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526 Chapter 9 SAMPLING RATE CONVERSION
response H(ω) isflat over the range 0 ≤|ω| ≤ω/I, each of the polyphase
subfilters possesses a relatively flat response over the range 0 ≤|ω| ≤π
(i.e., the polyphase subfilters are basically allpass filters and differ primarily
in their phase characteristics). This explains the reason for using
the term polyphase in describing these filters.
The polyphase filter can also be viewed as a set of I subfilters connected
to a common delay line. Ideally, the kth subfilter will generate a
forward time shift of (k/I)T x , for k =0, 1, 2,...,I − 1, relative to the
zeroth subfilter. Therefore, if the 0th filter generates zero delay, the frequency
response of the kth subfilter is
p k (ω) =e jωk/I
A time shift of an integer number of input sampling intervals (e.g., kT x )
can be generated by shifting the input data in the delay line by I samples
and using the same subfilters. By combining these two methods, we can
generate an output that is shifted forward by an amount (k + i/I)T x
relative to the previous output.
By transposing the interpolator structure in Figure 9.37, we obtain
a commutator structure for a decimator based on the parallel bank of
polyphase filters, as illustrated in Figure 9.38. The unit sample responses
of the polyphase filters are now defined as
p k (n) =h(k + nD); k =0, 1,...,D− 1, n =0, 1,...,K − 1 (9.65)
where K = M/D is an integer when M is selected to be a multiple of
D. The commutator rotates in a counterclockwise direction, starting with
the filter p 0 (n) atm =0.
Although the two commutator structures for the interpolator and the
decimator just described rotate in a counterclockwise direction, it is also
possible to derive an equivalent pair of commutator structures having a
clockwise rotation. In this alternative formulation, the sets of polyphase
filters are defined to have impulse responses
and
p k (n) =h(nI − k), k =0, 1,...,I − 1 (9.66)
p k (n) =h(nD − k), k =0, 1,...,D− 1 (9.67)
for the interpolator and decimator, respectively,
□ EXAMPLE 9.17 For the decimation filter designed in Example 9.11, determine the polyphase
filter coefficients {p k (n)} in terms of the FIR filter coefficients {h(n)}
Solution
The polyphase filters obtained from h(n) have impulse responses
p k (n) =h(2n + k) k =0, 1; n =0, 1,...,14
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