ECE 3323 Section 7.2 Noise

EE 3323

Principles of Communication

Systems

Section 7.2

Noise

1

Communication Systems

A typical (simplified) communication system is

illustrated below.

Transmitter Transmission

Medium Receiver x(t)

n(t)

(t) y(t)

(t) + n(t)

The message signal x(t) is applied to a Transmitter

where the signal is perhaps conditioned and used to

Modulate a carrier signal. The modulated signal

(t) is transmitted through a medium (free space, coaxial cable, fiber optic cable, etc.).

2

Communication Systems

In the course of transmission, the modulated signal

is corrupted by the addition of noise. The noise

corrupted, modulated signal is applied to a Receiver

that Demodulates the signal and perhaps conditions

the resulting signal. The output y(t) is related to the

input x(t) in a predictable way. Often the desire is

for y(t) to be a replica of x(t).

A model is need to access the effects of noise at the

input of the receiver and at the output of the

receiver.

3

White Gaussian Noise

The first model for Noise is White, Gaussian Noise.

This model is termed White because the Power

Spectral Density contains all frequencies equally.

This is an analogy to White Light, that contains all

visible wavelengths.

4


This model is termed Gaussian because the

instantaneous value of 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 many

noise sources (due to the central limit theorem).

The mean squared represents the DC power of the

noise, and the variance represents the total average

power in the noise.

5


The Power Spectral density of White Gaussian

Noise is depicted below.

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

S nn ( f )

f

N 0

6


Notice that this Power Spectral Density implies a

noise source of infinite power. The one-half factor

is a convention that will make sense when Band-

limited noise is discussed.

The Autocorrelation of Gaussian White Noise is

Rnn() = F 1

{Snn( f )}

7


Rnn() = N0 2

()

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

R nn ()

N 0

8


Observe that this Autocorrelation implies an

infinitely rapid changing noise signal. The White

Noise signal is un-correlated with itself after the

most minute time shift. Obviously such a noise

model does not reflect any physical process. A

more realistic noise model follows.

9

Band-limited Noise

Consider passing White Gaussian Noise through an

ideal Low-pass Filter of bandwidth B. The Power

Spectral Density of such Noise is shown below.

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

S nn ( f )

f

N 0

B B

10

Band-limited Noise

Snn( f ) = N0 2

rect

f

2B

The average power in this noise signal is

Pn = N0B

The Noise Power is directly proportional to the

bandwidth of the low-pass filter.

The Autocorrelation is

Rnn() = N0 2

2B sinc(2B)

11

Band-limited Noise

Rnn() = N0B sinc

1/2B

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

R nn ()

N 0 B

1/ 2B

1/ 2B

12

Thermal Noise

The thermal noise in a resistor (due to random

motion of the electrons in the resistor) is described

by the following Power Spectral Density.

Snn( f ) = 2 R h| | f

exp

h| | f

kT 1

where:

R = Value of the resistor (Ohms)

h = Planks Constant = 6.625 10 34 (joule sec)

k = Boltzmanns constant = 1.38 10 23 (joules / K) T = Temperature of the resistor in K

13

Thermal Noise

103

106

109

1012

1015

103

106

109

1012

1015

f

S nn ( f )

This is essentially constant for frequencies typically

used in electronic systems.

Snn( f ) = 2 k T R

14

Thermal Noise

A noise model for a resistor is:

Noiseless

R

Snn( f ) = 2 k T R

15

Thermal Noise

Example: Find the RMS voltage due to thermal

noise that may be measured in the following circuit

with R = 1 k, C = 1 F and T = 300 K.

