ee 3323 section 8.2 noise

8/7/2019 EE 3323 Section 8.2 Noise

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EE 3323

Principles of CommunicationSystems

Section 8.2Noise

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Communication Systems

A typical (simplified) communication system isillustrated below.

Transmitter TransmissionMedium Receiverx(t)

n(t)

J(t)y(t)

J(t) + n(t)

The message signalx(t) is applied to a Transmitter

here the signal is perhaps conditioned and used toodulate a carrier signal. The modulated signal

J(t) is transmitted through a medium ( ree space,coaxial cable, iber optic cable, etc.).

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Communication Systems

In the course o transmission, the modulated signalis corrupted by the addition o noise. The noise

corrupted, modulated signal is applied to a Receiver

that Demodulates the signal and perhaps conditions

the resulting signal. The outputy(t) is related to theinputx(t) in a predictable ay. O ten the desire is

ory(t) to be a replica o x(t).

model is need to access the e ects o noise at theinput o the receiver and at the output o the

receiver.

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White Gaussian Noise

The irst model or Noise is White, Gaussian Noise.

This model is termed White because the Po er

Spectral Density contains all requencies equally.

This is an analogy to White Light, that contains allvisible avelengths.

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This model is termed Gaussian because theinstantaneous value o the noise signal is a Gaussian

distributed random variable completely described by

the mean and variance. This is a convenient

distribution to use. It adequately represents manynoise sources (due to the central limit theorem).

The mean squared represents the DC po er o the

noise, and the variance represents the total average

po er in the noise.

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The Po er Spectral density o White GaussianNoise is depicted belo .

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Snn ( f )

f

N0

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Notice that this Po er Spectral Density implies anoise source o in inite po er. The one-hal actor

is a convention that ill make sense hen Band-

limited noise is discussed.

The utocorrelation o Gaussian White Noise is

Rnn

(X) = F1 {Snn

( f)}

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Rnn(X) N02 H(X)

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

R nn (X)

X

N0

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Observe that this Autocorrelation implies aninfinitely rapid changing noise signal. The White

oise signal is un-correlated with itself after the

most minute time shift. Obviously such a noise

model does not reflect any physical process. Amore realistic noise model follows.

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Band-limited Noise

Consider passing White aussian oise through anideal Low-pass Filter of bandwidth B. The ower

pectral ensity of such oise is shown below.

-8 -6 -4 -2 0 2 4 6 8

Snn ( f )

f

N0

BB

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Band-limited Noise

Snn( f) = N0

2rect

f2B

The average po er in this noise signal is

Pn = N0B

The Noise Po er is directly proportional to the

band idth o the lo -pass ilter.

The utocorrelation is

Rnn(X) =N0

22B sinc(2BX)

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Band-limited Noise

Rnn(X) N0B sinc

X1/2B

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

R nn (X)

X

N0B

1/2B

1/2B

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Thermal Noise

The thermal noise in a resistor (due to randommotion of the electrons in the resistor) is described

by the following ower pectral ensity.

Snn( f)2 Rh

| |f

exp

h| |f

kT 1

where:

Value of the resistor (Ohms)h lanks Constant 6.625 v 10

34(joule sec)

k oltzmanns constant 1.38 v 1023

(joules / K)

T Temperature of the resistor in K13


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Thermal Noise

103

106

109

1012

1015

103

106

109

1012

1015

f

Snn (f )

This is essentially constant for frequencies typically

used in electronic systems.

Snn( f) 2 k TR

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Thermal Noise

A noise model for a resistor is:

Noiseless

R

nn( f) 2 k TR

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Thermal Noise

Example: Find the RM voltage due to thermalnoise that may be measured in the following circuit

with R 1 k;, C 1 QF and T 300 K.

