Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Time-domain Analysis
s[3]= h[0]+h[1]+h[2]+h[3]=1+0.6+0.6´0.6+0.6´0.6´0.6
1
s[∞]=1+0.6+0.6´0.6+0.6´0.6´0.6+… = = 2. 5 ,
1−
0.6
where the following series summation formula is applied.
2 3 1
1+ a + a + a + ... =
a
1−
a
< 1
(3.13)
3.5 Convolution
In order to derive the convolution formula based on clear understanding, a signal is expressed by impulse functions as
following:
For a signal x [ n],
−∞ < n < ∞ , using the rules of the signal shifting and scaling described Section 2.3, it is decomposed
into a series of unit impulses scaled by the sample values:
x[ n]
= ... + x[
−2]
d[
n + 2] + x[
−1]
d[
n + 1] + x[0]
d[
n]
+ x[1]
d[
n − 1] + x[2]
d[
n − 2] + ...
or
∑ ∞
k = −∞
x [ n]
= x[
k]
d[
n − k]
(3.14)
The above expression can be illustrated as following. For example, a signal x [n]
has 2 non-zero samples, which can be
represented as a sum of shifted and scaled unit impulses x[ 1] d [ n −1]
+ x[2]
d[
n − 2]
, as illustrated in Figure 3.6.
X[n]
X[1]d[n-1]
X[2]d[n-2]
=
+
0 1 2 0 1 2 0 1 2
Figure 3.6 A signal can be decomposed into simple sequences.
(Except those non-zero samples, all other samples have zero values.)
In general cases, assuming the system is causal, if the input x[n] is
δ[n]: ... 0 0 1 0 0 ...
↑
and ,the impulse response is
33
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