noise in modulation

8/22/2019 Noise in Modulation

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Noise in Modulation


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The primary figure of merit for signals in analog

systems is signal-to-noise rat io. The signal-to-noise ratio is the ratio of the signal power to the

noise power.

While the amount of external ambient noise is not

always within the control of a communicationsengineer, the way in which we can distinguish

between the signal an the noise is.

The demodulation process for any modulationsystem seeks to recreate the original modulating

signal with as little attenuation of the signal and as

much attenuation of the noise as possible.


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The signal-to-noise ratio changes as the modulated

signal goes through the demodulation process. Wewish the signal-to-noise ratio to increaseas a result

of the demodulation process.

The extent to which the signal-to-noise ratio

increases is called the signal-to-noise improvement,

or SNRI. This signal-to-noise improvement is equal

to the ratio of the demodulated signal-to-noise ratioto the input signal-to-noise ratio.


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The calculation of the signal-to-noise improvement

(SNRI) is shown in the following diagram.

Demodulator

si(t)so(t)

ni(t) no(t)

22

22

onoini

osoisi

nPnP

sPsP


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..

no

soo

ni

sii

P

PSNR

P

PSNR

.no

ni

si

so

si

ni

no

so

i

o

P

P

P

P

P

P

P

P

SNR

SNRSNRI

The notation < >means t ime average.


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In determining the ability of a demodulator to

discriminate signal from noise, we calculate Psi, Pso,Pniand Pnoand plug these values (or expressions)

into the expression

.

no

ni

si

so

P

P

P

PSNRI


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Noise in Amplitude

Modulated Systems The SNRI will be calculated for DSB-SC

and DSB-LC.

A DSB-SC demodulator is linear: thesignal and the noise components canbe considered separately.

A DSB-LC demodulator is non-linear:the signal and the noise componentscannot be treated separately.


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X so(t)

cosct

si(t) = xc(t)

.cos)()( ttmtx cc w

The expression for the modulated carrier for DSB-

SC is

Demodulation of this signal is performed by

multiplying xc(t) by cos wct and low-pass filtering theresult.

LPFd(t)


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The result of this demodulation can be seen by

analysis:

].2cos1[)(

cos)(

cos)()(

21

2

ttm

ttm

ttxtd

c

c

cc

w

w

w

Low-pass filtering the result eliminates the cos 2wct

component.


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).()( 21 tmtso

Based upon these relationships, we can find theratio of the signal input power and the signal output

power.

21222 )(cos)( tmttmP csi w


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4122

212

)()( tmtmPso

.2

1

)(

)(

212

412

tm

tm

P

P

si

so

So, it seems, half of our SNRIcalculation is done.


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The other half of the calculation deals with the

noise.

Let us use the quadrature decompositionto

represent the noise:

.sin)(cos)()( ttnttntn cscc ww

It is this signal which will represent ni(t).


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The noise input to the demodulator (in the above

form) is demodulated along with the signal.Because the demodulator is linear, we can treat the

noise separately from the signal.

X no(t)

cosct

ni(t) = nccosctnss inct

LPFdn(t)


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tntnttntn

ttntnttntd

cscc

ccscc

ccscc

cn

wwwww

wwww

2sin]2cos1[cossincos

cossincoscos)()(

21

21

2

After dn(t )passes through the low-pass filter, all we

have left is


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).()( 21

tntn co

Now, we calculate the power in the noise.

.)()()( 241

4122

212 tntntnP ccno


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The ratio of the noise powers becomes

.4)(

)(

2

41

2

tn

tn

P

P

no

ni

Finally, our signal-to-noise improvement becomes


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.2421

no

ni

si

so

PP

PPSNRI

Thus, the DSB-SC demodulation process improves

the signal-to-noise ratio by a factor of two.


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.cos)(1)( ttmtx cc w

The expression for the modulated carrier for DSB-

LC is

(This is a simplified versionof the more accurate

expression which takes into account the modulation

index. In this simplified version, the modulation

index is equal to one.)

The demodulation process extracts m(t)from xc(t).