Noiseless

R

nn( f) 2kTR

R C Cv(t)

v(t)

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Thermal Noise

The RC circuit forms a filter with transfer function

H( f)1

1 +j 2TRC f

The magnitude squared of the transfer function is

| |H( f)2

1

1 + (2TRC)2 f2

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Thermal Noise

H( f) 2 = 11 (2TR )2 f2

f

H( f ) 2

0 101

102

103

101

102

103

B NB N

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Thermal Noise

The po er spectral density at the terminals due tothermal noise is

Syy( f) = Snn( f) H( f)2

Syy( f) =2 k R1

1 (2TR )2 f2

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Thermal Noise

And the total noise po er appearing at the terminalsis

Py = 20

g

Syy ( f) df

Py = 20

g

2 k R 11 (2TR )2 f2df

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Thermal Noise

Using the inde inite integral

g

g

dx

a2 b

2x

2 =1

abtan

1

bx

a

Py = 4 k R1

2TRtan

1(2TR f)

g

0

Py = 4 k

R

1

2TR

T

2

Py =k

C

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Thermal Noise

The S voltage appearing at the terminals is

Vrms =k

C

or the speci ic values given above

Vrms =1.38

v10

23(300)

1 v 10 6

Vrms = 0.06 QV22


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Equivalent Noise Bandwidth

Assuming the input to a ilter is Gaussian White Noise ith constant noise po erN0/2, and the

trans er unction o the ilter is kno n, e ish to

de ine an ideal ilter that passes the equivalent noise

po er.

f

H( f )2

0 101

102

103

101

102

103

B NB N

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Py = 2

0

g

N0

2 H( f) 2df= N0 H(0)

2BN

BN =

20

g

N0

2 H( f)2

df

N0 H(0)2

BN =

0

g

H( f) 2df

H(0)2

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If the filter is a simple lo -pass C filter as sho nabove,

H

(f

)

2

=

1

1 (2TRC)2 f2

and

BN =

0

g

1

1 (2TRC)2 f2df

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BN =1

2TRCtan

1(2TRC f)

g

0

BN =

1

2TRC

T

2

BN =1

4RC

is the equivalent noise band idth of the C lo -

pass filter.

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BandpassWhite Noise

Consider passing White Gaussian Noise through aBand-pass ilter ith band idth B. The Po er

Spectral Density of the filtered noise is:

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Snn ( f )

f

N0

B

f0f0

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BandpassWhite Noise

Snn( f) = N02rect

fB

* [H( ff0) H( f+f0)]

The total average po er is

P= N0B

Again, the average po er is proportional to the

band idth of the filter.

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BandpassWhite Noise

The Autocorrelation is

Rnn() =N0

2B sinc(BX) 2 cos(2Tf0X)

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BandpassWhite Noise

Rnn() = N0B sinc X1/B

cos(2Tf0X)

R nn (X)

X

N0B

1/B

1

/f0

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Noise Power ofBand-limitedWhite Noise

The Po er Spectral Density of Band-limited Noiseis often defined using an Ideal o -pass filter as

illustrated belo .

-8 -6 -4 -2 0 2 4 6 8

Snn ( f )

f

N0

BB

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The average po er in this noise signal is

Pn = N0B

easured in Watts across a one-ohm resistance. Ingeneral, the noise voltage ill be measured across a

resistance as follo s.

n(t)

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For such a Band-limited noise source, the averagepo er dissipated in the resistance is

n2(t) = N0BR

So if 100 mW of Noise, Band-limited to 1000Hz is

dissipated across a 50; resistance, the Noise po eris

N0 =n2(t)

BR=

.1

1000(50)= 2 QW/Hz

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Narrowband Noise

If Gaussian White noise is passed through a band-pass filter here the bandwidth of the filter is small

compared to the centerfrequency, it is possible to

develop a time-representation of the random noise

signal.

This effect is illustrated below.

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Narrowband Noise

0 0.02 0.04 0.06 0.08 0.1-4

-2

0

2

4

Time ( )

n(t)

Gaussian White Noise signal

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Narrowband Noise

Autocorrelation of aussian White Noise ignal

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5

1

1.5

Time ( )

Rx

x(tau)

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Narrowband Noise

Power pectral ensity of aussian White Noise

ignal

-1000 -500 0 500 10000

1

2

3

4

5x 10

-3

Frequency ( z)

xx(f)

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Narrowband Noise

This aussian White Noise is passed through aand-pass Filter as illustrated below.

and-pass Filter

Center Frequency =f0Bandwidth = B

nw(t) n(t)

For this examplef0 = 200 z andB = 40 z.

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Narrowband Noise

Narrow and Noise

0 0.02 0.04 0.06 0.08 0.1-1

-0.5

0

0.5

1

Time ( )

n

bn(t)

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Narrowband Noise

The narrow-band noise signal appears to be asinusoid with a slowly varying amplitude and

phase.

