS

The incremen"a1 noise function may be r e l a t e d t o t h e p ha se n o i s ef u n c t i o n as

From equa t i on (6) t h e n o i s e s p e c t r a l d e n s i t y of t h e i n cr e m en t al n o i s e

fu nc t io n may be i n f e r r e d as

F-33



Dividing equat ion (7) by the value of th e a c tu a l ground system nois e

s p e c t r a l d e n s i t y g i v e s

then

rS,

Imn,/ 4 2 2BL

SI pnigj = x *'m "LS

( 9 )

S -

and

o f t he s igna l - to -no i se r a t i o due t o the a c t ua l g round sys tem noi se

s p e c t r a l d e n s it y , t ak e n i n a bandwidth equa l t o the spacecra f t l imi t e r .

I f the normal thermal inpu t noise s t o t he spac ecra f t l im i t e r and groundreceiver are assumed t o be uncorre l a t ed , a t o t a l e q ui v a le n t n o is e s pe ct r a l d e n s it y i n t h e ground re ce iv er may be pos tul ate d as

F-34



--- -

*L1 + AT,

2-

si fl Sl + x -NiS

-

BLS

Equat ion (12) i s an approx imation usab le w i th in a bandwidth narrowerthan the spac ecra f t l im i t e r bandwid th and cen te red on the g round rece ived

c a r r i e r e e q u en c y.

F.5.2 Equiva len t S igna lFrom equa t ion F .5 .1 (2), page F-32, t h e s i g n a l p ha se f u n c t i o n o f t h e

turned-around channel as r e c e iv e d a t the ground i s given as

E m p lo y in g t h e i d e n t i t i e s of appendix A, t h i s s i g n a l i s seen to be

F-35



The turned-around s ignal w i l l be approximated by on ly th e primary code

term and the f i r s t o r d e r s u b c a r r i e r terms. Then

f L

L 7 The equiva lent no is ele ss s i gn al received on th e ground i s expressed

f i n a l l y as

2-0. .

T SL

s (t) = A e2

co s t + Acpreff c t ( t ) + C Acp.eff s i n w . t + cp.(t)g g

j =1 J iJ 5

K + c AT, s i n pit + ( 4 )i=1 -

where

...

and

L

h = j

As i n sec t i on A . 3 , t h e r e s i d u a l c a r r i e r term may be written as

F-36

1



APPENDIX G

PHASE M O D U L A W SIGNAL DESIGN

T hi s s ec t i on t r e a t s t he des i gn and op t i m i za ti on o f nar row dev i a t i on

phase modula ted s in us o id a l ca r r i e rs . The modula t ion func t io ns a r e taken

t o be baseband func t ion s and/or sub car r i e rs . Only the determinat ion oft he va r i ous m odul at ion i n d i ce s i s t r e a t e d , s i n c e s e l e c t i o n o f s u b c a r r i e rf r equenc i e s i s a separable problem and i s a f u n c t i o n of t h e s p e c t r a l ext e n t o f th e modulated sub ca r r i e r s themselves .

The s i g n a l t o b e t r e a t e d i s taken from equat ion A . 3 (l), page A-6.

K

+s ( t ) = A cos t + ~ v ~ c , ( t ) c nrpi s i n kit +i=1

where

A = ca r r i e r am p l i t ude

w = c a r r i e r r a d i a n fr eq u en c yC

c t ( t ) = rangin g code, having only valu es fl

ATr= peak phase dev iat ion of c a r r ie r by ranging code

ATi= peak phase deviat ion of c a r r i e r by t h e i

t hs u b c a r r i e r

u)

i= rad ian f requency of t h e ith s u b c a r r i e r

q i ( t ) = ef fec t ive phase modula t ion on the ith s u b c a r r i e r

It i s assumed t h a t t h e r e s t r i c t i o n t o s m a l l phase dev ia t ions in su res th a t mos t o f the s i gn a l power w i l l be con cent ra t ed i n the zeroand f i r s t orde r s i g na l produc t s.

It i s assum ed t h a t s i g na l de t e c t i on i s performed using phase

cohe ren t p roduc t de t ec t o r s , t r ea t e d i n append ix D, or a s p e c i a l i z e dr an g in g r e c e i v e r , t r e a t e d i n . s e c t i o n F. 1.

There are two b a s i c s i g n a l d es ig n c r i t e r i a , s u bj e ct t o c e r t a i n .

boundary condi t ions on the re s i du a l c a r r i e r and the amount o f a l lowablei n t e m o d u l at ion.

G-1



The f i r s t c r i t e r i o n i s a s e t of minimum design goals for t h e i n f o r

mation channels. T hese des i gn goa l s a r e gene ra l l y spec i f i ed by a minimum

channel s igna l - to -no i se r a t i o and the bandwidth i n which it i s taken.

The s i g n a l m u s t be designed s o t h a t as ca r r i e r pow er i s dec reased i n

the presence of add i t i ve w h i te G auss ian channel no ise , t he m i n i ” de

s i g n g o a l s are m e t simultaneously.

The s e c o n d c r i t e r i o n i s t h a t when the channel de s ig n go a l s ares imult aneous ly ach ieved, the channel s igna l - to -no i se r a t i o s must be maxim iz ed w i t h i n t h e c a p a b i l i t i e s of a v a i l a b l e c a r r i e r p ower.

A boundary condi t ion i s that s a t i s f a c t i o n of t h e basic d e s i g n c r i

t e r i a s ho uld n o t r ed uc e t h e s i gn a l- to - n oi se r a t i o i n t h e r e s i d u a l c a r

r i e r channel below i t s minimum design goal.

