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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