1. EE322 Digital Communications
Communication Theory
1/15/03
EE322 Digital Communications
Lecture #2
Random Processes
EE A. Al-Sanie
Spring 2003

2. Random Processes Random processes have the following properties:
Communication Theory
1/15/03
Random Processes
Random processes have the following properties:
Random processes are functions of time.
Random processes are random in the sense that it is not possible to predict exactly what waveform will be observed in the future.
Suppose that we assign to each sample point s a function of time with the label
And the sample function as
EE A. Al-Sanie
Spring 2003

3. Communication Theory
1/15/03
EE A. Al-Sanie
Spring 2003

4. Communication Theory
1/17/03
Random Variables
Random variables map the outcome of a random experiment to a number. S
heads
tails X
EE A. Al-Sanie
Spring 2003

5. Random Processes Random Processes map the outcome of a random
Communication Theory
1/17/03
Random Processes
Random Processes map
the outcome of a random
experiment to a signal
(function of time).
signal associated
with the outcome:
A random process evaluated at a
particular time is a random variable
sample function
ensemble S
heads
tails
Spring 2003

6. Communication Theory
1/15/03
Typical ensemble members for four random processes commonly encountered in communications (a) thermal noise (b) uniform phase. (c) Rayleigh fading process and (d) binary random data process.
EE A. Al-Sanie
Spring 2003

7. Random Process Terminology
Communication Theory
1/17/03
Random Process Terminology
The expected value, ensemble average or mean of a random process is:
The autocorrelation function (ACF) is:
Autocorrelation is a measure of how alike the random process is from one time instant to another.
EE A. Al-Sanie
Spring 2003

8. Mean and Autocorrelation
Communication Theory
1/17/03
Mean and Autocorrelation
Finding the mean and autocorrelation is not as hard as it might appear!
Why: because oftentimes a random process can be expressed as a function of a random variable.
We already know how to work with functions of random variables.
Example:
This is just a function g() of :
We know how to find the expected value of a function of a random variable:
To find this you need to know the pdf of .
a random variable
EE A. Al-Sanie
Spring 2003

9. An Example If  is uniform between 0 and , then: Communication Theory
1/17/03
An Example
If  is uniform between 0 and , then:
Spring 2003

10. Communication Theory
1/17/03
Stationarity
A process is strict-sense stationary (SSS) if all its joint densities are invariant to a time shift:
in general, it is difficult to prove that a random process is strict sense stationary.
A process is wide-sense stationary (WSS) if:
The mean is a constant:
The autocorrelation is a function of time difference only:
If a process is strict-sense stationary, then it is also wide-sense stationary.
EE A. Al-Sanie
Spring 2003

11. Communication Theory
1/15/03
Illustrating the autocorrelation functions of slowly and rapidly fluctuating random processes
EE A. Al-Sanie
Spring 2003

12. Properties of the Autocorrelation Function
Communication Theory
1/17/03
Properties of the Autocorrelation Function
If x(t) is Wide Sense Stationary, then its autocorrelation function has the following properties:
Examples:
Which of the following are valid ACF’s?
this is the second moment
even symmetry
EE A. Al-Sanie
Spring 2003

13. Power Spectral Density
Communication Theory
1/17/03
Power Spectral Density
Power Spectral Density (PSD) is a measure of a random process’ power content per unit frequency.
Denoted S(f).
Units of W/Hz.
S(f) is nonnegative function.
For real-valued processes, (f) is an even function.
The total power of the process if found by:
The power within bandwidth B is found by:
EE A. Al-Sanie
Spring 2003

14. Power Spectral density (PSD) of Random Process
Communication Theory
1/17/03
Power Spectral density (PSD) of Random Process
We can easily find the PSD of a WSS random processes.
Wiener-Khintchine theorem:
If x(t) is a wide sense stationary random process, then:
i.e. the PSD is the Fourier Transform of the ACF.
Example:
Find the PSD of a WSS R.P with autocorrelation:
EE A. Al-Sanie
Spring 2003