A second boundary condit ion i s t ha t t he amount of unusable power

or i n t erm odu l a ti on p roduct s r e su l t i n g f r o m t h e s a t i s f a c t i o n of t h e two

d e s i g n c r i t e r i a s h o u l d n o t be ove r l y l a rge .

G. 1 So l u t i on for Modulat ion Indices

For a coheren t p roduct de t ec t o r , t he ou t pu t s i gna l - to -no i se r a t i o

for t h e jth modula ted sub car r i e r , t aken i n a bandwidth Bo , i s obtained

3*an equa t i on D. 1 . 4 . 1 (6 ) , page D-7 , as

where

= r a t i o of t o t a l s i g n a l power t o i n p u t n o i s e p ower i n abandwidth Bo

0

J j

F o r e i t h e r a coheren t p roduct de tec tor or for t h e range c lock re c e i v e r of se c t i on F.1, t he output s i g n a l - t o - n o i s e r a t i o f o r t he range

code, tak en i n a bandwidth Bo , i s taken from equat ion F . l . 3 (6),m

page F-12, i n th e form

G- 2



For a produc t de tec to r

$ = % = 1 I n terms of t h e n o t a t i o n u sed i n s e c t i o n F.1.3,

r-

- - - .sC 1Y Bo = B-&jN

0r

NC1

r

Equat ions (1)and (2 ) may be rearranged as

V

Equations ( 5 ) and (6) may be combined as

( 3 )

(4)

G-3



I l l l l l I I ll1 I

Equation (7) may be s im pl i f ie d t o

I n genera l , for range code p l u s K s u b c a r r i er s , t h e r e a r e K equa t ions

of t h e form of equat ion (8). T hese may be w r i t t e n i n s m e d form a s

0i

G- 4



.The bandwidths are t r e a t e d as system constants , and the s ignal- to-noise

r a t i o s as independent var iables . ATi and Acpr

are dependent var iables .

When th e bandwidths and signal-t o-no ise r a t i o s a re ass igned as m i n i "

design goals , repeated s imul taneous solut ion of the K s e t s o f e q u a t i on s

y i e l d s s e t s of s o l u t i o n s (ATr, Acpi) s a t i s f y i n g t h e f i r s t d e s ig n c r i t e r i o n .The s olu t io ns a re n ot unique as the re a re an in f in i t e number o f so lu t ions .

For a s igna l hav ing K sub car r ie rs only , w i th range code dele ted,

the equa t ions analagous t o equa t ions (9) are

K-1 c i=1

(10) For t h e s p e c i a l c a s e o f a subcarr ier which i s phase-shif t keyed

&go0, it i s e a s i l y se en t h a t

BO i No= Rm[: E

- BOi

where

R = k ey in g b i t r a t e

E = energy per b i t

!@I=va lue o f channe l no i se sp ec t r a l dens i ty

G- 5



For t he spe c i a l c a se of a quadraphase su bc ar r i er which i s phase-

s h i f t keyed by two t e l emet ry channel s , it may be determined that

where

R and RY

= t e l em e t ry b i t ra tesX

E and E = t e l em e t r y e n er g i e s p e r b i tX Y

IQI = value of channel no i se s pe c t r a l dens i t y

Fo r t h e sp ec ia l case of the tu rnaround ranging channel, a r e l a t i o n b e

tween th e ef fe ct iv e turned-around phase i nd ice s of t he range code and

up subcarr iers on the dawn ca r r i e r may be obtained by dividing equa

t i o n s ~ . 5 . 2(6) by ~ . 5 . 2( 5 ) , page F-36. Then

L

where

ATj = d e v i a t i o n o f t h e j th subca r r i e r on t he up ca r r i e r

Acpr = dev ia t io n of the range code on the up ca r r i e r

G-6



Substitu t i o n of equat ion (14) i n equa t ion (8) g i v e s

J

where , fo r th e tu rnaround channel , co r r e l a t io n l o s s 4( a nd d e t e c t i o n

loss % are t aken as i d e n t i c a l l y u n it y.

G . 2 Maximization of Su bc ar rie r Channel Signa l-to-nois e Ra tio s

Inspec t ion o f equa t ion G.l (l), page G-2, shows tha t the s igna l

t o- no is e r a t i o i n t h e j th s u b c a r r i e r c ha nn el i s p r o p o rt i o n al t o a fulic

t i o n o f modula tion ind ice s g iven by

For a signal composed of K su bc ar r i er s and ranging code, maximizat ionof a l l t h e s u b c a r r i e r c ha n ne l s i g n al - t o- n o i se r a t i o s i s obtained bymaximizing equation (1) f o r a ny a r b i t r a r y j , u si ng s e t s o f (LITr,

w hich a r e s o lu t i o n s of e q u a t io n G . 1 (9), page G-4 . Since simultaneous

s a t i s f a c t i o n o f e q ua t io n G . l ( 9 ) s e t s a l l t h e s u b c a r r i e r ch an ne l s i g n a l t o - n oi s e r a t i o s p r o p o r t i o n a l t o e ac h o th e r by c o n s ta n t s , maximization

of one su b ca rr ie r channel maximizes a l l subcar r ie r channe ls .

