Communication Systems Lecture 25class.icc.skku.ac.kr/~dikim/teaching/3032/notes/EEE3032p... ·...

1

Communication SystemsLecture 25

Dong In KimSchool of Info/Comm Engineering

Sungkyunkwan University

2

Outline

Noise in Angle Modulation Phase deviation

Large SNR Small SNR

Output SNR PM FM

3

Review of Angle Modulation

General form of angle modulated signal:( ) ( )cosc c cx t A t tw fé ù= +ë û

( ) ( )

( ) ( )

PM: cos

: normalized message with max 1.c c c p n

n n

x t A t k m t

m t m t

wé ù= +ë û=

( ) ( )FM: cos 2t

c c c d nx t A t f m dw p a aé ù= +ê úê úë ûò

Transmitted Power: Demodulation output:

( ) ( ) ( )( )1PM: , FM:

2D D D D

d ty t K t y t K

dtf

fp

= =

2T

1P = .2 cA

4

Noise in Angle Modulation

( ) ( )From cos ,c c cx t A t tw fé ù= +ë û

( )cx t +

Noise w(t):

AWGN N0/2

PredetectionFilter

( )1e tDemodulation

Bandwidth of the predetection filter: Carson’s rule: BT = 2(D+1)W.

To analyze the effect of the noise, we need to write

( ) ( )( ) ( )

1( ) ( )cos ,

where is the noise added to .c e

e

e t R t t t t

t t

w q f f

f f

é ù= + + +ë û

FilterPostdetection

Reduce noise(explained later)

5


Predetection filter output:( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û

( ) ( )( )cos ( )cosc c n c nA t t r t t tw q f w q fé ù= + + + + +ë û

( ) ( ) ( ) ( )( )cos ( )cosc c n c nA t t r t t t t tw q f w q f f fé ù= + + + + + - +ë û

( ) ( )( ) ( ) ( )( )( )( ) ( ) ( )( )

cos ( )cos cos

( )sin sinc c n c n

n c n

A t t r t t t t t

r t t t t t

w q f w q f f f

w q f f f

é ù= + + + + + -ë û- + + -

( ) ( )( )( ) ( )

( )( ) ( ) ( )( )

( )cos cos

( )sin sin

c n n c

n c n

A r t t t t t

r t t t t t

f f w q f

w q f f f

é ù= + - + +ë û

- + + -

6


( ) ( )( )cos c eR t t t tw q f fé ù= + + +ë û

( ) ( )( )( ) ( )

( )( ) ( ) ( )( )1( ) ( )cos cos

( )sin sin

c n n c

n c n

e t A r t t t t t

r t t t t t

f f w q f

w q f f f

é ù= + - + +ë û

- + + -

( )( ) ( )( )

( ) ( )( )1 ( )sin

tan( )cos

n ne

c n n

r t t tt

A r t t tf f

ff f

--

=+ -

Phase deviation from the carrier:( ) ( )( ) et t ty f f= +

Only phase deviation affects the demodulation. Amplitude can be kept constant by a limiter.

7

Angle Modulation with High SNR

( )( ) ( )( )

( ) ( )( )( ) ( )( )

( ) ( )( ) ( ) ( )( )

1 1

1

( )sin ( )sintan tan

( )cos

( )sin ( )sin sin

n n n ne

cc n n

n n n n

c c

r t t t r t t tt

AA r t t t

r t t t r t t tA A

f f f ff

f f

f f f f

- -

-

- -= »

+ -

- -» »

High SNR: ( )c nA r t most of the time.

( ) ( ) ( ) ( ) ( )( )( )( ) sinne n

c

r tt t t t t tA

y f f f f f= + » + -

increases, then effect of ( ) reduces!c nA r t

Still dominated by the signal ( )tf

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Angle Modulation with Low SNRLow SNR: ( )c nA r t most of the time.

