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Frequency-band Transformations 457
This highpass filter design procedure can be easily extended to other
frequency-selective filters using the transformation functions in Table 8.2.
These design procedures are explored in Problems P8.34, P8.36, P8.38,
and P8.40. We now describe MATLAB’s filter design functions for designing
arbitrary frequency-selective filters.
8.6.2 MATLAB IMPLEMENTATION
In the preceding section we discussed four MATLAB functions to design
digital lowpass filters. These same functions can also be used to design
highpass, bandpass, and bandstop filters. The frequency-band transformations
in these functions are done in the s-plane, that is, they use
Approach-1 discussed on page 386. For the purpose of illustration we
will use the function butter. Itcan be used with the following variations
in its input arguments.
• [b,a] = BUTTER(N,wn,’high’) designs an Nth-order highpass filter
with digital 3-dB cutoff frequency wn in units of π.
• [b,a] = BUTTER(N,wn,)designs an order 2N bandpass filter if wn is a
two-element vector, wn=[w1 w2], with 3-dB passband w1 < w < w2 in
units of π.
• [b,a] = BUTTER(N,wn,’stop’) is an order 2N bandstop filter if wn=[w1
w2]with 3-dB stopband w1 < w < w2 in units of π.
To design any frequency-selective Butterworth filter, we need to know
the order N and the 3-dB cutoff frequency vector wn. Inthis chapter we
described how to determine these parameters for lowpass filters. However,
these calculations are more complicated for bandpass and bandstop filters.
In their SP toolbox, MATLAB provides a function called buttord to
compute these parameters. Given the specifications, ω p , ω s , R p , and A s ,
this function determines the necessary parameters. Its syntax is
[N,wn] = buttord(wp,ws,Rp,As)
The parameters wp and ws have some restrictions, depending on the type
of filter:
• Forlowpass filters wp < ws.
• For highpass filters wp > ws.
• For bandpass filters wp and ws are two-element vectors, wp=[wp1,
wp2] and ws=[ws1,ws2], such that ws1 < wp1 < wp2 < ws2.
• For bandstop filters wp1 < ws1 < ws2 < wp2.
Now using the buttord function in conjunction with the butter function,
we can design any Butterworth IIR filter. Similar discussions apply
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