Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 348

xlabel( ' Frequency (Hz) ' ); ylabel( ' Scaled Mag. Responses ' )
subplot(3,1,3), plot(f,phis, ' kx ' ,f, phiz, ' k- ' ); grid;
xlabel( ' Frequency (Hz) ' ); ylabel( ' Phases (deg.) ' );

The filter DC gain is given by

e j U U ¼ 0 ¼ Hð 1 Þ¼ 1 : 1031
We can further scale the filter to have a unit gain of

H

1
1 : 1031

0 : 2
1 0 : 8187 z 1 ¼

0 : 1813
1 0 : 8187 z 1

HðzÞ¼

The scaled magnitude frequency response is shown in the middle plot along with that of analog filter in
Figure 8.28 , where the magnitudes are matched very well below 1.8 Hz.
Example 8.15 demonstrates the procedure for using the impulse-invariant design. The filter
performance depends on the sampling interval (Lynn and Fuerst,1999). As shown in Figure 8.27 , the
analog impulse response hðtÞ is not a band-limited signal whose frequency extends to infinity, which is
certainly larger than the Nyquist limit (folding frequency); hence, sampling hðtÞ could cause aliasing.
Figure 8.29 (a) shows the analog impulse response ThðtÞ in Example 8.15 and its sampled version
ThðnTÞ , where the sampling interval is 0.125 second. The analog filter and digital filter magnitude
responses are plotted in Figure 8.29 (b). The aliasing occurs because the impulse response contains
frequency components beyond the Nyquist limit, that is, 4 Hz in this case. Furthermore, using the
lower sampling rate of 8 Hz causes less accuracy in the digital filter magnitude response, so more
aliasing develops.
Figure 8.29 ( c) shows the analog impulse response and its sampled version using a higher
sampling rate of 16 Hz. Figure 8.29 ( d) displays the more accurate magnitude response of the digital
filter. Hence, we can obtain a reduced aliasing level. Note that the aliasing cannot be avoided, due to
sampling of the analog impulse response. The only way to reduce the aliasing is to use a higher
sampling frequency or design a filter with a very low cutoff frequency to reduce the aliasing to
a minimum level.
Investigation of the sampling interval effect leads us to the following conclusions. Note that the
analog impulse response for an analog highpass filter or bandstop filter contains frequency up to
infinity, which is larger than the Nyquist limit (folding frequency), even assuming that the sampling
rate is much higher than the cutoff frequency of a highpass filter or the upper cutoff frequency of
a bandstop filter. Hence, sampling the analog impulse response always produces aliasing. Without
using an additional anti-aliasing filter, the impulse invariant method alone cannot be used for designing
the highpass filter or bandstop filter.
Instead, in practice, we should apply the BLT design method. The impulse-invariant design
method is only appropriate for designing a lowpass filter or bandpass filter with a sampling rate much
larger than the lower cutoff frequency of the lowpass filter or the upper cutoff frequency of the
bandpass filter.
Next, let us focus on second-order filter design via Example 8.16.