Infinite Impulse Response Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 191

FIGURE 5.13. Relationship between analog and digital frequencies.

which relates the analog frequency w A to the digital frequency w D . This relationship

is plotted in Figure 5.13 for positive values of w A . The region corresponding to w A

between 0 and 1 is mapped into the region corresponding to w D between 0 and w s /4

in a fairly linear fashion, where w s is the sampling frequency in radians. However,

the entire region of w A > 1 is quite nonlinear, mapping into the region correspond-

w s /2. This compression within this region is referred
to as frequency warping . As a result, prewarping is done to compensate for this
frequency warping. The frequencies

ing to w D between w s /4 and

w A and w D are such that

() () Hs = Hz (5.33)

sj = w

j w

D T

ze =

A

5.3.1 BLT Design Procedure

The BLT design procedure makes use of a known analog transfer function for the
design of a discrete-time filter. It can be applied using well-documented analog filter
functions (Butterworth, Chebychev, etc.). Several types of filter design are available
with MATLAB, described in Appendix D. Butterworth filters are maximally flat in
the passband and in the stopband. Chebyshev types I and II provide equiripple
responses in the passbands and stopbands, respectively. For a given specification,
these filters are of lower order than Butterworth-type filters, which have monoto-
nic responses in both passbands and stopbands. Chebyshev filters have sharper
cutoff frequencies than Butterworth filters, but at the expense of ripples in the pass-
band (type I) or in the stopband (type II). They are useful in applications requiring
sharp transitions while tolerating the ripples. An elliptic design has equiripple in
both bands and achieves a lower order than a Chebyshev-type design; however, it
is more difficult to design, with a highly non-linear phase response in the passbands.
Although a Butterworth design requires a higher order, it has a linear phase in the
passbands.
Perform the following steps in order to use the BLT technique and find H ( z ).