Introduction,
Auto Correlation functions and Properties,
Cross Correlation functions and Properties.
Power spectral density and Properties,
Cross spectral density and Properties.

**CORRELATION AND SPECTRAL DENSITY**

*Introduction*

*Auto Correlation functions and Properties*

*Cross Correlation functions and Properties*

*Power spectral density and Properties*

*Cross spectral density and Properties*

**Introduction**

The power
spectrum of a time series x(t) describes how the variance of the data x(t) is
distributed over the frequency components into which x(t) may be decomposed.
This distribution of the variance may be described either by a measure µ or by a statistical cumulative
distribution function S(f) = the power contributed by frequencies from 0 upto
f. Given a band of frequencies [a, b) the amount of variance contributed to x(t)
by frequencies lying within the interval [a,b) is given by S(b) - S(a). Then S
is called the spectral distribution function of x.

The
spectral density at a frequency f gives the rate of variance contributed by
frequencies in the immediate neighbourhood of f to the variance of x per unit
frequency.

**1 Auto Correlation of a Random Process**

Let X(t1)
and X(t2) be the two given random variables. Then auto correlation is

RXX (t1,
t2) = E[X(t1) X(t2)] Mean Square Value

Putting t1
= t2 = t in (1)

RXX (t,
t) = E[X(t) X(t)]

⇒
RXX (t, t) = E[X^{2}(t)]

Which is
called the mean square value of the random process.

**Auto Correlation Function**

Definition:
Auto Correlation Function of the random process {X(t)} is

RXX = (τ) = E{(t) X(t+τ)}

Note: RXX
(τ) = R(τ) = RX (τ)

**PROPERTY: 1**

The mean
square value of the Random process may be obtained from the auto correlation
function.

RXX(τ), by putting τ = 0.

is known
as Average power of the random process {X(t)}.

**PROPERTY: 2**

RXX(τ) is an even function of τ.

RXX (τ) = RXX (-τ)

**PROPERTY: 3**

If the
process X(t) contains a periodic component of the same period.

**PROPERTY: 4**

If a
random process {X(t)} has no periodic components, and E[X(t)] = X then

**PROPERTY: 5 **

**SOLVED PROBLEMS ON AUTO CORRELATION**

** **

**Example : 3**

Express
the autocorrelation function of the process {X'(t)} in terms of the auto
correlation function of process {X(t)}

**2 CORRELATION COEFFICIENT**

**3 CROSS CORRELATION**

Cross
correlation between the two random process {X(t)} and {Y(t)} is defined as RXY
(t1, t2) = E[X(t1) Y(t2)] where X(t1) Y(t2) are random variables.

**4 CROSS COVARIANCE**

Let
{X(t)} and {Y(t)} be any two random process. Then the cross covariance is
defined as

**4.1 CROSS CORRELATION COEFFICIENT**

**4.2 CROSS CORRELATION AND ITS PROPERTIES**

Let
{X(t)} and {Y(t)} be two random. Then the cross correlation between them is
also defined as

**PROPERTY : 1**

RXY (τ) = RYX (–τ)

**PROPERTY : 2**

**PROPERTY : 3**

If {X(t)}
and {Y(t)} are two random process then,

**SOLVED PROBLEMS ON CROSS CORRELATION**

**Example:4.4.1**

Two
random process {X(t)} and {Y(t)} are given by X(t) = A cos (ωt+θ), Y(t) =
A sin (ωt + θ) where A and ω are constants and 'θ' is a uniform random variable
over 0 to 2π. Find
the cross correlation function.

**5 SPECTRAL DENSITIES (POWER SPECTRAL DENSITY)**

**INTRODUCTION**

(i) Fourier
Transformation

(ii) Inverse
Fourier Transform

(iii) Properties
of Auto Correlation Function

(iv)Basic
Trigonometric Formula

(v) Basic
Integration

**5.1 SPECIAL REPRESENTATION**

Let x(t)
be a deterministic signal. The Fourier transform of x(t) is defined as

**Definition**

The
average power P(T) of x(t) over the interval (-T, T) is given by

**Definition**

The
average power PXX for the random process {X(t)} is given by

**6 POWER SPECTRAL DENSITY FUNCTION**

**Definition**

If {X(t)} is a stationary process (either in the strict sense or wide sense) with auto correlation function RXX(τ), then the Fourier transform of RXX(τ) is called the power spectral density function of {X(t)} and is denoted by SXX(ω) or S(ω) or SX(ω).

SXX(ω)= Fourier Transform of RXX (τ)

**6.1 WIENER KHINCHINE RELATION**

**7 PROPERTIES OF POWER SPECTRAL DENSITY FUNCTION**

**Property 1:**

The value
of the spectral density function at zero frequency is equal to the total area
under the group of the auto correlation function.

**WORKEDOUT EXAMPLES**

**Example : 3**

Express the autocorrelation function of the process {X'(t)} in terms of the auto correlation function of process {X(t)}

Solution

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