impact of noise on communications systemsihayee/teaching/ee5765/ece... · impact of noise on...

Department of Electrical and Computer Engineering

Impact of Noise on Communications Systems

Transmitter Channel Receiver)(tm TS

Channel Noise

ii NS ,

)(),( tnts oo

Input Output

oo NS ,

Baseband System

io SS

B

no dffSN0

)(2

NBdfN

B

0

22


NB

S

N

S i

o

o

NB

Si

o

o

N

S

Let

Then

B

mo dffSmS0

2 )(2Note that


Impact of Noise on DSB-SC System

Basband

Filter (LPF)

)(tm

Channel

Noise

ii NS ,

)(tyo

oo NS ,

2]cos)(2[ ttmS ci

tccos2

Bandpass

Filter

Bc 2

tccos2

)(tyi

Demodulator

ReceiverChannelTx

2]cos)([2 ttm c

222

)(2

)(2 mtm

tm


)(cos)(2)( tnttmty ci

To determine So , we start at demodulator input

ttnttntn cscc sin)(cos)()( We know that

ttnttntmty cscci sin)(cos)]()(2[)(

)(2

1)()( tntmty co

Hence, at the output of demodulator


)(2

1)()( tntmty co

io SmS 2

And2)(

2

1tnN co

For white noise with PSD = N/2

NBtntnc 2)()( 22

NBNBNo 22

1

NB

S

N

S i

o

o


Impact of Noise on AM System – Synchronous Demodulation

Basband

Filter (LPF)

)(tmA

Channel

Noise

ii NS ,

)(tyo

oo NS ,

tccos2

Bandpass

Filter

Bc 2

tccos2

)(tyi

Demodulator

ReceiverChannelTx

Please note that So and No will be same as in DSB-SC system

2mSo

NBNo


)(2)( 22 tmAtmA

22 )(tmA

2cos)]([2 ttmAS ci

2cos)]([2 ttmA c

2

)]([2

2tmA

2)]([ tmA


2mSo

NBNo

22 )(tmASi

NB

m

N

S

o

o

2

NB

S

mA

m i

22

2

22

2

mA

m

Remember A ≥ mp

22

2

mm

m

N

S

po

o

1/

122

mmp

2max

o

o

N

S


Impact of Noise on AM System – Envelop Detection

Envelop detector

)(tmA

Channel

Noise

ii NS ,

)(tyo

oo NS ,

tccos

Bandpass

Filter

Bc 2 )(tyi

Demodulator

ReceiverChannelTx

22

)]([ 222 mAtmASi

ttmAts ci cos)]([)(


)](cos[)()( tttEty icii

22 )()]()([)( tntntmAtE sci

Where

)(cos)]([)( tnttmAty ci

ttnttntmA cscc sin)(cos)]()([

Writing in envelop form


Small Noise Case:

)()]([ tntmA for all tif


2)]()([ tntmA c

)()( tntmA c

2mSo

NBtnNand co 2)( 2

22

2

22

22

2 mA

m

NB

S

mA

m

NB

m

N

S i

o

o


Large Noise Case:

)()]([ tntmA for all tif


222 )()]()[(2)()]([ tntmAtntntmA scc

22 )()]()[(2)( tntmAtntn scc

)]()[(2)()( 22 tmAtntntn csc


)()()( 22 tntntE scn where

)(

)(tan)( 1

tn

tnt

c

sn

and

)(cos)()( ttEtn nnc

)(sin)()( ttEtnand nns

)]()[(2)()()( 22 tmAtntntntE csci

)(cos)(

)]([21)( t

tE

tmAtE n

n

n


)(cos)(

)]([21)()( t

tE

tmAtEtE n

n

ni

)(cos

)(

)]([1)( t

tE

tmAtE n

n

n

)(cos)]([)( ttmAtE nn

Further simplification

For realistic case, SNR turns out to be

222916.0 mAN

S

o

o


)](cos[)( ttAt cEM

Impact of Noise on Angle Modulated Signals

)()( tmkt p

dmkt

t

f

)()(

Wherefor PM

for FM

PM)(tm dxxm )(

)(tFM


Phase

Demodulator

)(tm

Channel

Noise

ii NS ,

)(tyo

oo NS ,Bandpass

Filter)(tyi

ReceiverChannelTx

Pjase

Modulator

Lowpass

Filter

At demodulator input:

)(2 BfBW BBW

)](cos[)( ttAts ci

ttnttntn cscc sin)(cos)()(

)](cos[)( tttE ncn

)()()( tntsty ii


PM – Wide Bandwidth Case

)()](cos[)( tnttAty ci

)](cos[)()](cos[)( tttEttAty ncnci

)]()(cos[)( ttttR c

)()( tmkt pWhere for PM

At demodulator output:

)()()( tttyo

)()( ttmkp


)(

)]()(sin[)()](sin[

tR

tttEt nn

)(

)]()(sin[)()(

tR

tttEt nn

2/)( t

A

tttE nn )]()(sin[)(

)]()(sin[)(

ttA

tEn

n

By looking at the phasor diagram

For small noise case


At demodulator output:

)()()( tttyo

)]()(sin[)(

)( ttA

tEtmk n

np

])(sin[)(

)( tA

tEt n

n

BW of >> BW of)(tn )(t

sin)(cos)(

cos)(sin)(

tA

tEt

A

tEn

nn

n

sin)(

cos)(

A

tn

A

tn cs


sin)(

cos)(

)(A

tn

A

tnt cs

)(sin

)(cos

tnA

tnA

cs

)(sin

)(cos

)(2

2

2

2

cs nn SA

SA

S

)(sin

)(cos

2

2

2

2

ss nn SA

SA

2

22

2

)(]sin[cos

)(

A

S

A

Sss nn


{)(

)(0

2

)(Bff

AN

Bff

fS

2

)()(

A

SS sn

{2

0

)(Bf

AN

Bfn fS

o

22

2)(2

A

NB

A

NBNo

After the low pass filter

After the phase demodulator


22 mkS po

2

2

A

NBNo

)]()(sin[)(

)()( ttA

tEtmkty n

npo

Remember the starting point

NB

mAk

ANB

mk

N

Sp

p

o

o

2)(

2

22

2

22

NB

A

NB

A

NB

Si

2

2/ 22

We know that


NB

mAk

N

Sp

o

o

2)(

22

NB

A

2

2

22 mkN

Sp

o

o

22 )( pppp mkmk max])([

tmmp

Also for PM

where

2

22)(

po

o

m

m

N

S


Example of Tone Modulation

ttm m cos)(

2/22 m

mpm

22

22

2

22 2/

)()(mpo

o

m

m

N

S

2

2

m

2

2

mf

f


Phase

Demodulator

)(tm

Channel

Noise

ii NS ,

)(tyo

oo NS ,Bandpass

Filter)(tyi

PM ReceiverChannelPM Tx

Phase

Modulator

Lowpass

Filter

)(2 BfBW BBW

Impact of Noise on WBFM

Phase

Demodulator

)(tm

Channel

Noise

ii NS ,

)(tyo

oo NS ,Bandpass

Filter)(tyi

FM ReceiverChannelFM Tx

Phase

Modulator

Lowpass

Filter

)(2 BfBW BBW

dt

d


)](cos[)( ttAt cFM

dmkt

t

f

)()(Where

2222 )( mktmkS ffo

Noise Spectral density just before differentiator is

22

)()(

A

N

A

SS sn


Noise Spectral density just before differentiator is

22

)()(

A

N

A

SS sn

Noise Spectral density just after the differentiator is

2

2

2

)()(

A

N

A

SS sn

dA

NN

B

o

2

0

2

2

2

2

32

0

2

2

3

8)2(2

A

NBdf

A

fNB


22 mkS fo

2

32

3

8

A

NBNo

32

222

2

32

22

8

3

3

8 NB

mkA

A

NB

mk

N

S ff

o

o

NB

A

B

mk f 2/

43

2

22

22

2

22

)2(3

B

mk f


2

22

)2(3

B

mk

N

S f

o

o

22

222

)2(3

p

pf

mB

mmk

2

2

2

2

)2(3

pm

m

B

2

22

2

22

33pp m

m

m

m

B

f


Example of Tone Modulation

ttm m cos)(

2/22 m

pm

5.02

2

pm

m

5.03 2o

o

N

S

2

2

3


Comparison of SNR for WBPM and WBFM

2

22

3pFMo

o

m

m

B

f

N

S

2

22

2

22 )2()(

ppPMo

o

m

mf

m

m

N

S

2

22

2

22

3

)2(

p

p

FMo

o

PMo

o

m

m

B

f

m

mf

N

S

N

S


2

22

2

22

3

)2(

p

p

FMo

o

PMo

o

m

m

B

f

m

mf

N

S

N

S

2

222

3

4

p

p

m

mB

impact of noise on communications systemsihayee/teaching/ee5765/ece... · impact of noise on...

Documents