G.3 Boundary Condition on Residual C ar ri er

The s i g n a l - t o - n o i s e r a t i o f o r t h e r e s i d u a l c a r r i e r ch an ne l i n i t s

bandwidth B may be determined from equa t ion A . 3 (7),page A-8, t o beC

G -7



Rearranging equat ion (1)and equa t ing t o equa t ion G. 1 ( 5 ) , page G - 3 , wehave

K a

NJ

o r

X

The boundary condition i s ob tained by s t ip u la t in g th a t when the s igna l

t o- no is e r a t i o i n t h e j th subcarr ier cha.nne1 f a l l s t o i t s des ign goal

t h e s i g n a l- t o - no i s e r a t i o i n t h e r e s i d u a l c a r r i e r c ha nn el sh ou ld be

e q ua l t o or g r e a t e r t ha n i t s design goal . Then,

- 1(4)

2BC

C

G- 8



G. 4 S i g n a l E f f i c i e n c y

From equa t i ons A. 3 (6) and ( 7 ) , page A-8, it may be determinedth a t t he powers res id i ng i n the r es id ua l c a r r i e r component, p rime codecomponent, and f i r s t order subcarr ier components of the modulated s ignala re gi ven , r e s pec t i ve l y , by

P code

; K

P s u b c a r r i e r s = {2 cos2 (ATr )

L- j=1

The t o t a l usable power, or ef fec t ive power , iS given as -the sum of the

three component powers as

+ 2 cos

i=1

The percent, or dec imal , e f f ec t iv e power, re fe r r ed to t he t o t a l c a r r i e r

power i s

% Peff =

The di f f e ren ce be tween e f fe c t iv e power and t o t a l power i s unusable

power composed of ot he r modu lation pro duc ts. $ Peff may be used as a

boundary con dit ion on ch annel maximizat ion.

G-9



APFEXDIX H

SuPPI;EMENTARY THlCORY

H.l The Eq ui va le nt Noise Bandwidth of L in ea r Networks

Figure H . l - 1 shows a diagram of a two-port l i n e a r network which i s

d e s c r ib ed b y an i n p u t- o u tp u t v o l t a g e t r a n s f e r f u n c t i o n G ( s ) i n t h ecomplex variable s .

Figure H . l - 1 . - Linear network model

The t r a n s f e r f u n c t i o n i s def ined as

where

V ( s ) = u n i l a t e r a l L a p l a c e0

vol tage func t ion

V.(s) = u n i l a t e r a l L a p l a c e1

vol tage func t ion

transform of t h e outpu t

v i ( t )

t ransform of t h e i n p u tv o ( t )

Since the network i s L ine a r, t h e p r i n c ip l e o f s u p e r p o s i t i o n a p p l i e s andt h e t r ea tment o f t he network fo r no i se inpu ts may be made independentlyof con s ider a t ion s abou t th e p resence o f s igna l .

The inpu t t o th e network i s t aken a s n . ( t ) , a sample M c t i o n of1

a Gaussian process . n . ( t ) i s a func t ion wi th f i n i t e non-zero power.1

T h a t i s,

where the bar denotes "average." It i s assmned tha t the inpu t no i se

H-1



has some nois e s pe ct ra l den si ty , an ( j W ) , which i s a r e a l , e ve n f u n c t i o n

of the imaginary var iable j w . i

Papoul i s ( r e f . 15) has shown that for f i n i t e power inputs to l i n e a r

sys tems, the ou tpu t s pe c t ra l dens i ty may be wr i t t en as

where

The t o t a l noise power out of t h e l i n e a r n etw ork i s obta ined by in teg ra t

i n g t h e o u tp ut s p e c t r a l d e n s i t y w i th r e s p e c t to frequency.

For the case where the inpu t sp ec tr a l den si ty i s cons tan t or f l a t

with va lue 1 'nil

, i n t h e r eg i on s of G ( j w ) o f i n t e r e s t , t h e i n t e g r a l

may be rewritten as

Note t h a t t h e u se o f t r a n s f e r f u n c t i o n s w hich e x i s t f o r p o s i t i v e

and negat ive f requencies impl ies the use o f i n pu t s p e c t r a l d e n s i t i e s

which are a l s o " two-sided." This i s no cause for alarm and i s merely

a consequence of the use of Fo uri er transform s. The Fou rier transform

o f a real - t ime func t ion i s always two-sided. I f , i n t h e p h y s i c a l w or ld

where only po si t i ve freq uen cies have meaning, a "r ea l" one-s ided spec

t r a l d e ns it y N i s given as

N = K T (7)

H-2



- - - -

where

K .= Boltzmann's constan t

T = temperature

then the equ iva len t " two- sided" sp ec t r a l dens i ty i s simply

N K T / pn j- 2 - 2

rt i s p o s s i b l e to def ine an equ iva len t r rs qu a re " t r a n s f e r f u n c t i o nhaving constant ampli tude Gr, some reference amplitude of t h e o r i g i n a l

t r an sf er funct ion , and a t ra nsmiss ion bandwidth ( two-s ided) of j2A%.

This equ iva len t t r ans f e r func t ion i s def ined such t h a t the power t ran sm i t t e d through it from a white, Gaussian input densi ty i s e x a c t l y equal

to t he power t r ansmi t t ed th rough the o r ig in a l t r a ns f e r func tion . Theequivalence i s made by equating output noise powers

then

The treatment w i l l now be confined to t r ans f e r func t ions wh ich a r er a t i o s of polynominals i n s, o f t h e form

G ( s ) = 5l.3where the degree of Q ( s ) i s a t l e a s t one g r e a t e r t h an P(s) and a11t h e c o e f f i c i e n t s of s a r e r e a l .