15. Example: Random Binary wave
Communication Theory
1/15/03
Example: Random Binary wave
Sample function of random binary wave.
EE A. Al-Sanie
Spring 2003

16. Autocorrelation function of random binary wave
Communication Theory
1/15/03
Autocorrelation function of random binary wave
Power spectral density of random binary wave.
EE A. Al-Sanie
Spring 2003

17. Noise in Communication Systems
Communication Theory
1/15/03
Noise in Communication Systems
The term noise refers to unwanted electrical.
The noise limits the receiver’s ability to make correct symbol decisions.
Noise arises from a variety of sources:
Man-made noise: such as spark-plug ignition noise, switching transients, and other radiating electromagnetic signals.
Natural noise: such as sun and other galactic sources, thermal noise.
EE A. Al-Sanie
Spring 2003

18. Communication Theory
1/15/03
Thermal Noise
Thermal noise is caused by the thermal motion of electrons in all dissipative components (resistors, wires ).
The same electrons that are responsible for electrical conduction are also responsible for thermal noise.
Thermal noise cannot be eliminated.
We can describe the thermal noise as a zero mean Gaussian random process.
EE A. Al-Sanie
Spring 2003

19. Communication Theory
1/15/03
A Gaussian process n(t) is a random function whose value n at any arbitrary time t is statistically characterized by the Gaussian probability density function
The Gaussian distribution is often used as the system noise model because of the theorem called central limit theorem.
EE A. Al-Sanie
Spring 2003

20. Communication Theory
1/15/03
EE A. Al-Sanie
Spring 2003

21. Communication Theory
1/15/03
White Gaussian Noise
The power spectral density Sn(f) of the thermal noise is flat for all frequencies as shown in the figure and is denoted as
Where the factor of 2 is included to indicate that Sn(f) is two-sided power density.
Because the noise has such Sn(f) we refer to it as white noise.
EE A. Al-Sanie
Spring 2003

22. The autocorrelation function of white noise is
Communication Theory
1/15/03
The autocorrelation function of white noise is
Any two different samples of white noise, no matter how close together in time they are taken, are uncorrelated.
Since thermal noise is a Gaussian process and the samples are uncorrelated, then noise sample are also independent.
Therefore, the noise affects each transmitted symbol independently.
EE A. Al-Sanie
Spring 2003

23. The parameter No may be expressed as
Communication Theory
1/15/03
The parameter No may be expressed as
were k is Boltzmann’s constant =1.38×10-23 J/K
And Te is the equivalent noise temperature of the receiver. The equivalent noise temperature of a system is defined as the temperature at which a noisy resistor has to be maintained such that, by connecting the resistor to the input of noiseless version of the system, it produced by all the sources of noise in the actual system.
EE A. Al-Sanie
Spring 2003

24. Thermal noise is present in all communication systems.
Communication Theory
1/15/03
Thermal noise is present in all communication systems.
The thermal noise characteristics : White, Gaussian, and additive.
We call it Additive White Gaussian Noise (AWGN).
In this course we shall assume that the signals are corrupted by AWGN.
EE A. Al-Sanie
Spring 2003

25. Communication Theory
1/15/03 n n
Characteristics of white noise. (a) Power spectral density. (b) Autocorrelation function.
EE A. Al-Sanie
Spring 2003

26. Communication Theory
1/17/03
Linear Systems
The output of a linear time invariant (LTI) system is found by convolution.
However, if the input to the system is a random process, we can’t find X(f).
Solution: use power spectral densities:
This implies that the output of a LTI system is WSS if the input is WSS.
x(t)
y(t)
h(t)
EE A. Al-Sanie
Spring 2003

27. Ideal low-pass filtered white noise:
Communication Theory
1/15/03
Ideal low-pass filtered white noise:
Low-pass filter H(f)
White noise
n(t)
H(f) 1
-B B
(a) Power spectral density. (b) Autocorrelation function
EE A. Al-Sanie
Spring 2003