**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

Autocorrelation

Definition: If the process {X(t)} is stationary either in the strict sense or in the wide sense, then E{X(t).X(t-τ)} is a function of τ, denoted by $R_{xx}$ (τ)or R(τ) or $R_x$ (τ). This function R(τ)is called the autocorrelation function of the process {X(t)}

Properties:

R(t) is a even function of τ

i.e. R(τ)=R(-τ)

R(τ)is maximum at τ=0

If the autocorrelation function R(t) of a real stationary process {X(t)} is continuous at τ=0, it is continuous at every other point

If R(τ) is the autocorrelation function of a stationary process {X(t)} with no periodic component, then lim┬(τ→∞)⁡ R(τ)=$μ_x^2$ , provided the limit exists.

Power Spectral Density

Definition:

If {X(t)} is a stationary process (either in a strict sense or wide sense) with autocorrelation function R(τ), then the Fourier transform of R(τ) is called the power spectral density function of {X(t)} and denoted as $S_{xx}$ (ω) or $S_x$ (ω).

Thus $S_x$ (ω)=$∫_{-∞}^∞ R(τ) {e^{-iωτ}} dτ$

Or $S_x$ (f)=$∫_{-∞}^∞ R(τ) {e^{-i2πfτ}} dτ$

Properties:

The value of the spectral density at zero frequency is equal to the total area under the graph of the autocorrelation function The mean square value of a wide sense stationary process is equal to the total area under the graph of the spectral density.

The spectral density function of a real random process is an even function

i.e. $S_x$ (ω)=$S_x$ (-ω)

The Spectral density of a process {X(t)}, real or complex, is a real function of ω and non negative.

The spectral density and the autocorrelation function of a real WSS process form a Fourier Cosine transform pair

Íf X$_T$ (ω)is the Fourier transform of the truncated random process defined as

$X_T$ (t)=X(t) for |t|≤T

=0 for |t|>T

where{X(t)} is a real WSS process with power spectral density function S(ω) then

S(ω)=lim┬(T→∞)⁡1/2T E{$|X_T (ω) |^2$} Q