Random Processes & Variables | Communication System - Electronics and Communication Engineering (ECE) PDF Download

Introduction to Random Process and variables

• The concept of a random process allows us to study systems involving signals that are not entirely predictable.
• A random process is the natural extension of the concept of a random variable when dealing with signals.
• A random process is a collection of time functions or signals corresponding to various outcomes of a random experiment.
• Random process represents the mathematical model of these random signals.
• A random process (or stochastic process) is a collection of random variables (functions) indexed by time.
• Random process can be denoted by X(t,s) or X(t), where s is the sample point of the random experiment and t is the time.
• A random variable is an outcome is mapped to a number whereas the Random process is an outcome is mapped to a random waveform that is a function of time.
• These random signals play a fundamental role in the fields of communications, signal processing, control systems, and many other engineering disciplines.

Types of classes of signals in deterministic signals

1. Continuous-time and continuous amplitude signals: These are a function of a continuous independent variable, time. The range of the amplitude of the function is also continuous.
2. Continuous-time and discrete amplitude signals: These are a function of a continuous independent variable, time—but the amplitude is discrete.
3. Discrete-time and continuous amplitude signals: These are functions of a quantized or discrete independent time variable, while the range of amplitudes is continuous.
4. Discrete-time and discrete amplitude signals: These are functions where both the independent time variable and the amplitude are discrete.

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Mean Function

The mean function of a random process is the expected value of the process. 

Auto-correlation function

• This represents the relation of a function with its shifted version.
• The autocorrelation function of the random process X(t), denoted by R_xx(t₁ ,t₂).
• Auto-correlation function is defined as the expected value of the product of X(t₁) ,X(t₂)
• R_xx(t₁, t₂) = E [X(t₁) X(t₂)]

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Properties of Auto Correction Function

• The mean square value of a random process can be obtained from the auto-correlation function R(Z).
• R(Z) is even function Z.
• R(Z) is maximum at Z = 0 e.e. |R(Z)| ≤ R(0). In other words, this means the maximum value of R(Z) is attained at Z = 0.
• If R(Z) is the auto-correlation of a stationary random process {x(t)} with no periodic components and with non-zeros means then
  
  The short time auto-correlation function is usually computed using the following equation:
  

Stationary Process

A process whose statistical properties are independent of a choice of time origin is called as a stationary process.

• A random process is considered stationary (in the strict sense) if its specification is independent of time, i.e., if the joint probability density function of its sample values is independent of time.
• A process X(t) is called as strict-sense stationary, if the statistics of the process are invariant to a time shift.
• A process X(t) is called as wide-sense stationary (WSS), if the mean function and autocorrelation function are invariant to a time shift.
  • m_x (t) = E [X(t)] = constant (independent of t)
  • R_xx (t₁t₂) depends only on the time difference τ = (t₁ – t₂) and not on t₁ and t₂ individually. R_xx (t + τ, t) = R_xx (τ)
• All strict sense stationary random processes are also WSS, provided that the mean and autocorrelation function exist.
• If mx(t) and R_xx (t + τ, t) are periodic with period t, the process is called as the cyclostationary process.
• R_x(0) gives power content of a power signal (if signal is power signal).
• R_x(0) gives energy content of the signal (if a signal is energy signal).
• Fourier transform of R_xx(τ) gives power spectral density for power signal and energy spectral density for energy signal.

Power and Energy of Random Signal

Power content of a random signal X is given by

Energy content of random signal is given by

For the case of stationary processes, only power type process is of theoretical and practical interest.

Cross-Correlation

The cross correlation function between two random processes X(t) and Y(t) is

R_x,y(t₁, t₂) = E[X (t₁) Y (t₂)];

R_x,y(t₁, t₂) = R_y,x (t₂, t₁)

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Random Processes and Linear Systems

Let random process X(t) is input to an LTI system having impulse response h(t) and y(t) is output of LTI system then

• If X(t) is stationary process then X(t) and Y(t) will be jointly stationary with

R_xy(τ) = R_x(τ) * h(–τ)

R_x(τ) * h(–τ)

R_y(τ) = R_x(τ) * h(–τ) * h(τ)

S_y(f) = S_x(f)|H(f)|²

where * represents convolution

Power Spectral Density

• A quantity that is related to the auto-correlation function is the power spectral density [PSD, S_xx(f)].
• Note that the power spectral density is only defined for a stationary random process.
• The PSD and the auto-correlation function form a Fourier transform pair.
• Power spectral density of random process X(t):

• If the two processes X(t) and Y(t) are uncorrected, then R_xy(τ) = E[X(t)] E[Y(t)], where X(t) and Y(t) are jointly stationary processes.