# ELEC 303 – Random Signals - Rice University_4_

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```					ELEC 303 – Random Signals

Lecture 21 – Random processes
Dr. Farinaz Koushanfar
ECE Dept., Rice University
Nov 19, 2009
Lecture outline
•   Basic concepts
•   Gaussian processes
•   White processes
•   Filtered noise processes
•   Noise equivalent bandwidth
Things to remember
• Stationary
– A random process is stationary if time shift does
not affect its properties
– For all T, and for all sets of sample times, (t0,…,tn),
P(X(t0)x0,…,X(tn)xn) = P(X(T+t0)x0,…,X(T+tn)xn)
– Stationary random processes have constant mean,
defined as E[X(t)] = mX
– For stationary RPs, autocorrelation depends on
the time difference between the samples
– RX(t1,t2)=E[X(t1)X(t2)] = RX(=t1-t2)
Exact definition WSS
• A process is wide sense stationary if its expected
power is finite |E[X2(t)|<, its mean is constant,
and its autocorrelation depends only on the time
difference between samples
• WSS processes: stationary in 1st and 2nd moment
• Stationary processes are WSS, but not vice versa
• Power spectral density (PSD)
– Defined only for WSS processes
– The Fourier transform of the autocorrelation function
– Expected power is the integral of the PSD
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
White processes
Definition 3: A random process X(t) is a called a white
process if it has a flat spectral density, i.e., SX(f) is constant
for all f

• White processes are those where all frequency
components appear with equal power
• Thermal noise can be modeled as a white noise
over a wide range of frequencies
• A wide range of information sources can be
modeled as the output of LTI systems driven by a
white process
Power of a white process
• SX(f)=C, (C is a constant), then

• Obviously, no real physical process can have an
infinite power
• Thus, the white process is not a meaningful
physical process.
• Quantum mechanical analysis of natural noise
shows it has a power spectral density given by
White processes
• Quantum mechanical analysis of natural noise
shows it has a power spectral density given by
White noise

• Thermal noise, though not precisely white, can be
modeled as a white process for all practical purposes
• PSD is Sn(f) = kT/2 (denoted by N0) = N0/2
• Autocorrelation Rn() = -1[N0/2]=N0/2 (t)
• For all 0, we have RX()=0
• Thus, two samples of noise at t1 and t2 will be
uncorrelated
• If the RP is white and Gaussian, any pair of RVs X(t1)
and X(t2) are independent for t1t2
Example 1
• A stationary RP passes through a quadrature filter defined
by h(t)=1/t
• What are the mean and autocorrelation functions of the
output?
• What is the cross correlation between input and output?

Using the fact that     and that RX() has no DC component.
Properties of thermal noise
•   Thermal noise is a stationary process
•   Thermal noise has a zero mean process
•   Thermal noise is a Gaussian process
•   Thermal noise is a white process with a power
spectral density Sn(f) = kT/2
– Thermal noise increases with increasing ambient
temperature, cooling circuits lowers the noise
Filtered noise process
• In many cases, the noise in one stage of the process gets
filtered by a bandpass filter
• Frequency of bandpass is fc, away from zero
• The bandpass filters can be expressed in terms of the
inphase and quadrature components:
– E.g., single frequency signal is an extreme case:
– x(t) = A Cos(2fct + ) = A Cos()Cos(2fct)–A Sin() Sin(2fct)
= xc Cos(2fct) - xs Sin(2fct)       {Phasor: Aej = xc + j xs}
– More generally: x(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)
– In phase component:         xc(t) = A(t) Cos ((t))
– Quadrature component: xs(t) = A(t) Sin ((t))
Bandpass Filter
• X(t) is the output of an ideal bandpass filter of
bandwidth W centered at fc
• Examples:
Filtered noise
• Filtered thermal noise is Gaussian but not
white                                               2
For ideal filter:|H(f)| =H(f)

• Power spectral density:
• For the examples on the last slides,
Filtered noise components
• All filtered noise signals have in-phase and
quadrature components that are lowpass, i.e.,
X(t) = Xc(t) Cos(2fct) - Xs(t) Sin(2fct)
• In-phase and quadrature components:
– Xc(t) and Xs(t) are zero-mean, low pass, jointly
stationary, and jointly Gaussian random processes
– If the power in process X(t) is PX, then the power
in each of the processes Xc(t) and Xs(t) is also PX
Properties of Xc and Xs
• Both have a common amplitude
– Shifting the positive frequencies to the left by fc
– Shifting the negative frequencies to the right by fc
– If H1(f) and H2(f) are used, then
– P1=4WN0/2=2N0W,              P2=2WN0/2=N0W
Noise equivalent bandwidth
• A white Gaussian noise passing through a
filter would be Gaussian but not white
• We have        SY(f) = SX(f)|H(f)|2=.5 N0|H(f)|2
• We have to integrate SY(f) to get the power

• Define Bneq, the noise equivalent bandwidth
Hmax is the maximum of |H(f)| in the
Filter’s passband
Noise equivalent bandwidth
Hmax is the maximum of |H(f)| in the
Filter’s passband

• Thus, given Bneq, finding the output noise becomes a
• The of filters and amplifiers are usually given by the
manufacturers
Example
• Find the noise equivalent bandwidth of a low
pass filter
=RC
Summary: Gaussian processes
• X(t) is a Gaussian process if Yg=0T g(t) X(t) dt is
Gaussian for any T and function g
• Linear filtering of a Gaussian process results in a
Gaussian process
• Samples of a Gaussian process are jointly
Gaussian random variables
• Uncorrelated samples of a Gaussian process are
independent
Summary: white noise
• White noise is defined as a WSS random
processes with a flat PSD: Sn(f) = N0/2
• The autocorrelation of white noise is N0/2 (t)
• White noise is the most random form of noise
since it decorrelates randomly!

http://www.stanford.edu/class/ee179/multi/lecture16-multi.pdf
Summary: filtered noise
• Filtered thermal noise is Gaussian but not white
• The bandpass filters can be expressed in terms of
the inphase and quadrature components
x(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)
– In phase component: xc(t) = A(t) Cos ((t))
– Quadrature component: xs(t) = A(t) Sin ((t))
• Define Bneq, the noise equivalent bandwidth

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