( ) : dominated by ( ).nt ty f

( )ta ( ) ( ) ( ).nt t ty f a= -

Need expression of ( )tain terms of ( ) and ( ).n t tf f

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Angle Modulation with Low SNRLow SNR:

( )ta

( ) ( ) ( ).nt t ty f a= -( ) very small if ( ).c nt A r ta

( ) sin ( ) tan ( )sin ( )

( )c

n

t t tA t

r t

a a aq

» »

»

( )

( ) ( )

sin ( ) ( )( )

n

cn

n

t tA t t

r t

y f

f f

»

- -

10

Angle Modulation with Low SNR

Low SNR:

( )( ) ( ) sin ( ) ( )( )c

n nn

At t t tr t

y f f f» - -

Message signal is lost at low SNR! Threshold effect.

11

Angle Modulation

Low SNR:

( )( ) ( ) sin ( ) ( )( )c

n nn

At t t tr t

y f f f» - -

High SNR:( ) ( )( )( )( ) ( ) sinn

nc

r tt t t tA

y f f f» + -

( )1( ) ( )cos ce t R t t tw q yé ù= + +ë û

(1)

(2)

Note: Eq (2) can be obtained from (1) by switching

Phase deviation

( ) ( ), and ( ).n c nt t A r tf f« «

Small most of the time

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Outline


Large SNR Small SNR

Output SNR PM FM

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PM Output SNR

( ) ( ) ( ) ( )

( ) ( ) ( )( )

( )( )

( ) sin

( )P

D D D D e

nD D n

cn t

D p n P

y t K t K t K tr tK t K t tA

K k m t n t

y f f

f f f

= = +

= + -

= +

( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û

( ) ( ) ( )( )cos ( )cosc e cR t t t t R t t tw q f f w q yé ù é ù= + + + + +ë û ë û

( ) ( ) ( )( )( )( ) sinnn

c

r tt t t tA

y f f f» + -High SNR:

PM Demodulation output:

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PM Output SNR

( ) ( ) ( ) ( )( )( )

( ) sin

P

nD D p n D n

cn t

r ty t K k m t K t tA

f f= + -

PM Demodulation output:

Demod Signal power: 2 2nDP D p mS K k P=

To compute noise power: let ( ) 0 (message is 0)tf =

( )( ) ( )( ) sinn sP D n D

c c

r t n tn t K t KA A

f= =( )cn t

( )sn t( )nr t

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PM Output SNR

ffc fc+B/2

Sn(f)N0/2

f

Sn(f-fc)

f

Sn(f+fc)

( ) ( ) ( )Lpsn n c n cS f S f f S f fé ù= - + +ë û

B/2

f

Sns(f)

B/2

N0

Recall: NB Noise

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PM Output SNR

( ) ( ) ( )D D p n Py t K k m t n t= + PM Demodulation output:( )( ) s

P Dc

n tn t KA

=2

02( )P

Dn

c

KS f NA

=

Spectrum of ns(t) is in [-1/2 BT, 1/2 BT] BT: Bandwidth of predetection filter (by Carson’s rule).

Output noise power: 2

02D

DP Tc

KN N BA

=

17

PM Output SNR Post-detection filter: Since BT= 2(D+1)W > 2W, the output noise power

can be reduced by applying a post-detection filter with bandwidth W

PredetectionFilter

( )1e tDemodulation

FilterPostdetection

2

022 DDP

c

KN N WA

=

2

02D

DP Tc

KN N BA

=

18

PM Output SNR

122

020

2 .D TDP D

c

K PN N W KA N W

-æ ö÷ç= = ÷ç ÷ç ÷è ø

Transmitted Power: 2T

1P = .2 cA

Output noise power:

2T

0 0

P =N W 2N W

cA

Increasing transmitted power reduces output noise power!Different from linear modulation.

( )( )( )

2 22

1200

/

n

n

D p m Tp mDP

D T

K k P PSNR k PN WK P N W

-= =

PM Demod Output SNR:

Transmitted power

Per unit message bandwidth

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PM Output SNR

( ) 2

0n

Tp mDP

PSNR k PN W

= PM Demod Output SNR:( ) ( )PM: cosc c c p nx t A t k m twé ù= +ë û

( ) sin or cosm mm t t tw w=

: Modulation Indexpkb ( ) sin sinp m mt k t tf w b w=

Single-tone Modulation

( ) 2

0

: increasing increases output SNR.n

TmDP

PSNR PN W

b b=

( )But bandwidth 2 1 is also increased.TB Wb= +

Input noise power will be increased.