Noiseless

R

Snn( f ) = 2kTR

R C C v(t)

v(t)

16

Thermal Noise

The RC circuit forms a filter with transfer function

H( f ) = 1

1 + j 2RC f

The magnitude squared of the transfer function is

| |H( f ) 2 = 1

1 + (2RC)2 f 2

17

Thermal Noise

| |H( f ) 2 = 1

1 + (2RC)2 f 2

f

|H ( f )| 2

0 101

102

103

101

102

103

B N B N

18

Thermal Noise

The power spectral density at the terminals due to

thermal noise is

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

Syy( f ) =2 k T R 1

1 + (2RC)2 f 2

19

Thermal Noise

And the total noise power appearing at the terminals

is

Py = 2

0

Syy ( f ) df

Py = 2

0

2 k T R 1

1 + (2RC)2 f 2 df

20

Thermal Noise

Using the indefinite integral

dx

a2 + b

2x

2 = 1

ab tan

1

bx

a

Py = 4 k T R 1

2RC tan

1 (2RC f )

0

Py = 4 k T R 1

2RC 2

Py = k T

C

21

Thermal Noise

The RMS voltage appearing at the terminals is

Vrms = k T

C

For the specific values given above

Vrms = 1.38 10 23 (300)

1 10 6

Vrms = 0.06 V

22

Equivalent Noise Bandwidth

Assuming the input to a filter is Gaussian White

Noise with constant noise power N0/2, and the

transfer function of the filter is known, we wish to

define an ideal filter that passes the equivalent noise

power.

f

|H ( f )| 2

0 101

102

103

101

102

103

B N B N

23


Py = 2

0

N02

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

BN

BN =

2

0

N02

| |H( f ) 2 df

N0| |H(0)2

BN =

0

| |H( f )2 df

| |H(0) 2

24


If the filter is a simple low-pass RC filter as shown

above,

| |H( f ) 2 = 1

1 + (2RC)2 f 2

and

BN =

0

1

1 + (2RC)2 f 2 df

25


BN = 1

2RC tan

1 (2RC f )

0

BN = 1

2RC 2

BN = 1

4RC

is the equivalent noise bandwidth of the RC low-

pass filter.

26

Bandpass White Noise

Consider passing White Gaussian Noise through a

Band-pass Filter with bandwidth B. The Power

Spectral Density of the filtered noise is:

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

S nn ( f )

f

N 0

B

f 0 f 0

27


Snn( f ) = N0 2

rect

f

B * [( f f0) + ( f + f0)]

The total average power is

P = N0B

Again, the average power is proportional to the

bandwidth of the filter.

28


The Autocorrelation is

Rnn() = N0 2

B sinc(B ) 2 cos(2 f0 )

29


Rnn() = N0B sinc

1/B

cos(2 f0 )

R nn ()

N 0 B

1/ B

1/ f 0

30

Noise Power of Band-limited White Noise

The Power Spectral Density of Band-limited Noise

is often defined using an Ideal Low-pass filter as

illustrated below.

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

S nn ( f )

f

N 0

B B

31


The average power in this noise signal is

Pn = N0B

Measured in Watts across a one-ohm resistance. In

general, the noise voltage will be measured across a

resistance as follows.

R n(t)

32


For such a Band-limited noise source, the average

power 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 power is

N0 = n

2(t)

BR =

.1

1000(50) = 2 W/Hz

33

Narrowband Noise

If Gaussian White noise is passed through a band-

pass filter where the bandwidth of the filter is small

compared to the center frequency, it is possible to

develop a time-representation of the random noise

signal.

This effect is illustrated below.

34

Narrowband Noise

0 0.02 0.04 0.06 0.08 0.1-4

-2

0

2

4

Time (S)

n(t

)

Gaussian White Noise signal

35

Narrowband Noise

Autocorrelation of Gaussian White Noise Signal

-0.1 -0.05 0 0.05 0.1-0.5

0

0.5

1

1.5

Time (S)

Rxx(t

au)

36

Narrowband Noise

Power Spectral Density of Gaussian White Noise

Signal

-1000 -500 0 500 10000

1

2

3

4

5x 10

-3

Frequency (Hz)

Sxx(f

)

37

Narrowband Noise

This Gaussian White Noise is passed through a

Band-pass Filter as illustrated below.

