Communication Systems

Signal Classification &

Fourier Review

Energy and Power Signals

Let be the voltage across a resistor producing a current . The instantaneous

power per ohm is defined as

tv R ti tp

tiR

titvtp 2

The total energy and power on a per-ohm basis are E P

joulestidtE 2

2

2

21lim

T

TT

wattstidtT

P

For an arbitrary continuous-time signal , the normalised energy

content of is defined as

Energy and Power Signals

E tx

joulestxdtE2

and the normalised average power of as P tx

2

2

21lim

T

TT

wattstxdtT

P

(1)

(2)

tx

Fourier Series

a sustained note is completely described by its fundamental frequency and amplitude (in addition to the harmonic-amplitudes.

i.e. we can write the infinite series sum

00

0202

0101

0

2sin2cos

4sin4cos

2sin2cos

nn bna

ba

ba

atF

0

Q: Why are sines as well as cosines required?

tnbtnatF nn

n 00 2sin2cos

tnBtnAAtF nn

n 0

1

00 2sin2cos2

1

(2)

Applying the identities:

coscos and sinsin gives

Q: How does the term arise? 20A

Expressions (1) and (2) are identical provided:

nnn aaA and nnn bbB

tnBtnAAtF nn

n 0

1

00 sincos2

1

Comment regarding

The amplitudes of the harmonics

If Fourier Synthesis was the process (by summation) of composing a waveform,

then Fourier Analysis is the converse process whereby we de-compose a

waveform into its constituent sinusoids i.e. determination of harmonic-

amplitudes.

tmBtmAAtF mm

m 0

1

00 2sin2cos2

1

?

Fourier Analysis has greater physical application. In applied physics

(engineering) is typically obtained experimentally. The aim is to fix the

and for as many values of as needed. mA

mB tF

m

Review Orthogonal Properties of Sinusoids

0

00

02

2sin

cos2

sin

cos

mn

T

ttmtndtI

If is one period, then:

0

1

T

02cos2sin 00

0 tmtndtIT

t

and

always!

Determination of the harmonic amplitudes and - continued mA mB

T

tm tmtFdt

TB

002sin

2

(and provided is defined in , can be found). tF T,0 mB

mA is similarly found to be:

T

tm tmtFdt

TA

002cos

2

if is not analytic, but experimentally determined, then and

are estimated from (computer) numerical integration.

integration may commence from any point but it must cover one period

tF mA mB

T

A useful mnemonic-form to remember finding the coefficients

in a Fourier series

periodonem

period

tmtFdt

periodA

2cos

2

periodonem

period

tmtFdt

periodB

2sin

2

Fourier Theory Applied to Comms

its usefulness is apparent in connection with the shift theorem

recall

so consequently

if two -functions are equally spaced either side of the origin, the FT is?

The Dirac delta-function

aFaxxFdxax

axax

1

0

pajpajpxj eaxeeaxdx 222

The Addition Theorem

ppxFxF 2121

The Shift Theorem

papaxFaxF

epaxF

epaxF

paj

paj

2cos2 111

2

11

2

11

e.g. let be the function, then xF1

paaxax

eax

eax

paj

paj

2cos2

2

2

ax

-3 -2 -1 1 2 3

-2

-1

1

2

ax ax 0x

ax

a

1

p

p

xF

Interference between Satellite Communication Channels and

Terrestrial Systems

Interference must be avoided:

(a) Between adjacent satellites and two earth stations

(b) When a terrestrial link sees a satellite

Interference is avoided by using high quality antennas.

m

Wavelength

Rad

iant

Em

itta

nce

121 ... mmsJ

1

11

2,

3

1

3

321

5

2

x

TkTk

hc

x

c

hcTM

BB

CKT 10001273

KT1073

KT873

Plancks Blackbody Radiation Law

Types (1) & (2) arise from Black-Body radiators

These are described by Plancks radiation law W

kT

h

BhP

1exp

h

B

k

T

Plancks constant Js3410626.6~ Bandwidth

Frequency

Hz Hz

Boltzmanns constant 1241038.1~ JKAbsolute temperature K

For radio frequencies:

kTBPkT

h

kT

h

1exp

This is the Rayleigh-Jeans Law

18

Product modulation spectra

19

AMPLITUDE MODULATION: PRODUCT MODULATION

Modulated signal is given by:-

)()(4

1)(

)()(2

1

)()()()(

:

)()(Let

cos)()(

cmcm

cmcm

ccm

m

FT

m

cm

PPPPSD

VV

VV

spectra

VtA

ttAtv

20

AM DETECTION IN PRESENCE OF CHANNEL NOISE: PRODUCT DETECTION -4

For product detection in presence of noise, detected signal is:-

CSSP

P

P

CPP

P

Pdtt

tA

T

P

P

dttAT

P

txtA

tttyttxttA

s

n

s

sn

s

s

T

T

cm

T

n

s

T

T

mT

s

m

ccccm

2 P

4

4 ratio noise-to-signal (detected)Output

2P

2

ratio noise-to-Carrier

2 = )2cos1(

2

)(

2

1limpower carrier Modulated

P= P n(t)power noise channel

4 =power noiseoutput Total

4 =power signaloutput Total

)(2

1limpower signal Total

contribute termsphasein only termsfreqhigher ))()((2

1

cossin)(cos))(cos)((

n

n

2

n

c

n

2

22

ANGLE MODULATION

signal modulating as varies)(

modulation Phase

signal modulating as varies)(

modulationFrequency

))(2cos()(

t

dt

td

ttAtV ccc

FM

PM

Am(t)

dAm(t)/dt Mod

signal

23

Frequency modulation (FM) Spectrum of an FM carrier with sinusoidal modulation

Let a unit peak amplitude carrier of frequency c be represented by f(t) where:-

)('=-=

-:is frequency carrier dunmodulate fromDeviation

)('2

))(2()(=2frequency instaneous the

))(2cos()(

cid

d

t

t

dt

ttd

dt

phased

tttf

c

ci

c

If the modulating signal is cos(2mt)

tt m 2cos)('

