ELEC 303 – Random Signals
Lecture 20 – Random processes
Dr. Farinaz Koushanfar
ECE Dept., Rice University
Nov 11, 2010
Lecture outline
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Basic concepts
Random processes and linear systems
Power spectral density of stationary processes
Power spectra in LTI systems
Power spectral density of a sum process
Gaussian processes
RP and linear systems
• When a RP passes a linear time-invariant system the
output is also a RP
• Assuming a stationary process X(t) is input, the linear
time-invariant system with the impulse response h(t),
output process Y(t)
• Under what condition the output process would be
stationary?
• Under what conditions will the input/output jointly
stationary?
• Find the output mean, autocorrelation, and
crosscorrelation
X(t)
h(t)
Y(t)
Linear time invariant systems
• If a stationary RP with mean mX and autocorrelation
function RX()
• Linear time invariant (LTI) system with response h(t)
• Then, the input and output process X(t) and Y(t) will be
jointly stationary with
X(t)
h(t)
Y(t)
The response mean
• Using the convolution integral to relate the
output Y(t) to the input X(t), Y(t)=X()h(t-)d
This proves that mY is independent of t
X(t)
h(t)
Y(t)
Cross correlation
• The cross correlation function between output
and the input is
This shows that RXY(t1,t2) depends only on =t1-t2
Output autocorrelation
• The autocorrelation function of the output is
This shows that RY and RXY depend only on =t1-t2,
 Output process is stationary, and input/output are jointly stationary
Power spectral density of a stationary
process
• If the signals in the RP are slowly varying, then
the RP would mainly contain the low
frequencies in its power concentration
• If the signal changes very fast, most of the
power will be concentrated at high frequency
• The power spectral density of a RP X(t) is
denoted by SX(f) showing the strength of the
power in RP as a function of frequency
• The unit for SX(f) is Watts/Hz
Wiener-Khinchin theorem
• For a stationary RP X(t), the power spectral
density is the Fourier transform of the
autocorrelation function, i.e.,
Example 2
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
• From the previous lecture, we know
Example 3
• X(t)=X
• Random variable X~U[-1,1]
• In this case
• Thus,
• For each realization of the RP, we have a different
power spectrum
Power spectral density
• The power content of a RP is the sum of the powers at
all frequencies in that RP
• To find the total power, need to integrate the power
spectral density across all frequencies
• Since SX(f) is the Fourier transform of RX(), then RX()
will be the inverse Fourier transform of SX(f), Thus
• Substituting =0, we get
Example 4
• Find the power in the process of example 2
Translation to frequency domain
• For the LTI system and stationary input, find the translation
of the relationships between the input/output in frequency
domain
• Compute the Fourier transform of both sides to obtain
• Which says the mean of a RP is its DC value. Also, phase is
irrelevant for power. Only the magnitude affects the power
spectrum, i.e., power dependent on amplitude, not phase
Example 5
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If a RP passes through a differentiator
H(f)=j2f
Then, mY=mX H(0) = 0
Also, SY(f) = 42 f2 SX(f)
Cross correlation in frequency domain
• Let us define the cross spectral density SXY(f)
• Since RYX() = RXY(-), we have
• Although SX(f) and SY(f) are real nonnegative
functions, SXY(f) and SYX(f) can generally be
complex functions
Example 6
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
• The X(t) goes thru a differentiator H(f)=j2f
Example 7
• X(t)=X
• Random variable X~U[-1,1]
• If this goes through differentiation, then
SY(f) = 42 f2 ((f)/3) = 0
SXY(f) = -j2f ((f)/3) = 0
Power spectral density of a sum
process
• Z(t) = X(t)+Y(t)
• X(t) and Y(t) are jointly stationary RPs
• Z(t) is a stationary process with
RZ() = RX() + RY() + RXY() + RYX()
• Taking the Fourier transform from both sides:
SZ(f) = SX(F) + SY(f) + 2 Re[SXY(f)]
• The power spectral density of the sum process is the sum
of the power spectral of the individual processes plus a
term, that depends on the cross correlation
• If X(t) and Y(t) are uncorrelated, then RXY()=mXmY
• If at least one of the processes is zero mean, RXY()=0, and
we get: SZ(f) = SX(F) + SY(f)
Example 8
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X(t)=X
Random variable X~U[-1,1]
Z(t) = X(t) + d/dt X(t), then
SXY(f) = jA2f0 /2 [(f+f0) - (f-f0)]
Thus,
Re[SXY(f)] = 0
SZ(f)= SX(f)+SY(f) = A2(1/4+2f02)[(f+f0)+(f-f0)]
Gaussian processes
• Widely used in communication
• Because thermal noise in electronics is produced
by the random movement of electrons closely
modeled by a Gaussian RP
• In a Gaussian RP, if we look at different instances
of time, the resulting RVs will be jointly Gaussian:
Definition 1: A random process X(t) is a Gaussian process if
for all n and all (t1,t2,…,tn), the RVs {X(ti)}, i=1,…,n have a
jointly Gaussian density function.
Gaussian processes (Cont’d)
Definition 2: The random processes X(t) and Y(t) are jointly
Gaussian if for all n and all (t1,t2,…,tn), and (1,2,…,m)the
random vector {X(ti)}, i=1,…,n, {Y(j}, j=1,…,m have an n+m
dimensional jointly Gaussian density function.
• It is obvious that if X(t) and Y(t) are jointly
Gaussian, then each of them is individually
Gaussian
• The reverse is not always true
• The Gaussian processes have important and
unique properties
Important properties of Gaussian
processes
• Property 1: If the Gaussian process X(t) is
passed through an LTI system, then the output
process Y(t) will also be a Gaussian process.
Y(t) and X(t) will be jointly Gaussian processes
• Property 2: For jointly Gaussian processes,
uncorrelatedness and independence are
equivalent