1. ELEC 303 – Random Signals Lecture 21 – Random processes
Dr. Farinaz Koushanfar
ECE Dept., Rice University
Nov 19, 2009

2. Lecture outline Basic concepts Gaussian processes White processes
Filtered noise processes
Noise equivalent bandwidth

3. 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)

4. 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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. White processes
Quantum mechanical analysis of natural noise shows it has a power spectral density given by

11. 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

12. 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.

13. 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

14. 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))

15. Bandpass Filter
X(t) is the output of an ideal bandpass filter of bandwidth W centered at fc
Examples:

16. 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)

17. 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

18. 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

19. 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

20. Noise equivalent bandwidth
Hmax is the maximum of |H(f)| in the
Filter’s passband
Thus, given Bneq, finding the output noise becomes a simple task
The of filters and amplifiers are usually given by the manufacturers

21. Example
Find the noise equivalent bandwidth of a low pass filter
=RC

22. 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

23. 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!

24. 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