Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 578

Filter specifications for H 2 ðzÞ:
Passband frequency range: 0e3,400 Hz
Passband ripples: 0.05/2 ¼ 0.025 (0.212 dB)
Stopband frequency range: 4,000e12,000 Hz
Stopband attenuation: 0.005, d s dB ¼ 46 dB
Filter type: FIR, Hamming window

Note that

f stop f pass
f sM1

¼ 4;000 3;400
24;000

Df ¼ ¼ 0:025:

The length of the filter and the cutoff frequency can be determined by
N ¼ 3:3

Df ¼ 132

We choose N ¼ 133, and

f pass þ f stop
2

¼ 4;000 þ 3;400
2

f c ¼ ¼ 3;700 Hz

The reader can verify this case by using only one stage with a decimation factor of M ¼ 30. Using the Hamming
window for the FIR filter, the resulting number of taps is 1,321, and the cutoff frequency is 3,700 Hz. Thus, such
a filter requires a huge number of computations and causes a large delay during implementation compared with the
two-stage case.

The multistage scheme is very helpful for sampling rate conversion between audio systems. For
example, to convert CD audio at a sampling rate of 44.1 kHz to MP3 or Digital Audio Tape (DAT), in
which the sampling rate of 48 kHz is used, the conversion factor L=M ¼ 48 = 44 : 1 ¼ 160 = 147 is
required. Using the single stage scheme may cause impractical FIR filter sizes for interpolation and
downsampling. However, since L=M ¼ 160 = 147 ¼ð 4 = 3 Þð 8 = 7 Þð 5 = 7 Þ , we may design an efficient
three-stage system, in which stages 1, 2, and 3 use the conversion factors L=M ¼ 8 = 7, L=M ¼ 5 = 7,
and L=M ¼ 4 = 3, respectively.

12.2 POLYPHASE FILTER STRUCTURE AND IMPLEMENTATION
Due to the nature of the decimation and interpolation processes, polyphase filter structures can be
developed to efficiently implement the decimation and interpolation filters (using fewer multiplica-
tions and additions). As we will explain, these filters are all-pass filters with different phase shifts
(Proakis and Manolakis, 1996), thus we call them polyphase filters.
Here, we skip their derivations and illustrate implementations of decimation and interpolation
using simple examples. Consider the interpolation process shown in Figure 12.17 , where L ¼ 2.

x n () wm () ym ()

2

FIGURE 12.17
Upsampling by a factor of 2 and a four-tap interpolation filter.