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FIR Filter Structures for Sampling Rate Conversion 525
p 0 (n)
FIGURE 9.37
x(n)
Rate: F y = IF x
p 1 (n)
y(m)
Rate:
F y = IF x
p I − 1 (n)
Rate: F x Rate: F x
Interpolation by use of polyphase filters
to define a set of smaller filters, called polyphase filters, with unit sample
responses
p k (n) =h(k + nI); k =0, 1,...,I − 1, n =0, 1,...,K − 1 (9.63)
where K = M/I is an integer.
From this discussion it follows that the set of I polyphase filters can
be arranged as a parallel realization, and the output of each filter can be
selected by a commutator, as illustrated in Figure 9.37. The rotation of
the commutator is in the counterclockwise direction, beginning with the
point at m =0.Thus, the polyphase filters perform the computations at
the low sampling rate F x , and the rate conversion results from the fact
that I output samples are generated, one from each of the filters, for each
input sample.
The decomposition of {h(k)} into the set of I subfilters with impulse
response p k (n),k =0, 1,...,I− 1isconsistent with our previous observation
that the input signal was being filtered by a periodically time-variant
linear filter with impulse response
g(n, m) =h(nI +(mD) I ) (9.64)
where D =1in the case of the interpolator. We noted previously that
g(n, m) varies periodically with period I. Consequently, a different set of
coefficients is used to generate the set of I output samples y(m),m =
0, 1,...,I − 1.
Additional insight can be gained about the characteristics of the set
of polyphase subfilters by noting that p k (n) isobtained from h(n) by
decimation with a factor I. Consequently, if the original filter frequency
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