The 1ef-t;-hand qu a nt i ty o f equat ion (10) i s ca l l ed the " imaginarytwo-sided equ ival ent no ise r adi an bandwidth. ." The r e a l ozle-sided

equivale n t no ise bandwidth i n cycles i s r e l a t e d a s

H-3



- -BIT - 2Tc

i s t h e physical square bandwidth through which equal pwer w i l l be

t r ansmi t t ed as t h a t t r a n s m i t t e d t hr ou gh t h e r e l a t e d t r a n s f e r funetfon.

For t r a n s f e r f u n c t i o n s of t h e ty-pe spec i f i ed , contour in t eg ra t io n and

t.he t heo ry o f r e s i d u e s b y be used to solve equation (10).

Figure, H.1-2.- Gontoixr o f i n t e g r a t i o n

Along the contour shown i n the above f fgure , t h e eqm1f”r;yholds.

For la rge IRI the t r a n s f e r f u n ct i o n s t r e a t e d are of t h e orde r of l / sr-3

and the i n t eg rand G(s)G(-&) i s of t h e order l/s‘-. For these t r a n s f e rfunctfons and large 1 Rl the i n t e g r a l a l on g p a t h C2 approaches zero-

H-4



t hen , by the theo ry of r e s idues

of G(s)G(-s) G( jw)G(- j w ) d j w = 27cj

i n t h e l e f t - ha l f

and

of G ( s ) G ( - s )

(1-5)G i n t h e l e f t - h a l f p l aa er

or Residues of G( s)G( -S )

BN=%c{

i n t h e l e f t - h a l f p la ne-

H. 2 Equ ival ent Noise Temperature of L ine ar Systems

E.2.1 Single NetworksEvery l i n ea r ne twork , a c t iv e or p a s s i v e , c o n t r i b u t e s n o i s e t o a

s ig na l pass ing th rough it. For purposes o f p r e d i c t i o n , it i s importantt h a t t h e n o i s e p r o p e r t i e s o f th e ne tworks d ea l t w i t h be known.

It i s w e l l known t h a t a r e s i s t a n c e h av in g a physica l tempera tureTR

produces a whi te Gauss ian one- sided no i se sp ec t r a l den s i t y ( ava i l ab l epower). .

j o u l e smR(f) = mR

cycle per second

where

K = Boltzmann s cons tan t

It i s p o s s i b l e t o a t t r i b u t e n o i s e p ro du ce d by a l i n e a r n etw ork t o animag ina ry r e s i s t anc e a t th e ne twork input , matched t o the input ,and 'coconsider the network i t s e l f noi se les s. The temperature of 'chis imaginaryresistance which would be r equ i r ed to produce the network noise i f t h e

network w e r e n o i s e l e s s i s ca l l ed the ne twork ' s "equ iva len t no i se

H-5



t empera tu re ." Since the network i s assumed l inear , the presence of a

s i g n a l or other uncor re la ted no ise does no t in f luence the ne twork s e l f -genera ted no ise or i t s e q u i v a l e n t n o i s e temperature.

S .1 + Ni G so + No

A A

v v

N 8- ( n o i s y ) -

S.1 + Ni G so + No

PFigure H. 2.1-1. - Equivalent noise temperature of a n o i s y l i n e a r netw ork

Figure H.2.1-1 shows the resolut ion of a l inea r no i sy ne twork N

having power gain G i n t o a no ise les s ne twork fed by a re s i s t o r having

equivalent noise temperature TN. The summing ju nc ti o n s a r e a concep

t u a l ai d , in di ca t i ng the summing of tempera tures . They are not physicalc i r c u i t s m i n g p o in t s .

H-6



- -

__

I n f ig u re H.2.1-1 ( a ) ,

So = GSi iN = N C GN = N + GKT. ( p e r c y c l e o f e q u iv a l e n tO N i N 1 noise bandwidth) Iwhere

NN = noise con t r ibu te d by the no is y ne tworks

I n f i g u r e H.2.1-1 ( b ) ,

So = GSi

No = NN + GN - GKTN + GKTii

0 It i s seen th a t i n t erms o f equ iva len t no i se t empera tu re and on a p e rcyc le equ iva len t no i se bas i s , t h e r a t i o of i n p u t t o o ut pu t s i g na l -t o

n o i s e r a t i o s ( n o i s e f i g u r e ) i s given as:

KTTN

Ni -

i

= I + - (4)S GS; T;

GK(TN + T

NO

T h i s r a t i o i s a f i g u r e o f m e r i t , which, when equa l t o one, i n d i c a t e s anoise less sys tem.

H . 2 . 2 Cascaded NetworksThere a re ge nera l ly two s i tu a t io ns where it i s d e s i r a b l e to o b t a i n

an equ iva len t no i se t empera tu re fo r two or more networks i n cas cade.The f i r s t case i s fo r two noisy networks each having power gain s g re at er

t h a n u n i t y . I t i s d e s i r a b l e t o have an equ iva len t t empera tu re re fe rencedt o th e i n p u t of t h e f i r s t network. The Second case i s for a n o i s y

network wi th power ga in g re a t e r than u n i t y , f ed by a pass ive networkwi th power g ai n l e s s than uni ty , fe d by some " input" noise tempera ture .It i s d e s i r a b l e t o o b t a in a n e q u iv a le n t n o i s e t e m p er a tu r e r e f e r e n c edt o t h e i n p u t o f t h e n e tw ork w i t h g a i n g r e a t e r t h a n u n i t y .

