Random Processes & Variables Study Notes for GATE EC 2022 Exam

Properties of Auto-Correction Function

R_x (τ) is the auto-correction function of stationary process X(t) has the following properties.

• R_x (τ) is an even function i.e., R_x (τ) = R_x (–τ)
• The maximum absolute value of R_x (τ) is achieved at τ = 0 i.e., | R_x (τ)| ≤ R_x (0)
• If for some T₀ we have R_x(T₀) = R_x(0), then for all integers K, R_x(KT₀) = R_x (0).

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 strict-sense stationery if the statistics of the process are invariant to a time shift.
• A process X(t) is called wide-sense stationery (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 m_x(t) and R_xx (t + τ, t) is periodic with period t, the process is called the cyclostationary process.

• R_x(0) gives the power content of a power signal (if the signal is power signal).
• R_x(0) gives the energy content of the signal (if a signal is an 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

The power content of a random signal X is given by

The energy content of the random signal is given by

For the case of stationary processes, only the 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 (t_2, t₁)

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.

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