Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Time-domain Analysis
or
y[
n]
− y[
n −1]
+ ay[
n]
= bx[
n]
T
( 1+
Ta ) y[
n]
= y[
n −1]
+ Tbx[
n]
yielding a standard form difference equation:
y [ n]
= a1 y[
n −1]
+ b1
x[
n]
(3.4)
where
1
a = and
1
1 + Ta
b
Tb
= 1+ Ta
1
are constants.
For input’s derivative, we have similar digital form as
dx ( t)
x[
n]
− x[
n −1]
= .
dt T .
Further, the second order derivative in a differential equation contains can be discretised as
y[
n]
− y[
n −1]
y[
n −1]
− y[
n − 2]
2
−
d y(
t)
1
=
T
T
= y[
n]
− 2y[
n −1]
+ y[
n
2
dt
T
T
( 2] )
2
−
. (3.5)
When the output can be expressed only by the input and shifted input, the difference equation is called non-recursive
equation, such as
y[ n]
= b1 x[
n]
+ b2
x[
n −1]
+ b3
x[
n −
2]
(3.6)
On the other hand, if the output is expressed by the shifted output, the difference equation is a recursive equation, such as
y[ n]
= a1 y[
n −1]
+ a2
y[
n − 2] + a3
y[
n −
3]
(3.7)
where the output y [n]
is expressed by it shifted signals y [ n −1]
, y [ n − 2]
, etc. In general, an LTI processor can be
represented as
y[ n]
= a1 y[
n −1]
+ a2
y[
n − 2] + ... + b1
x[
n −1]
+ b2
x[
n − 2] + ...
or a short form
N
M
y[
n]
= ∑ ak
y[
n − k]
+ ∑ bk
x[
n − k]
k = 1
k = 0
(3.8)
27
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