H-7



I 1 I

Consider two a rb i t r a ry ne tworks i n cascade , hav ing power ga ins G1and G 2 and equiv a len t no i se tempera tures , re fe r r ed to t h e i n d i v i d u a lnetwork inputs of T1 and T2 , r e s p e c t i v e l y .

e e

Figure H. 2.2-1. - Cascaded l inear noisy networks

The available output noise power i s

No = G KT2 2 + G2G1KT1

e e

[2e + TI]

No = KG2G1 G1

Equation ( 2 ) shows tha t t he equiva len t no i se t empera ture re fe r red t o

the input of the cascaded networks i s

T2

T =T1 + - e (47n .1. e G1

Equation (3) shows tha t t he equiva len t t empera ture re fe r red t o t he ou tpu t

of the cascaded networks i s

I- 1T = G~ + G ~ T ~ ~ ]

n0 2e

H-8

I.---.-.,- ...,, I,,.,,.,.,, .1,111 I 1 1 1 1 11111111I 1111 1111II 1 1 1 1 1 I I I I 1111 I I I I I



It can be shown ( r e f . 16) t h a t t h e e q u iv a l e n t i n p u t n o i s e t em p e ra tur eof a l i ne a r , b i l a te ra l , pass ive network, having power ga in G

P'whose

phys ica l t empera tu re i s TPY

i s given a s

T'e = k - j T p

Case 1: For th e sp ec ia l case of two noisy networks wit h powerg a in s g r e a t e r t h a n u n i ty , e q ua t io n (4) shows tha t the equ iva len t no i seinpu t t empera ture i s

T,L

Tn = T1 +-e(7)

i e G1

For G1 s u f f i c i e n t l y h ig h, T1 denotes the express ion .

Case 2: For t h e s p e c i a l c a se of a nois y hi- gai n network, fed bya lo ss y passiv e network, fed by some in pu t temperature Ti, f igure H.2.2-2

a p p l i e s .

Te = T2+ GLTi + - G d T pe

If the los sy network i s def ined by i t s a t t e n u a t i o n or l o s s f a c t o r Lwhere

then

TiTe = T2 + L +E - q T p

e

H -9



__

GL GNON0

(lossy) ( n o i s e l e s s1 0

Figure H. 2.2-2. - Cascaded passive and noisy networks

H.3 The Band-pass Amplitude Limiter

T h i s s e c t i o n s e t s down, i n t h e n o t a t i o n u se d el se w he re i n t h i s

p ap er , c e r t a in p e r t i n e n t r e s u l t s o f th e c l a s s i c a n a ly s i s of D av en po rt( r e f . 4 ) . The analys is w a s p erf or me d f o r a n i d e a l l l s ~a - p a c t i o n "l i m i t e r

f ol lo w ed b y a n i d e a l ba nd -p as s f i l t e r . The d r iv ing s igna l w a s t a k e n asa co nst an t amplitude sin us oid embedded i n narrow-band Gaussian no ise .

Narrow-banding of the in p u t n o i s e i s t ak en t o i m pl y b an d-p as s f i l t e r i n ga t t h e i n p u t t o t h e l im i t e r . The model i s shown i n f ig u re H.3-1.

The output spectrum of the l i m it er i t s e l f c o n t ai n s s p e c t r a l c o n t r i b u t i o n s

cen te red no t on ly a t the fundamenta l inpu t cen te r f requency , bu t a l so a tharmonics of th e in put ce nt er frequency. The ac t i on of the outp ut band-

pass f i l t e r i s t o allow t r ansm iss ion o f on ly the energy cen te red on the

H-10



Figure H . 3 - 1 . - Band-pass l i m i t e r model

f iindamenta l inpu t f requency . For an ou tpu t f i l t e r su f f i c ie n t ly wide t opass a l l the zonal energy centered on the fundamental frequency, Davenp or t ' s an a ly s i s shows th a t the ou tpu t s i gn a l and ou tpu t no i se a re re l a t e d t o t h e i n p ut s i gn a l- to - no i se r a t i o (SNR) by ra ther complicatedexpress ions involving th e conf luent hypergeometr ic funct ion ( re f . 17).The important results are reproduce d below. Fig ure H . 3 - 2 i s a graph o f

output noise power and output si gn al power vers us inp ut SNR.

Observation o f f i g u r e H. 3-2 shows th a t t he t o t a l output power oft h e b an d-p as s f i l t e r i s constant and i s given by

2

Po = s0+ No = 8k]

where

VL

= v o l ta g e l i m i t i n g l e v e l

A t low inpu t SNR it i s ev iden t from f i g u r e H . 3 - 2 t h a t t h e o u tp u tsignal power i s suppressed from the value a t high inpu t SNR. T h i s suppression may be expressed through use of a s ig na l vo l tage suppress ionf a c t o r aL such tha t

= a 2 p so L 0

Mart in ( re f . 7) has approximated "L by

1

L

Niwhere - i s th e inpu t noise - to-s ignal ra t i o . The ac tu al and approx

s:

imate %* a r e p l o t t ed i n f i g u r e H.3-3 .

H-11



F i g w e ~.3-2.-L i m i t e r s i g n a l and noise suppress ion versus input sm



~

Y

-20 -15 - 1 -5 0 5 IO 15 20

Si- INTO LIMITER B W , db.Ni

Figure H.3-3 . - Exact and approximate s i gn al suppre ssion

D a v e np o r t' s a n a l y s i s w a s f o r t h e c as e of a n m o d u l a t e d s i nu s o id a ls i g n a l . F or t h e pu rp os e of s i m p l i o i n g t h e a n a l y s i s i n t h i s p ap e r, t h e

assumption w i l l be made th a t th e r e s u l t s c i t e d above apply equ a l ly to

angle modula ted s inu so ida l s ign a l s .