Eventually large input SNR assumption is invalid threshold effect.

20

Outline


Large SNR Small SNR

Output SNR PM FM

21

FM Output SNR

FM Demodulation output:( )

( )

( )( ) ( )( )

( )( )

2( ) sin

2 2

( ) ( )F

DD

nD Dn

c

n t

D d n F d f

d tKy tdt

d t r tK K d t tdt dt A

K f m t n t f k

yp

ff f

p p

=

ì üï ïï ï= + -í ýï ïï ïî þ

= + =

( ) ( ) ( ) ( )1 cos ( )cos ( )sinc c c c s ce t A t t n t t n t tw q f w q w qé ù= + + + + - +ë û

( ) ( ) ( )( )cos ( )cosc e cR t t t t R t t tw q f f w q yé ù é ù= + + + + +ë û ë û

( ) ( ) ( )( )( )( ) sinnn

c

r tt t t tA

y f f f» + -High SNR:

22

FM Output SNR

FM Demodulation output:

Demod Signal power: 2 2nDF D d mS K f P=

To compute noise power: let ( ) 0 (message is 0)tf =

( )cn t

( )sn t( )nr t

( ) ( ) ( ) ( )( )( )

( ) sin2

F

nDD D d n n

c

n t

r tK dy t K f m t t tdt A

f fp


( )( )sin ( )( )2 2

n n sD DF

c c

r t t dn tK Kdn tdt A A dt

fp p

ì üï ïï ï= =í ýï ïï ïî þ

How to find noise psd?( ) : quadrature component of noise.sn t

23

FM Output SNR

( )( )sin ( )( )2 2

n n sD DF

c c


fp p

ì üï ïï ï= =í ýï ïï ïî þ

( )sn t ddt

( )u t2( ) ( ) ( )u nsS f H f S f=

( )2 2

2 20 02

1 1( ) 2 , f , .2 2 2

D DNF T T

c c

K KS f f N N f B BA A

pp

æ ö é ù÷ç= = Î -ê ú÷ç ÷ç ÷ ê úè ø ë û

( ) 2 ( )dx t j fX fdt

p« d/dt is a linear filter

24

FM Output SNR2

202

1 1( ) , f , .2 2

DNF T T

c

KS f N f B BA

é ù= Î -ê ú

ê úë û

The noise has less effect on low-freq message signals.

After Post-detection filter with bandwidth W:2 2

2 30 02 2

2 .3

WD D

DF Wc c

K KN N f df N WA A-

= =ò

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FM Output SNR

Transmitted Power: 2T

1P = .2 cA

Output noise power:

2T

0 0

P =N W 2N W

cA

Noise power is inversely proportional to T

0

PN W

( )22 2

12 20T

0

3P

3 N W

n

n

D d m d TmDF

D

K f P f PSNR PW N WK W

-

æ ö÷ç= = ÷ç ÷ç ÷æ ö è ø÷ç ÷ç ÷ç ÷è ø

FM Demod Output SNR:

12 2 23 T

020

2 P3 3 N W

D DDF

c

K K WN N WA

-æ ö÷ç= = ÷ç ÷ç ÷è ø

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FM Output SNR

( )2

0

3n

d TmDF

f PSNR PW N W

æ ö÷ç= ÷ç ÷ç ÷è ø


max '( )peak frequency deviationbandwidth of m(t)

tD

Wf

= =

( )maxd n df m t fD

W W= =

Recall: Deviation Ratio for general m(t):

( ) ( )FM: cos 2t

c c c d nx t A t f m dw p a aé ù= +ê úê úë ûò

( ) 2

0

3n

TmDF

PSNR D PN W

=

27

FM Output SNR


( ) 2

0

3n

TmDF

PSNR D PN W

=

TFor D 1, B =2(D+1)W 2DW:»

( )2

0

34 n

T TmDF

B PSNR PW N W

æ ö÷ç= ÷ç ÷çè ø

Input noise power will be increased.

