Digital Signal Processing Reference
In-Depth Information
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0.2
0.25
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0.35
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Normalized frequency
Figure 4.21
Magnitude response of a digital bandpass filter.
A plot of the magnitude response of this function is shown in Figure 4.21. It is
found that the given specifications are met by this transfer function of the digital
filter.
The design of lowpass, highpass, and bandstop filters use similar procedures.
In contrast to the impulse-invariant transformation, we see that the bilinear trans-
formation can be used for designing highpass and bandstop filters as well. Indeed,
the use of bilinear transformation is the most popular method used for the design
of IIR digital filter functions that approximate the magnitude-only specifications.
4.7 DIGITAL SPECTRAL TRANSFORMATION
In the design procedure described above, we used the bilinear transformation to
convert the magnitude specification of an IIR digital filter to that of
H(jλ)
by
prewarping the frequencies on the
λ
axis. Then we either scaled the frequencies
on the
λ
axis or applied the analog frequency transformations
p
=
g(s)
to reduce
the frequency response to that of a lowpass, analog prototype filter function. There
is an alternative method for designing IIR digital filters. It replaces the analog
frequency transformation by a frequency transformation in the digital domain.
Constantinides [1] derived a set of digital spectral transformations (DSTs) that
convert the magnitude of a lowpass digital filter with an arbitrary bandwidth,
say,
θ
p
, to that of digital highpass, bandpass, and bandstop filters or digital
lowpass filters with a different passband. These transformations are similar to
the analog frequency transformations, and the parameters of the transformation
are determined by the cutoff frequencies of these filters just as in the case of the
analog frequency transformations. Let us denote the cutoff frequency of the new
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