H.4 Th e Range Equation

T h i s s e c t i o n w i l l de r i ve t he s i gna l - t o -no i se r a t i o , computed i n an

a rb i t r a ry bandw i d t h B a t some r e fe rence po i n t o f a r ad i o r ece i ve r , dueto t r ansm i s s i on of r a d i o e n e r gy f r o m a t r a n s m i t t e r w hich i s p h y s i c a l l ysepa ra t ed from t h e r e c e i v e r by a d i s t a n c e R.

Figure H . 4 - 1 shows t h e model of t h e c o m u n i c a t i o n s l i n k to be usedi n t h e d e r i va t io n . A t r a ns m i t t e r w i t h ou t pu t power PT f e e d s a n an

ten na wi t h power g ainGT

t h rough a lo s sy network having power gain

GLT. GLT i s a number l e s s t hen un i t y . The energy from t h e t r a n s m c t t i n g

an tenna p ropaga te s ac ros s t he d i s t ance R to t he r ec e i v i ng ant ennawhich has a power gain

GR.Only a p o r t i o n of t h e t r ansm i t t ed ene rgy

H-13



IFigure H. 4-1. - Communication l i n k model

i s i n t e rcep ted by the r ece iv ing an tenna. Th i s l o s s of power i s a t t r i b

u te d t o a propagation power gain GLP

l ess th an un ity . The power

in te rcep ted by the an tenna i s passed through a lo ss y network, having

power gain G” t o the r e fe rence po in t o f t he r ece ive r . Noise i n the

l i n k i s a t t r i b u t e d t o a n e x t e r n a l s ys te m n o i s e t em p er at ur eTS

summed

w i th t h e s i g n a l a t t h e r e f e r e n c e p o i n t.

The r e fe rence po in t f o r dete rm ining the s igna l - to -no i se r a t i o i sg e n e r a l l y t h e i n p u t o f a s t age which has su f f i c i e n t power ga in s o tha tadd i t io n of noise by subsequent s tag es i s n e g l i g i b l e .

Figure H. 4-2. - Antenna geometry

Figure H.4-2 d e t a i l s the geometry of th e antenna sys tem. The trans

m i t t ing an tenna ga in i s due t o a bunching of the t ransmi t ted energy in to

a beam. This beam e f f e c t then r a i s e s th e area power dens i ty of the re

ce iv ing antenna. The e f f e c t i s the same as i f a h i g h e r e f f e c t i v e povnr

’e f fhad been r ad ia t ed by the t r ansm i t t e r .

H-14



The receiving antenna has an e f f ec t i ve r ad i o f r equency a reaAeff

which i s th e a re a of th e p ass ing ra di o wave from which th e antenna, ext r a c t s a l l energy.

The ar ea power de ns i t y a t t he r ece i v i ng an t enna i s given by simplegeometry as

where

GLP

= e f fec t o f p ropaga t i on loss

The amount of power extracted by t he r ece i v i ng an t enna i s

~- G ~ G ~ Aef fG ~ P ~

'R -47rR

2

The ef fec t ive area of an an tenna f o r r ece i v i ng has been r e l a t ed t othe power gain f o r t r ansm i t t i ng by Friis ( re f . 18) and others as

where

A = wave len gth of th e ra di o energy

then P R = GLpGLTG GT R ["I2P T

The power reach ing th e r ec e iv er re fe rence po in t i s

r .1 2

H-15



The loss of power due t o geometry may be a t t r ib u t e d to a "space

l o ss" having power gainGLs'

2 2

G L s =b]=[&I where

C = v e l o c i t y of l i g h t

f = radio energy f 'requency

then

'i = G ~ p G ~ s G ~ ~ G ~ G ~ G ~ P ~ (9)

The effect of K d i f f e r e n t power gain s may be ex pressed compactly as

K 'i

= PT Gii=1

The power ga ins l e s s th an un it y may be expre ssed as

then equa t ion (10) may be written

K Gi

- i=1

'i - 'T L

The noise behavior of th e re ce iv er may be a t t r ib u te d to an input twc

s ide d n o i s e s p e c t r a l d e n s i t y mn ( f ) wherei

H-16

I 1 I I II



--. . . . . . . . ..... . . .

over some f requency range of in t e re s t where K i s Bol tzman's constant

and Ts i s t h e equiv a len t syst em noi se temperature. The noi se i n an

a r b i t r a r y b an dw id th B i s given as

o r

N J B = KTsB

T he s i gna l - t o -no i se r a t i o a t t h e r e f e r e n c e p o i n t i s now written

2

fiL; TB [A]

H . 5 Antenna Polar i za t ion Loss

The polar i za t ion o f a radio wave i s d e f i r e d a c co r di ng t o t h e s pa ce

o r i e n t a t i o n of i t s e l e c t r i c vec t o r . For an E a r t h bound r ece i v i ng s tat i o n , a ra d i o wave whose e l e c t r i c v ec to r i s p e r pe n d ic u la r t o t h e E a r t h ' ssu r f ace i s s a i d to be v e r t i c a l l y p o la r i z ed ; a wave whose e l e c t r i c ve ct or

i s p a r a l l e l t o t h e E a r t h ' s s u rf ac e i s hor i zon t a l l y po l a r ized . B oth a r es p e c i a l c a se s of l i n e a r p o l a r i z a t i o n , w herei n t he e l e c t r i c vec t o r a lw ays

l i e s i n one unique plane which i s pe rpend j cu l a r to t he p l ane of the wave.

