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1. Commutative Law: (Commutative Property of Convolution)
2. Associate Law: (Associative Property of Convolution)
3. Distribute Law: (Distributive property of convolution)

**PROPERTIES OF LINEAR CONVOLUTION **

x(n) =
Excitation Input signal y(n)

= Output
Response

h(n) =
Unit sample response

**1. Commutative Law: (Commutative Property of
Convolution)**

x(n) *
h(n) = h(n) * x(n)

**2. Associate Law: (Associative Property of
Convolution)**

[ x(n) *
h1(n) ] * h2(n) = x(n) * [ h1(n) * h2(n) ]

**3. Distribute Law: (Distributive property of
convolution) **

x(n) * [ h1(n) + h2(n) ] = x(n) * h1(n) + x(n) *
h2(n)

**CAUSALITY OF LSI SYSTEM**

The
output of causal system depends upon the present and past inputs. The output of
the causal system at n= n0 depends only upon inputs x(n) for n≤ n0. The linear
convolution is given as

The
output of causal system at n= n0 depends upon the inputs for n< n0 Hence
h(-1)=h(-2)=h(-3)=0

Thus LSI
system is causal if and only if

h(n) =0 for n<0

This is
the necessary and sufficient condition for causality of the system. Linear
convolution of the causal LSI system is given by

**STABILITY FOR LSI SYSTEM**

A System
is said to be stable if every bounded input produces a bounded output.

The input
x(n) is said to bounded if there exists some finite number Mx such that |x(n)|
≤ Mx < ∞. The output y(n) is said to bounded if there exists some finite
number My such that |y(n)| ≤ My < ∞.

Linear
convolution is given by

Taking
the absolute value of both sides

The
absolute values of total sum is always less than or equal to sum of the
absolute values of individually terms. Hence

The input
x(n) is said to bounded if there exists some finite number Mx such that |x(n)|
≤ Mx < ∞.

Hence
bounded input x(n) produces bounded output y(n) in the LSI system only if

With this
condition satisfied, the system will be stable. The above equation states that
the LSI system is stable if its unit sample response is absolutely summable.
This is necessary and sufficient condition for the stability of LSI system.

Example
1:

**SELF-STUDY: Exercise No. 1**

Q1) Show
that the discrete time signal is periodic only if its frequency is expressed as
the ratio of two integers.

Q2) Show
that the frequency range for discrete time sinusoidal signal is -∏ to ∏
radians/sample or -½ cycles/sample to ½ cycles/sample.

Q3) Prove
δ (n)= u(n)= u(n-1)

Q6) Prove
that every discrete sinusoidal signal can be expressed in terms of weighted
unit impulse. Q7) Prove the Linear Convolution theorem.

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