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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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White Gaussian Noise
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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White Gaussian Noise
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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White Gaussian Noise
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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White Gaussian Noise
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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White Gaussian Noise
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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Equivalent Noise Bandwidth
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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Equivalent Noise Bandwidth
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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Equivalent Noise Bandwidth
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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Noise Power ofBand-limitedWhite Noise
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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Noise Power ofBand-limitedWhite Noise
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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Signal to Noise Ratio
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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Signal to Noise Ratio
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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Signal to Noise Ratio
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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Signal to Noise Ratio
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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