ELEC 303 – Random Signals

Lecture 21 – Random processesDr. Farinaz Koushanfar

ECE Dept., Rice UniversityNov 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)

• 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

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.

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

• 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

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

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 t1t2

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(2fct + ) = A Cos()Cos(2fct)–A Sin() Sin(2fct)

= xc Cos(2fct) - xs Sin(2fct) {Phasor: Aej = xc + j xs}

– More generally: x(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)

– 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

• Power spectral density:• For the examples on the last slides,

For ideal filter:|H(f)|2=H(f)

Filtered noise components

• All filtered noise signals have in-phase and quadrature components that are lowpass, i.e.,

X(t) = Xc(t) Cos(2fct) - Xs(t) Sin(2fct)• 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

• Thus, given Bneq, finding the output noise becomes a simple task• The of filters and amplifiers are usually given by the

manufacturers

Hmax is the maximum of |H(f)| in the Filter’s passband

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 componentsx(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)

– In phase component: xc(t) = A(t) Cos ((t))

– Quadrature component: xs(t) = A(t) Sin ((t))

• Define Bneq, the noise equivalent bandwidth