24

Frequency modulation (FM) Spectrum of an FM carrier with sinusoidal modulation - 2

INDEX MODULATION theasknown is and = where

Re)(or

2sin+t2cos)(

2cos=

ddisregarde becan and 0= tat time phase thengrepresenticonstant a is

2sin=)( -:then

deviationfrequency peak theis where2cos=)('Let

2sin2

c

d

m

tjtj

m

m

o

om

m

m

mc eetf

ttf

t

tt

tt

25


)()()(2

1)(

)()()()(2

1)()(

for gSubtitutin

)( termThe

kindfirst theoffunction Bessel )( 2

1

-:so =)2( ,2Let

1 where

SERIES) (FOURIER

1 period with periodic is termthe

FT

FT22sin

)sin(

2

2

-

22sin

22sin

2sin

mcmc

n

n

mncc

n

mn

tjn

n

tj

n

nj

n

m

T

T

tjntj

m

n

tjn

n

tj

mm

tj

nnJF

nJFtf

C

nCeCe

JdeC

ddtt

dteeT

C

eCe

Te

mm

m

m

mm

mm

m

26


Sidebands with amplitude >1% of unmodulated carrier

Information signal contains may sinusoids: so combine

27

Frequency modulation (FM) Carsons Rule

Carsons Rule (an empirical rule) states that the sidebands of significant amplitude lie within a bandwidth of 2m(1+ ) Hz, located centrally about the carrier fc

In many practical cases this underestimates the bandwidth and the expression 2m(2+ ) is more accurate

28

FM Link performance in presence of channel noise

As for AM we want to find the relationship between s/n and c/n. A typical FM detector has the form

n(t)+Vc(t)

Post

Detection

filter

channel

noise

+ FM

signal Bandpass

filter

Amplitude

limiter

Phase

detect d/dt

Low

Pass

filter

Discriminator n(t)

IF

filter

29

Pre-emphasised FM

Since the output noise spectra from the discriminator is parabolic a non ideal post detection filter would reduce the high frequency noise components where the spectral density is greatest - so reducing the overall output noise

BUT the high frequency components of the demodulated information signal would also be reduced by the filter leading, to distortion.

However if prior to modulation these high frequency components were emphasised by a filter with complementary performance to the post detection filter then the detected signal would not be distorted but the noise would be reduced

In this case the FM system is said to be operating with de-emphasis.

P()

0

freq

freq

Output noise

after ideal

post-detection

filter

De-emphasis

post detection

filter

m

P()

0

m -m

-m

Lower total

output noise

freq

freq m 0

-m

Pre-emphasis

filter at

modulator

30

FM Link performance in presence of channel noise-3

whereelse 0= ;2

B+2

B- 4

=)( Thus

2=)(

tordifferentafor function transfer theis )( where)( )( =)(

is PSDoutput so ator,differenti he through tpasses then Signal

bandwidth ruleCarson theis B where

2B+

2B-for , watts/Hz=)(=

PSD noiseoutput noise, whiteis )( Since

=power noiseoutput

of valuesquaremean has )( now

2

22

2

1

2

2

21

2

BA

PP

jH

HPHP

BA

PP

ty

A

P

Pty

c

n

c

n

c

n

n

Note: spectra is PARABOLIC with frequency

Significant power outside the information signal bandwidth - so filter

B/2

P()

-B/2 0

freq

31

The change in noise spectral distribution from channel to noise in the detected output signal

Carrier+channel

noise

Phase detector output

noise power

Differentiator output

noise power

Only power in

this bit

contributes

when post

detection filter

used

32


bandwidth ruleCarson is B where2

3

2/

where22

3=

4

3

4

3 =S=/

'2

4=output detector at S/N

2

4

2

2 is signal ofpower average Total

2

2

2222

3

22

22

222

CB

S

P

ACNC

P

AB

P

BA

P

BANS

P

m

n

c

mn

c

mnm

c

nm

c

n

Compare to AM where S=2C, for FM

doubling increases S by factor of 4

33


above derived as 2

CB'3

'2

B'3''3=S' Thus

''2

BC=' to changemust ,' to change if

C hence and B increases increasing

PSD noise channelconstant For increased. is if happensWhat

-:MISLEADING IS THIS BUT 3=Sget then 2= takeIf

m

2

m

33

m

3

CC

B

BCCC

CB m

34

Using FM link noise performance equation

EXAMPLE :-- = 7, C = 16 dB over 140 MHz, m=10 MHz, find S.

What happens to S if increased to 8?

S=33 C = 46.15 dB, C=16 dB

Increase to 8

dBB

dBCB

S

m

m

42.15'2

CB=C' MHz 160 '2'

3.472

)8(3'

m

2

Thus S has improved at expense of increased Bandwidth and worse C.

By spreading the modulated signal over a wider bandwidth for a given

baseband signal more noise power is filtered out by the post detection filter at

the receiver. So S gets bigger

35

Threshold effect thus limits the minimum level of C/N for FM

For Conventional discriminator circuit which reacts rapidly to changes in instantaneous frequency, limit of C/N is about 10dB

However a discriminator with some inertia would react less readily to doubling of the instantaneous frequency and so the threshold effect occurs at a lower C/N

Such properties are exhibited by a phase-locked-loop circuit.

Communication Systems

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