Introduction,
Linear time invariant systems,
Linear systems with random inputs,
Auto Correlation and Cross Correlation functions of inputs and outputs,
System transfer function.

**LINEAR SYSTEM WITH RANDOM INPUTS**

*Introduction*

*Linear
time invariant systems*

*Linear
systems with random inputs*

*Auto
Correlation and Cross Correlation functions of inputs and outputs*

*System
transfer function*

**Introduction**

Mathematically
a "system" is a functional relationship between the input x(t) and
y(t). We can write the relationship as

y(f) =
f[x(t): –∞< +
<∞]

Let x(t)
represents a sample function of a random process {X(t)}. Suppose the system
produces an output or response y(f) and the ensemble of the output functions
forms a random process {Y(t)}. Then the process {Y(t)} can be considered as the
output of the system or transformation 'f' with {X(t)} as the input and the
system is completely specified by the operator "f".

**1 LINEAR TIME INVARIANT SYSTEM**

Mathematically
a "system" is a functional relationship between the input x(t) and
output y(t). we can write the relationship

**2 CLASSIFICATION OF SYSTEM**

**1. Linear System: **f is
called a linear system, if it satisfies

**2. Time Invariant System:**

form a
time invariant system.

**3. Causal System:**

Suppose
the value of the output Y(t) at t = t0 depends only on the past values of the
input X(t), t≤t0.

then such
a system is called a causal system.

**4. Memory less System:**

If the
output Y(t) at a given time t = t0 depends only on X(t0) and not on any other
past or future values of X(t), then the system f is called memory less system.

**5. Stable System:**

A linear
time invariant system is said to be stable if its response to any bounded input
is bounded.

**REMARK:**

i) Noted
that when we write X(t) we mean X(s,t) where s ∈ S, S is the sample space. If the
system operator only on the variable t treating S as a parameter, it is called
a deterministic system.

a) Shows a
general single input - output linear system

b) Shows a
linear time invariant system

**3 REPRESENTATION OF SYSTEM IN THE FORM OF
CONVOLUTION**

**4 UNIT IMPULSE RESPONSE TO THE SYSTEM**

If the
input of the system is the unit impulse function, then the output or response
is the system weighting function.

Y(t) =
h(t)

Which is
the system weight function.

**4.1 PROPERTIES OF LINEAR SYSTEMS WITH RANDOM INPUT**

**Property 1:**

If the
input X(t) and its output Y(t) are related by

**Property 2:**

If the
input to a time - invariant, stable linear system is a WSS process, then the
output will also be a WSS process, i.e To show that if {X(t)} is a WSS process
then the output {Y(t)} is a WSS process.

**Property 3:**

(ii)Equation
(c) gives a relationship between the spectral densities of the input and output
process in the system.

(iii)System
transfer function:

We call H
(ω ) = F {h (τ)} as the power transfer function
or system transfer function.

**SOLVED PROBLEMS ON AUTO CROSS CORRELATION FUNCTIONS
OF INPUT**

**AND OUTPUT**

**Example :5.4.1**

Find the
power spectral density of the random telegraph signal.

**Solution**

We know,
the auto correlation of the telegraph signal process X(y) is

**WORKEDOUT EXAMPLES**

**Example: 1**

Find
the power spectral density of the random telegraph signal.

**Solution**

We
know, the auto correlation of the telegraph signal process X(y) is

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