THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
College of Engineering
Department of Electrical and Computer Engineering
332:322
Principles of Communications Systems
Spring 2003
Problem Set 1
1. Derive the convolution intergal from first principles (as outlined in class) given by
y t
∞
∞
x
τ
h t
τ
d
τ
, where
h t
is the impulse response of a LTI system,
x t
the input and
y t
the output.
Convolution Integral: In time domain a linear system is described in terms of its impulse
response which is defined as the response of the system (with zero initial conditions) to a
unit impulse or delta funtion
δ
t
applied to the input of a system.
If the system is time
invariant,then the shape of the impulse response is same no matter when the impulse is
applied to the system.
Let h(t) denote the impluse response of a LTI system. Let this system be subjected to an
arbitrary excitation x(t).
To determine the output y(t) we first approximate the input x(t)
by staircase function composed of narrow rectangular pulses,each of duration
∆τ
. The ap
proximation becomes better for smaller
∆τ
. As
∆τ
approaches zero, each pulse in the limit
approaches a delta funtion weighed by a factor equal to the height of the pulse times
∆τ
.
Consider a pulse which occurs at t
n
∆τ
. By defenition,the response of the system to a unit
impulse or a delta funtion
δ
t
occuring at t
0
is h(t). Due to the time invariance property
the response of the system to a delta funtion weighed by a factor x n
∆τ ∆τ
occuring at
t
n
∆τ
,must be
∆
y t
lim
∆τ
0
x n
∆τ
h t
n
∆τ ∆τ
Above is the response to one component of input occuring at t
n
∆τ
.
Since the system
is linear we can apply superposition principle to find the response to the sum of the input
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Rose
 LTI system theory, §¡

Click to edit the document details