ece 3323 section 7.2 noise
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EE 3323
Principles of Communication
Systems
Section 7.2
Noise
1
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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
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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
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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
-
White Gaussian Noise
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
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White Gaussian Noise
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
-
White Gaussian Noise
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
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White Gaussian Noise
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
-
White Gaussian Noise
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
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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
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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
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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
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Thermal Noise
A noise model for a resistor is:
Noiseless
R
Snn( f ) = 2 k T R
15
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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
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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
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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
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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
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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
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Equivalent Noise Bandwidth
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
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Equivalent Noise Bandwidth
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
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Equivalent Noise Bandwidth
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
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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
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Bandpass White Noise
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
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Bandpass White Noise
The Autocorrelation is
Rnn() = N0 2
B sinc(B ) 2 cos(2 f0 )
29
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Bandpass White Noise
Rnn() = N0B sinc
1/B
cos(2 f0 )
R nn ()
N 0 B
1/ B
1/ f 0
30
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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
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Noise Power of Band-limited White Noise
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
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Noise Power of Band-limited White Noise
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
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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
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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
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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
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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
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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
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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
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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
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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
45
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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
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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
-
Signal to Noise Ratio
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
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Signal to Noise Ratio
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
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Signal to Noise Ratio
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
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Signal to Noise Ratio
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
54
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