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The input signal power is

212

2

12

212

22

)(1

)()(21

)(1

cos)(1

tm

tmtm

tm

ttmP csi

w


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.)(1 212

tm

The modulating signal m(t)is assumed here to have

zero mean.

If m(t) = coswmt, then

.1 43

21

21 siP


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The output power is simply

)(2 tmPso

which is equal to if m(t) = coswmt.

.32

43

21 si

so

PP

Thus, when m(t)is a sinewave, our signal power

ratio is


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Now, we deal with the noise.

As mentioned previously, since the demodulation

process is non-linear, we cannot treat the signal and

the noise separately. To deal with the noise, we

retain the carrier, but set the modulat ing s ignal tozero. Thus, the signal that we demodulate is

).(cos1 tntc w


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The resultant output from demodulating this signal

will be the output noise power.

To demodulate this signal (analytically), we use the

quadrature decomposition for n(t). The input to our

demodulator becomes

.sin)(cos)(cos ttnttnt csccc www


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We then work with this expression:

.sin)(cos)(1

sin)(cos)(cos

ttnttn

ttnttnt

cscc

csccc

ww

www

At this point we make an assumpt ion(which turns

out to be quite reasonable in many cases):

.)(1)( tntn cs


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In words, the noise (either the in-phase or

quadrature component) is much smaller inmagnitude that that of the signal coswct.

With this assumption the input to the demodulator

becomes

.cos)(1 ttn cc w


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We recall that when the input

is applied, the output is

ttm cwcos)(1

).(tm


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Similarly, when the input is

is applied, the output is

ttn cc wcos)(1

).(tnc


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Thus, the noise output is

).()( tntn co

The output noise power is

.)()( 22 nicno PtntnP


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.132

32

no

ni

si

so

PP

PPSNRI

,1no

ni

P

P

Thus,

and


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We see that the signal-to-noise improvement for

DSB-LC is not as good as that of DSB-SC. The

reason for using DSB-LC is that it is relatively easy

to demodulate. (Standard broadcast AM [530-1640

kHz], uses DSB-LC.)


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Noise in Frequency

Modulated Systems The frequency modulation and demodulation process

is non-l inear.

As with DSB-LC, we cannot treat the signal and the

noise separately.


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].)(cos[)( tmkttx fcc w

The expression for the modulated carrier for FM

As with DSB-LC AM, he demodulation process

extracts m(t)from xc(t). This extraction process

consists of three steps:


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1. Extract argument from cos().

2. Subtract wct.

3. Differentiate what is left to get kfm(t).

Based upon these three steps we can quickly get

the input signal power and the output signal power.

).somethingcos()()( txts ci

.)something(cos212 siP


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).()( tmkts fo

.)()( 2222 tmktmkP ffso

Without much loss of generality, we can assume that

m(t) = cosmt.

.2

21

fso kP


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.

2

21

2

21

f

f

si

so

k

k

P

P

We now have the signal power ratio:


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As with the DSB-LC AM, the effective noise input

comes along with an unmodulated carrier:

).(])(0cos[ tntmtc w

If use the quadrature decomposition of the noise, we

have, as the effective noise input

,sin)(cos)(cos ttnttnt csccc www


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.sin)(cos)(1 ttnttn cscc ww

or,

This expression can be combined into a single

sinewave

),cos( w tR c


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where

.)(1

)(tan

).()](1[

1

222

tn

tn

tntnR

c

s

sc

At this point we make a simliar assumption to what

we made with DSB-LC AM:

.1)(1)( tntn cs


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(The new part is the 1.)

Using this assumption we have

).()(tan 1 tntn ss

(This last is true because tan-1x xfor small values

ofx.)


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))(cos( tntR sc w

With the assumptions given, our effective noise input

becomes

We may now apply our three demodulation steps:

1. Extract argument from cos().2. Subtract wct.

3. Differentiate what is left to get kfm(t).


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1.

2.

3.

)(tntsc

w

)(tns

)(tns


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Thus, the output noise is

).()( tntn so

All that remains to be done is to find the noise power

from the noise signals [ni(t) and no(t)].

The noise power will be found from the powerspectral densities of the input noise and the output

noise.


