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Unformatted text preview: igan) Fall 2012 September 7, 2012 89 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Mean
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0 EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 91 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density The power spectral density function of a random signal is the
amount of power in the signal as a function of frequency.
The autocorrelation measures the correlation between the noise
at different points in time.
For noise like signals the autocorrelation does not depend on the
time but just the time difference between two samples.
In this case (and assuming zero mean) the process is called
widesense stationary. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 92 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Sample Path and Frequency Content of Noise
4 x 10
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10 8 EECS 455 (Univ. of Michigan) 6 4 2 0
frequency Fall 2012 2 4 6 8 10 September 7, 2012 93 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Random Process Deﬁnitions
The autocorrelation of a random process X (t ) is
RX (τ ) = E [X (t )X (t + τ )] The power of a (widesense stationary) random process X (t ) is
PX = E [X 2 (t )] = RX (0).
To see the meaning of this deﬁnition we start with the time
average power of a particular realization of a random process over
a ﬁnite interval. Then take the (probablistic) average to get the
average power.
EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 94 / 174 Lecture Notes 2 PX =E
= 1
T →∞ 2T EECS 455 (Univ. of Michigan) T lim 1
T →∞ 2T T lim 1
T →∞ 2T
= RX (0) = Random Variables, Random Processes, Noise −T
T −T X (t )2 dt E [X (t )2 ]dt
RX (0)dt lim −T Fall 2012 September 7, 2012 95 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density
Deﬁne the power spectral density as the Fourier transform of the
autocorrelation function.
SX (f ) =
RX (τ ) = ∞ RX (τ )e−j 2πf τ d τ ∞
∞ SX (f )ej 2πf τ df ∞ Then the average power of a random process is
∞ PX = RX (0) = SX (f )df . −∞ EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 96 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Power Spectral Density Thus the power spectral density is a measure of how the power is
distributed over frequency.
Another way to see that the Fourier transform of autocorrelation is
power spectral density is to put a random process through a linear
timeinvariant ﬁlter which has a very narrow bandpass transfer
function.
Then the power at the output of the ﬁlter is the power spectral
density of the input evaluated at the frequency that the ﬁlter
passes. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 97 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Example: White Noise RN (τ ) = N0
δ(τ ),
2 RN (τ ) SN (f ) = N0
2
SN (f ) τ EECS 455 (Univ. of Michigan) f Fall 2012 September 7, 2012 98 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise White Gaussian Noise White Gaussian Noise (WGN) sometimes also called thermal
noise or Johnson noise has a ﬂat power spectral density (same
noise power at all frequencies).
The power spectral density is N0 /2 where N0 = kTo ,
k = 1.38 × 10−23 Joules/◦ K (Boltzman’s constant) and To is the
temperature in Kelvin.
For room temperatures To = 290K which makes N0 = 4 × 10−21
Watts/Hz.
In dBs this is N0 = −174 dBm/Hz or N0 = −204 dBW/Hz.
The distribution of the noise (at any time) is Gaussian. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 99 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise White Gaussian Noise
When using power spectral density the noise has equal power at
positive and negative frequencies. “Onesided” power spectral
density is the sum of the power at the positive and negative
frequencies. The onesided power spectral density of white noise
is N0 .
Notice that the power (not the power spectral density) of white
noise is inﬁnite. Since there is the same power at all frequencies
the total power must be inﬁnite. White noise is only a model for
what happens in practice. When operating at frequencies below
the optical band this is a very accurate model.
Since we always ﬁlter the signal and noise, the noise at the output
of the ﬁlter becomes ﬁnite. EECS 455 (Univ. of Michigan) Fall 2012 September 7, 2012 100 / 174 Lecture Notes 2 Random Variables, Random Processes, Noise Noise Power Example What is the noise power in the frequency band from 2400 MHz to 2483
MHz at room tem...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.
 Fall '08
 Stark
 Frequency

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