## random signals & Stochastic process

16EC404

UNIT I
PROBABILITY: Probability introduced through Sets and Relative Frequency: Experiments
and Sample Spaces, Discrete and Continuous Sample Spaces, Events, Probability Definitions
and Axioms, Mathematical Model of Experiments, Probability as a Relative Frequency, Joint
Probability, Conditional Probability, Total Probability, Bays’ Theorem, Independent Events:
THE RANDOM VARIABLE : Definition of a Random Variable, Conditions for a Function
to be a Random Variable, Discrete and Continuous, Mixed Random Variable, Distribution
and Density functions, Properties, Binomial, Poisson, Uniform, Gaussian, Exponential,
Raleigh, Conditional Distribution, Methods of defining Conditioning Event, Conditional
Density, Properties.

UNIT II
MULTIPLE RANDOM VARIABLES: Vector Random Variables, Joint Distribution
Function, Properties of Joint Distribution, Marginal Distribution Functions, Conditional
Distribution and Density – Point Conditioning, Conditional Distribution and Density –
Interval conditioning, Statistical Independence, Sum of Two Random Variables, Sum of
Several Random Variables, Central Limit Theorem, (Proof not expected). Unequal
Distribution, Equal Distributions.
OPERATIONS ON MULTIPLE RANDOM VARIABLES: Expected Value of a Function
of Random Variables, Joint Moments about the Origin, Joint Central Moments, Joint
Characteristic Functions, Jointly Gaussian Random Variables: Two Random Variables case,
N Random Variable case, Properties, Transformations of Multiple Random Variables, Linear
Transformations of Gaussian Random Variables.

UNIT III
RANDOM PROCESSES – TEMPORAL CHARACTERISTICS: The Random Process
Concept, Classification of Processes, Deterministic and Nondeterministic Processes,
Distribution and Density Functions, concept of Stationary and Statistical Independence. FirstOrder Stationary Processes, Second- Order and Wide-Sense Stationary, (N-Order) and StrictSense Stationary, Time Averages and Ergodicity, Mean-Ergodic Processes, CorrelationErgodic Processes, Autocorrelation Function and Its Properties, Cross-Correlation Function and its Properties, Covariance Functions, Gaussian Random Processes, Poisson Random Process.

UNIT IV
RANDOM PROCESSES – SPECTRAL CHARACTERISTICS: The Power Spectrum:
Properties, Relationship between Power Spectrum and Autocorrelation Function, the CrossPower Density Spectrum, Properties, Relationship between Cross-Power Spectrum and
Cross-Correlation Function.

UNIT V
LINEAR SYSTEMS WITH RANDOM INPUTS: Random Signal Response of Linear
Systems: System Response – Convolution, Mean and Mean-squared Value of System
Response, autocorrelation Function of Response, Cross-Correlation Functions of Input and
Output, Spectral Characteristics of System Response: Power Density Spectrum of Response,
Cross-Power Density Spectrums of Input and Output, Band pass, Band-Limited and
Narrowband Processes, Properties.

TEXT BOOKS:
1. Random Variables & Random Signal Principles Peyton Z. Peebles,“Probability,”, TMH,
4th Edition,
2. Probability, Random Variables and Stochastic Processes, Athanasios Papoulis and
Unnikrishna Pillai, PHI, 4th Edition, 2002.

REFERENCES:
1. Communication Systems Analog & Digital R.P. Singh and S.D. Sapre, TMH, 1995.
2. Probability and Random Processes with Application to Signal Processing Henry Stark
and John W.Woods, Pearson Education, 3rd Edition.
3. Probability Methods of Signal and System Analysis George R. Cooper, Clave D. MC
Gillem, Oxford, 3rd Edition, 1999.
4. Statistical Theory of Communication S.P. Eugene Xavier, Statistical Theory of
Communication, New Age Publications, 2003.
5. Signals, Systems & Communications B.P. Lathi, , B.S. Publications, 2003.