Digital Signal Processing Reference
In-Depth Information
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
w
W
0
0
−
p
−0.8
p
−0.6
p
−0.4
p
−0.2
p
0
0.2
p
0.4
p
0.6
p
0.8
pp
−60
−40
−20
2 0
4 0
6 0
(a)
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
W
W
0
0
−
p
−0.8
p
−0.6
p
−0.4
p
−0.2
p
0
0.2
p
0.4
p
0.6
p
0.8
pp
−
p
−0.8
p
−0.6
p
−0.4
p
−0.2
p
0
0.2
p
0.4
p
0.6
p
0.8
pp
(c)
(d)
Fig. 16.3. Impulse invariance
transformation used to derive
digital representations of the
analog filter specified in Example
16.2. Magnitude spectra of
(a) the analog filter with transfer
function
H
(
s
); (b) the digital
filter with sampling interval
T
= 0.1 s; (c) the digital filter
with
T
= 0.0348 s; (d) the
digital filter with
T
= 0.01 s;
(e) the digital filter with
T
1
0.8
0.6
0.4
0.2
0.6
p
0.8
p
W
0
−
p
−0.8
p
−0.6
p
−0.4
p
−0.2
p
0
0.2
p
0.4
p
(e)
(2) Among the digital implementations, Fig. 16.3(b) results in the highest
gain (i.e. lowest attenuation) at the stop-band frequency
Ω
=π
radians/s.
Since the sampling interval (
T
=
0.1 s) is greater than the Nyquist bound
(
T
=
0
.
0348 s), Fig. 16.3(b) suffers from aliasing, which increases the gain
within the pass band. In using impulse invariance transformation, it is crit-
ical that the effects of the aliasing be considered within the stop band.
= 0.001 s.
16.2.1 Impulse invariance transformation using M
ATLAB
M
ATLAB
provides a library function
impinvar
to transform CT transfer
functions into the DT domain using the impulse variance method. We illustrate
the application of
impinvar
for Example 16.2 with the sampling interval
T
set to 0.1 s. The M
ATLAB
code for the transformation is as follows:
>> num = [0 0 81.6475]; % numerator of CT filter
>> den = [1 12.7786 81.6475]; % denominator of CT filter
>> T = 0.1;
>> Fs = 1/T; % sampling rate
>> [numz,denz] = impinvar (num,den,Fs);
% numerator & denominator
% of DT filter
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