Digital Signal Processing Reference
In-Depth Information
Let us examine convolving a finite long sequence with an infinite long sequence.
EXAMPLE 3.13
A system representation using the unit-impulse response for the linear system
yðnÞ¼0:25yðn 1ÞþxðnÞ
for
n 0 and
yð1Þ¼0
was determined in Example 3:8as
yðnÞ¼
N
k ¼
N
xðkÞhðn kÞ
n
uðnÞ. For a step input xðnÞ¼uðnÞ, determine the output response for the first three output
samples using the table method.
Solution:
Using
Table 3.5
as a guide, we list the operations and calculations in
Table 3.7
. As expected, the output values are
the same as those obtained in Example 3.8.
where hðnÞ¼ð0:25Þ
Table 3.7
Convolution Sum in Example 3.13.
k :
L
2
L
1
0
1
2
3
.
xðkÞ :
1
1
1
1
.
yð
0
Þ¼
1
1
¼
1
hðkÞ :
0.0625
0.25
1
hð
1
kÞ
yð
1
Þ¼
1
0
:
25
þ
1
1
¼
1
:
25
0.0625
0.25
1
hð
2
kÞ
0.0625
0.25
1
yð
2
Þ¼
1
0
:
0625
þ
1
0
:
25
þ
1
1
¼
1
:
3125
Stop as required
3.6
SUMMARY
1.
Digital signal samples are sketched using their encoded amplitude versus sample numbers with
vertical bars topped by solid circles located at their sampling instants, respectively. The impulse
sequence, unit-step sequence, and their shifted versions are sketched in this notation.
2.
The analog signal function can be sampled to its digital (discrete-time) version by substituting time
t ¼ nT
into the analog function, that is,
xðnÞ¼xðtÞj
t¼nT
¼ xðnTÞ
The digital function values can be calculated for the given time index (sample number).
3.
The DSP system we wish to design must be a linear, time-invariant, causal system. Linearity means
that the superposition principle exists. Time invariance requires that the shifted input generate the
corresponding shifted output in the same amount of time. Causality indicates that the system output
depends on only its current input sample and past input sample(s).
4.
The difference equation describing a linear, time-invariant system has a format such that the current
output depends on the current input, past input(s), and past output(s) in general.
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