The nominal frequency is the same as the centerfrequency of the band-pass filter.

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Narrowband Noise

-0.1 -0.05 0 0.05 0.1-0.05

0

0.05

Time ( )

Rxx(tau)

Autocorrelation ofNarrowband Noise

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Narrowband Noise

Power pectral ensity ofNarrowband Noise

-1000 -500 0 500 10000

1

2

3

4x 10-6

Frequency (k z)

xx(f)

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Narrowband Noise

This Pow

er Spectral Density is relatively narrow

(looking somewhat like a delta function). Perhaps a

time representation is available.

A phasor representation of narrowband noise is asfollows

nc(t)

ns(t)

an

Un

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Narrowband Noise

n(t) = [ ]nc(t)+jns(t) (j Tf0t)

n(t) = [ ]nc(t)+jns(t) [ ](2Tf0t)+j (2Tf0t)

n(t) =

nc(t) (2Tf0t)+jnc(t) (2Tf0t)

+jns(t) (2Tf0t)+jjns(t) (2Tf0t)

n(t) = nc(t) (2Tf0t) ns(t) (2Tf0t)

wherenc(t) ns(t) reran omnoise rocesses.

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Narrowband Noise

Both nc(t) and ns(t) are low-pass (relatively low-frequency) random signals.

nc(t) in-phase component

ns(t) quadrature component

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Narrowband Noise

An alternative expression is

foun

dby letting

nc(t) = a(t)cos[J(t)]

ns(t) = a(t)sin[J(t)]

n(t) = a(t)cos[J(t)]cos(2Tf0t) a(t)sin[J(t)]sin(2Tf0t)

n(t) =

1

2a(t) cos[J(t) + 2Tf0t] +1

2a(t) cos[J(t) 2Tf0t]

1

2a(t) cos[J(t) 2Tf0t] + cos[J(t) + 2Tf0t]

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Narrowband Noise

n(t) a(t) cos[2Tf0t+ J(t)]

where a(t) is a randomly varying amplitude and J(t)is a randomly varying phase angle.

a(t) nc2(t) + ns

2(t)

J(t) tan 1

ns(t)

nc(t)

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Narrowband Noise

The random amplitude is described by a RayleighPDF

fA(a)a

2TWA2 exp

a2

WA2 , au 0.

where WA2 is the RM power in the narrow-bandnoise signal.

0

0.8

-1 0 1 2 3 4 5a

fA (a ) WA 1

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Narrowband Noise

The random phase angle is described by a uniformdistribution

f*(J)1

2T, 0 eJ 2T

0

0.1

0.2

-2 0 2 4 6 8J

f*

(J)

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Signal to Noise Ratio

Recall the simplified communication system shownbelow.

TransmitterTransmission

Medium Receiverx(t)

n(t)

J(t) y(t)J(t) + n(t)

Sin ,Nin Sout ,Nout

The signal at the input of the receiver is corruptedby noise. We make these assumptions about the

noise.

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1. The noise is zero-mean, aussian distributed,white noise with power spectral density

Snn( f)N0

2

2. The noise is uncorrelated with the modulated

signal J(t).

3. The noise is additive.

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Under these conditions, the signal power input to thereceiver is

E{ }[ ]J(t) + n(t) 2 E{ }J2 (t) + E{ }2J(t)n(t)

+ E{ }n2(t)

ince the noise is zero-mean

E{ }[ ]J(t) + n(t)2

E{ }J2

(t) + E{ }n2

(t)

E{ }[ ]J(t) + n(t) 2 Sin +Nin

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The quality of

the signal can be measured

byforming the signal-to-noise ratio

S

N

in

=Sin

Nin

=E{ }J2 (t)

E{ }n2

(t)

The larger the signal-to-noise ratio, the better the

received signal quality

The signal-to-noise ratio is often measured indecibels

S

N

in dB

= 10 log10

Sin

Nin

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In like manner, the signal o

fthe receive

dmessage isis given by

S

N out =

Sout

Nout =

E{ }y2(t)

E{ }n2(t)

S

N

out dB

= 10 log10

Sout

Nout

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ee 3323 section 8.2 noise

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