Eventually large input SNR assumption is invalid threshold effect.

Increasing bandwidth can improve the output SNR.

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Pre-emphasis and De-emphasis for Noise

Unmodulated carrier:

( ) ( ) ( ) ( )( )( )

( ) sin2

F

nDD D d n n

c

n t

r tK dy t K f m t t tdt A

f fp


( ) 0 (message is 0)tf =

( )( )sin ( )( )2 2

n n sD DF

c c


fp p

ì üï ïï ï= =í ýï ïï ïî þ2

202

1 1( ) , f , .2 2

DNF T T

c

KS f N f B BA

é ù= Î -ê ú

ê úë û( ) 2 ( )dx t j fX f

dtp«

Demod output:

+

Noise

Pre DemoddetectionPost

detectionModPre

emphasis

f

De-emphasis

f

29


22

02

1 1( ) , f , .2 2

DNF T T

c

KS f N f B BA

é ù= Î -ê ú

ê úë û Assume de-emphasis filer is a 1st-order lowpass RC

filter:( )2

3

1( )1 /

DEH ff f

=+

Noise power after de-emphasis filter:2( ) ( )

W

DF DE NFWN H f S f df

-= ò

3

f W

+

Noise


detectionModPre

emphasis

f

De-emphasis

f

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Noise power after de-emphasis filter:

( )

2

2 2 222

0 0 322 2 2 233

2 2 23 1 3 2

0 3 0 3 0 32 2 23 3 3

( ) ( )

1 /

2 tan 2 22

W

DF DE NFW

W WD D

W Wc c

D D D

c c c

N H f S f df

K K ffN df N f dfA A f ff f

K K KW W WN f N f N f WA f f A f A

p

-

- -

-

=

= =++

æ ö æ ö÷ ÷ç ç= - » - »÷ ÷ç ç÷ ÷ç ç÷ ÷è ø è ø

ò

ò ò

+

Noise


detectionModPre

emphasis

f

De-emphasis

f

12 2

1 1 tan udua u a a

-=+ò

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Noise power after de-emphasis: Output SNR:

22

0 322 DDF

c

KN N f WA

»

Demod Signal power:2 2

nDF D d mS K f P=( ) ( ) ( )D D d n Fy t K f m t n t= +

( ) ( )2

30

/n

Td mDF

PSNR f f PN W

=

2T

0 0

P =N W 2N W

cA

( ) ( )2

0

3 /n

Td mDF

PSNR f W PN W

=

SNR without deemphasis:

+

Noise


detectionModPre

emphasis

f

De-emphasis

f

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Output SNR with de-emphasis:( ) ( )2

3_0

/n

Td mDF DE

PSNR f f PN W

=

Output SNR without de-emphasis:

SNR can be improved significantlyby de-emphasis filter.

3

f W

( ) ( )2

0

3 /n

Td mDF

PSNR f W PN W

=

+

Noise


detectionModPre

emphasis

f

De-emphasis

f

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Example:

SNR with de-emphasis:

SNR without de-emphasis:

n3 m

FM: 75 , W=15kHz, D = 5.

2.1 , P =0.1.df kHz

f kHz

=

=

( ) ( )2

3_0

/n

Td mDF DE

PSNR f f PN W

=

( ) ( )2

0

3 /n

Td mDF

PSNR f W PN W

=

( ) ( )2

_0 0

75 / 2.1 0.1 128T TDF DE

P PSNRN W N W

= =

( ) ( )2

0 0

3 75/15 0.1 7.5T TDF

P PSNRN W N W

= =

Transmitted power can be reduced by 17 times, with the same noise performance.

Communication Systems Lecture 25class.icc.skku.ac.kr/~dikim/teaching/3032/notes/EEE3032p... ·...

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Transcript of Communication Systems Lecture 25class.icc.skku.ac.kr/~dikim/teaching/3032/notes/EEE3032p... ·...