Band-pass Filter

Center Frequency = f0

Bandwidth = B nw(t) n(t)

For this example f0 = 200 Hz and B = 40 Hz.

38

Narrowband Noise

Narrow Band Noise

0 0.02 0.04 0.06 0.08 0.1-1

-0.5

0

0.5

1

Time (S)

nbn(t

)

39

Narrowband Noise

The narrow-band noise signal appears to be a

sinusoid with a slowly varying amplitude and

phase.

The nominal frequency is the same as the center

frequency of the band-pass filter.

40

Narrowband Noise

-0.1 -0.05 0 0.05 0.1-0.05

0

0.05

Time (S)

Rxx(t

au)

Autocorrelation of Narrowband Noise

41

Narrowband Noise

Power Spectral Density of Narrowband Noise

-1000 -500 0 500 10000

1

2

3

4x 10

-6

Frequency (kHz)

Sxx(f

)

42

Narrowband Noise

This Power Spectral Density is relatively narrow

(looking somewhat like a delta function). Perhaps a

time representation is available.

A phasor representation of narrowband noise is as

follows

nc(t)

ns(t)

an

n

43

Narrowband Noise

n(t) = Re{ }[ ]nc(t) + j ns(t) exp( j 2 f0 t)

n(t) = Re{ }[ ]nc(t) + j ns(t) [ ]cos(2 f0 t) + j sin(2 f0 t)

n(t) = Re

nc(t) cos(2 f0 t) + j nc(t) sin(2 f0 t)

+ j ns(t) cos(2 f0 t) + j j ns(t) sin(2 f0 t)

n(t) = nc(t) cos(2 f0 t) ns(t) sin(2 f0 t)

where nc(t) and ns(t) are random noise processes.

44

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

45

Narrowband Noise

An alternative expression is found by letting

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

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

n(t) = a(t)cos[(t)]cos(2 f0 t) a(t)sin[(t)]sin(2 f0 t)

n(t) = 1

2 a(t) cos[(t) + 2 f0 t] +

1

2 a(t) cos[(t) 2 f0 t]

1

2 a(t) cos[(t) 2 f0 t] + cos[(t) + 2 f0 t]

46

Narrowband Noise

n(t) = a(t) cos[2 f0 t + (t)]

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

a(t) = nc2(t) + ns

2(t)

(t) = tan 1

ns(t)

nc(t)

47

Narrowband Noise

The random amplitude is described by a Rayleigh

PDF

fA(a) = a

2A2 exp

a

2

A2 , a 0.

where A2 is the RMS power in the narrow-band

noise signal.

0

0.8

-1 0 1 2 3 4 5a

f A (a ) A = 1

48

Narrowband Noise

The random phase angle is described by a uniform

distribution

f() = 1

2 , 0 2

0

0.1

0.2

-2 0 2 4 6 8

f ()

49

Signal to Noise Ratio

Recall the simplified communication system shown

below.

Transmitter Transmission

Medium Receiver

x(t)

n(t)

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

Sin , Nin Sout , Nout

The signal at the input of the receiver is corrupted

by noise. We make these assumptions about the

noise.

50


1. The noise is zero-mean, Gaussian distributed,

white noise with power spectral density

Snn( f ) = N02

2. The noise is uncorrelated with the modulated

signal (t).

3. The noise is additive.

51


Under these conditions, the signal power input to the

receiver is

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

+ E{ }n2(t)

Since the noise is zero-mean

E{ }[ ](t) + n(t) 2 = E{ }2 (t) + E{ }n2(t)

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

52


The quality of the signal can be measured by

forming the signal-to-noise ratio

S

N

in =

SinNin

= E{ }2 (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 in

decibels

S

N

in dB = 10 log10

SinNin

53


In like manner, the signal of the received message is

is given by

S

N

out =

SoutNout

= E{ }y2(t)

E{ }n2(t)

S

N

out dB = 10 log10

SoutNout

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ECE 3323 Section 7.2 Noise