A more ge ne ra l type of pola Yizat io n i s e l l i p t i c p o l a r i z a t i o n where t h ee l e c t r i c vec t o r r o t a t e s i n t he p l ane o f t he wave and va r i e s i n am p li tude

as a func t i on o f ro t a t i on ang l e . F i gu re H . 5 - 1 a p p l i e s .

Figure H . 5 - l shows a plan e wave pr opa gati ng out of t h e page, whose

p la ne l i e s i n the page. The sense of ro t a t i on o f t he e l e c t r i c vec t o rw i t h r e s p e c t t o th e d i r e c t i o n of p ro p ag a ti on i s clockwise. This i s def i n e d as " r igh t-hand" po l a r i za t i on . The l ocus o f e l e c t r i c vec t o rampli tude i s an e l l i p s e , hence t he name e l l i p t i c po l a r i za t i on . The

r a t i o o f minor a x i s l e n g t h t o major a x i s l e n gt h f o r t h e e l l i p s e i sc a l l e d t h e " a x i a l r a t i o " or " e l l i p t i c i t y r a t i o" . It i s s e e n t h a t l i n

e a r p o l a r iz a t io n i s a s p e c i a l c a s e of e l l i p t i c a l p o l a r i z a t i o n where t h ea x i a l r a t i o i s zero. The spe c i a l c a se f o r ax ia .1 r a t i o of one i s c a l l e dc i r c u l a r p o l a ri z a ti o n .

H-17



Plane of waveI

Figure H. 5-1. - E l l i p t i c p o l ar i za t io n

I t can be shown ( r e f s . 19 and 20) t h a t an an tenna which t ransmi t se l l i p t i c a l l y p o l a r iz e d w aves can be u se d t o r e c e iv e e n e r gy from a n i n c i d e nt e l l i p t i c a l l y p o l ar i z ed wave. If t h e a n t e n n a p o l a r i z a t i o n e x a c t l ymatches th e wave po la ri z a ti o n , maximum energ y i s ex t rac ted f rom the

passing wave. If n o t , t h e r e i s an e f fec t ive power loss or r e f l e c t i o n .The pol ar iz at io n power loss factor , which i s e s s e n t i a l l y an e f f i c i en c y

f a c t o r , i s given as

+4 % + (1 - d(1- %’) cos 2a

P 2 +(1+ 4T2)(1+ %2) (1)

where

= a x i a l r a t i o of t h e i n c i d e n t wave

% = a x i a l r a t i o o f t h e r e c e i v i n g a nt en na

a, = ang le be tween major axes o f the po la r iza t ion e l l ipses

3- = used for same sense ro ta ti on

- = used fo r opposi te sense ro t a t io n

KP

= 1 i n d i c a t e s maximum power ex tr ac te d from th e wave. KP

= 0 i n d i

c a t e s no power extracted from the wave. It can be easily shown that

K = 1 for circul-ar po la r i za t io n with th e same senseP or l i n e a r p o l a r i z a t i o n w i t h p a r a l l e l a xes . ( 2 )

H-18



- -K f o r c i r c u l a r to l i n e a r p o l a r iz a t io n .P - 2 ( 3 I

KP = 0 f o r c i r c u l a r p o l a r i z a t i o n w i t h o p p o s it e s en s e ( 4 )o r l i n e a r p o l a r i z a t i o n w i t h p e r p e n d i cu l a r ax es .

H.6 I n t e l l i g i b i l i t y o f C lip pe d Voice

S tu d ie s o f th e i n t e l l i g i b i l i t y of peak cl ipped voice waveforms as

a fun ct io n of c l ip pin g depth have been performed by se ve ra l auth ors and

agenc i es . The r e su l t s o f i nve s t i g a t i o ns by Shyne ( r e f . 21) and Lick l ider( r e f . 22) have been summarized and manipulated by Kadar of Grumman A i r

craf t and Engineer ing Corpora t ion ( re f . 23) . It i s not intended to re-

summarize here the resu l t s of t h e r e fe rences . R a the r , t he app l i ca b l e

r e s u l t s f r o m t h e r e f e r e n c e s w i l l be s t a t ed .

P l o t s of e m p i r i ca l ly d e ri ve d d a t a r e l a t i n g p e r ce n t i n t e l l i g i b i l i t y

f o r s i ng l e words and pe rcen t word a r t i c u l a t i o n t o pos t de t ec t i on peakspeech t o root-mean-squared noise , wi th peak cl ip pin g depth a s a param

e t e r , have been p l o t t ed i n t he r e fe rences . The o rd i na t e s o f t he p l o t sa r e l i n e a r i n p e rc e nt i n t e l l i g i b i l i t y . The a b s c is s a s are l i ne a r i n

dec ib e l s , peak speech t o r . m . s . no i se . S i nce t he dec i b e l va lue o f apeak t o r . m . s . r a t i o i s t h e same as t h a t o f a peak-squared t o mean-

squa red r a t i o , and s ince th e r a t i o of peak-squared s i gn a l t o mean-squa red no ise has been de r i ved f o r s eve r a l de t e c t o r s i n append ices D

and E, t he i n t e l l i g i b i l i t y p l o t s a r e d i r e c t l y u se ab le i n p re d i c ti n g

the performance o f voice channels .

mnned Spacecraft CenterNat ional Aeronaut ics and Space Administ rat ion

Houston, Texas, December 3 0 , 1965

H - 1 9



REFlERENCES

1. G i aco l e t t o , L. J . : Generalized Theory of Multitone Amplitude andFrequency Mod ulation. Proc. IRE, Ju ly 1947, p. 680.