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The input noise is additive white Gaussian noise.

The power spectral density of the input noise is

.2)(

0N

fSni

The output noise is the derivative of the quadraturecomponent of the (input) noise [ns(t)].


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We know, from a previous exercise, that

.)( 0NfSns

We need to find the power spectral density of the

derivat ive of the quadrature component. To find this

we multiply Sns(f) by the square of the transfer

function for the differentiation operation.


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The effect of the differentiation operation is shown in

three ways:

d

dt

j f

ns(t) ns(t).

Ns(f) No(f)

| f|2Sns(f) Sno(f)


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Thus, the power spectral density of the output noise

is

.2)( 02

0 NffSn p

Now that we have the power spectral densities of

the input and the output spectra, all that remains to

find thepoweris to integrate the respective powerspectral densitiesover the appropriate frequencies.


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The input noise is a bandpass process. Let BTbe

the bandwidth.

f

Sni(f)

N0/2

BT


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Integrating the power spectral density, we have

.)(2

20

0

TTni BNB

NP


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The input noise is a lowpass process. Let Wbe the

bandwidth.

f

Sno(f)=N04p2f2

W-W


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Integrating the power spectral density, we have

W

W

W

W

no

f

N

dffNP

34

4

32

0

22

0

p

p


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.3

8

342

3

0

2

3

20

WN

WN

p

p


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.8

332

3

80 320 W

BBN

P

PT

WNT

no

ni

pp

Thus, the ratio of the noise powers is


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Finally, the signal-to-noise improvement is

.

8

332

2

W

Bk

P

P

P

PSNRI Tf

no

ni

si

so

p

kf peak frequency deviation

BT modulated carrier bandwidth

W demodulated bandwidth


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Exercise: Suppose that the modulated carrier

bandwidth is given by Carsons rule:

).1(2 mT fB

Further suppose that the demodulated bandwidth isthe bandwidth of the modulating signal:

mfW

Show that the resultant signal-to-noise improvement

is

).1(3 2 SNRI


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Use

.2 m

f

fkp


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Exercise: Suppose that the modulated carrier

bandwidth is simply twice the modulating frequency:

.2 mT fB

Further suppose that the demodulated bandwidth isthe bandwidth of the modulating signal:

mfW

Show that the resultant signal-to-noise improvement

is

.3 2SNRI


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Improvement of FM SNRI

Using De-Emphasis The signal-to-noise improvement for FM, while good,

can be improved by minimizing the noise.

The weakest link where it comes to the noiseamplification is the differentiator: the noise increasesas the cube of the bandwidth.

If the high-pass effect of the differentiator can beoffset by a low-pass filter, the noise will beattenuated, and the SNRI will be improved.


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The weakest link:

f

Sno(f)=N04p2f2

W-W

.3

84

30

222

0

W

W

no

WNdffNP

pp


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Now, let us insert a low-pass filter whose transfer

function is

.1

1

)(1f

jfLP fH

Our output noise power becomes


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.14

2

22

0'

1

W

W f

fno df

fN

P

p

We can integrate this function by letting

.tan1 ff

1tan W


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1

1

1

1

1

1

1

1

1

1

1

tan

tan

22

0

3

1

tan

tan

22

2

20

31

tan

tan

2

2

22

03

1

'

]1[sec4

secsectan4

sec

tan1

tan4

fW

f

W

fW

fW

f

fW

dNf

dNf

dN

fPno

p

p

p


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].tan)[2(411

12

0

3

1 f

W

f

WNf p

We now define a quantity called the signal-to-noise

improvement improvement (no mistake). This is theimprovement in the SNRI as a result of de-

emphasis:

.'

no

no

PP


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The factor becomes

.]tan[3

]tan)[2(4

38

11

11

13

1

3

12

0

3

1

3

0

2

'

fW

fW

fW

fW

no

no

f

W

Nf

WN

P

P

p

p

A plot of this factor is plotted on the following slide.


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10-1

100

101

0

2

4

6

8

10

12

14

16

Increase in SNRI Using De-Emphasis

W / f

(d

B)

noise in modulation

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