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FIR Filter Designs for Sampling Rate Conversion 519
4
Amplitude Response
Amplitude
0
0 0.2 0.275 1
Frequency in π units
Log–magnitude Response
0
Decibels
−40
−60
0 0.2 0.275
Frequency in π units
1
FIGURE 9.29 The filter design plots in Example 9.16
9.5.6 FIR FILTERS WITH MULTIPLE STOPBANDS
We now discuss the use of multiple stopbands in the design of FIR integer
interpolators when the low sampling rate is more than two times
that required. Let us refer back to the Figure 9.22b on page 506, which
illustrates a typical spectrum V (ω) ininteger interpolators. We could
use a lowpass filter with multiple stopbands of bandwidth ω s /I centered
at 2πk/I for k ≠ 0. For I = 4, such a spectrum is shown in
Figure 9.30(a), and the corresponding filter specifications are shown in
Figure 9.30b.
Clearly, these filter specifications differ from those given in (9.52) on
page 506 in that the stopband is no longer one contiguous interval. Now
if ω s π/2,
the single-band lowpass filter specification (9.52) is easier and works as
well.
Similar advantages can be obtained for FIR integer decimators. We
again find that we can substitute a multiple stopband lowpass filter for the
single stopband design given in (9.57). With reference to the signal specifications
on page 513, we note that only part of the bands [π/D, 3π/D],
[3π/D, 5π/D], ...etc. will get aliased into [−ω s , +ω s ]. Therefore, the multiple
stopbands are given by [(2π/D) − ω s , (2π/D) +ω s ], [(4π/D) − ω s ,
(4π/D)+ω s ], etc., centered at 2πk/D, k ≠0.Once again there are practical
advantages when ω s