2. Eennet t , W i l l i a m R. : E le c tr i c a l Noise. McGraw-Hill Book Co., Inc.,1960, p. 234.

3. Davenport, Wilbur B., Jr. ; and Root, W i l l i a m L. : An In tr od uc ti ont o t h e T he ory o f Random Signals and Noise. McGraw-Hill Book Co.,

Inc . , 1958, p. 158.

4. Davenport, W. B., Jr. : Signal - to-Noise Ra t io s i n Band-Pass L imi ters .J o u r n a l of Applied Phy sics, vo l . 24, no. 6, June 1953, p. 720.

5 . Middleton, David: A n I nt ro d uc t i on t o S t a t i s t i c a l C m u n i c a t i o nTheory . McGraw-Hill Book Co., In c. , 1960, p. 669.

6. Kuo, Benjamin C. : Automatic Con trol Systems. Pren t ice-H al l , In c. ,1962, p. 123.

7. Martin, Benn D .: The Pioneer JV Lunar Probe: A Minimum-PowerFM/PM System Design. Tech. Rept. 32-21?, J e t Pro pu lsio n Labor

atory, Pasadena, C a I T f . , March 15, 1962, P. 1 4

8. Develet , Jean A. J r . : An An al yt ic Ap proximation of Phase-LockReceiver Threshold. Space Technology La bo rato rie s, I nc. ,

Redondo Beach, C a l i f . , Apr. LO, 1962.

9. Korn, Granino A , ; and Korn, The resa M . : Mathematical Handbook forS c ie n t i s t s and Engineers . McGraw-Hill Book Co., I n c . , 1961,p. 18.8-3.

10. Titsworth, R. C . ; and Welch, L. R. : Power Spectra of Si gna l sModula ted by Random and Pseudo Random Sequences. Tech. Rept.32-140, J e t Propulsion Laboratory, Pasadena, C a l i f . , Oct. 10,

1961, p. 21.

11. Mart in , Jan W .: Apollo Pseudo-Random Noise Ranging System. To bepubl ished as a NASA TN i n 1965.

12. E a s t e r l i n g , M . ; and Schot t l e r , P . H: Maximum Likelihood Acquisitionof a Ranging Code. Research Surmnary 36-12, vol. I, J e t P r op u ls io n

Laboratory, Pasadena, C a l i f . , Ja n 2, 1962, p. 81.

13. P a i n t e r , J. H. ; and Hondros, G.: Uni f i ed S-Band Telecmunica t ionsTechniques for Apol lo - Volume I. NASA TN D-2208, 1965.

R - 1



1 4 .

15*

16.

18. 19.

20.

21.

22.

23-

Middleton, David: A n I n tr od u ct io n t o S t a t i s t i c a l C m u n i c a t i o n

Theory. McGraw-Hill Book Co., In c . , 1960, p. 606.

Papoul is , Athanansios: The Four i e r In t e gr a l and i t s Appl ica t ions .

McGraw-Hill Book Co., Inc., 1962, p. 247.

Livingston, Marvin L.: The Effect of Antenna Character i s t ics on

Antenna Noise Temperature and System SNR. IRE Transact ions onSpace El ec t ro nic s and Telemetry, Sept . 1961, p. 71.

Korn, Granino A , ; and Korn, T her esa M .: Ma them atical Handbook f o r

Sc i e n t i s t s and E ng ineer s. McGraw-Hill Book Co., I n c . , 1961,p. 9.3-10.

Schelkunoff , Sergie A , ; and f i i i s , Harold T.: Antennas, Theory

and Pr ac t ic e. John Wiley and Sons, In c. , New York, 1952, p. 43.

Sichak and Milazzo, S. : Antennas f o r Ci rc ula r Po lar iza t io n.Proc. I R E , vol. 36, A u g . 1948, p. 997.

Rumsey, V. H. ; e t a l : T echn iques fo r H andling E l l i p t i ca l l y Po l a r

iz ed Waves, wi th Sp ec ial Reference t o Antennas. Proc. I R E ,

~01.39, May 1951, P. 533.

Shyne, N. A,: Speech Sign al Proc essing and App l icat ions t o SingleSideband. ERL Tech. Re pt. , Montana S ta t e College, Boseman,

Mont., 1962.

L i ck l i de r , J. C. R.; and Pol lack, I . : E f f e c t s of D i f f e r e n t i a ti o n ,

I n t e g r a t i o n , a nd I n f i n i t e Peak C li pp in g Upon t h e I n t e l l i g i b i l i t yof Speech. Journal of the Acoust ical Society of America, no. 20,

1948.

Kadar, I.: S p e e c h I n t e l l i g i b i l i t y Cri te r i a for Apollo VHF AM

C m u n i c a t i o n L ink s. L;ED-380-4, G r m a n A ir cr af t Engineering

Corp., Bethpage, L.I . , Oct. 18, 1963.

R-2 NASA-Langley, 1966 S-89



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Unified S-Band Telecommunications Techniques for Apollo Volume II Mathematical Models and Analysis

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