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FIR Filter Structures for Sampling Rate Conversion 527
x(n)
p 0 (n)
y(m)
Rate: F y = F x
D
p 1 (n)
p I − 1 (n)
FIGURE 9.38
Decimation by use of polyphase filters
Note that p 0(n) =h(2n) and p 1(n) =h(2n + 1). Hence one filter consists of
the even-numbered samples of h(n), and the other filter consists of the oddnumbered
samples of h(n).
□
□ EXAMPLE 9.18 For the interpolation filter designed in Example 9.8, determine the polyphase
filter coefficients {p k (n)} in terms of the filter coefficients {h(n)}.
Solution
The polyphase filters obtained from h(n) have impulse responses
p k (n) =h(5n + k) k =0, 1, 2, 3, 4
Consequently, each filter has length 6.
□
9.6.3 TIME-VARIANT FILTER STRUCTURES
Having described the filter implementation for a decimator and an interpolator,
let us now consider the general problem of sampling rate conversion
by the factor I/D. Inthe general case of sampling rate conversion
by afactor I/D, the filtering can be accomplished by means of the linear
time-variant filter described by the response function
g(n, m) =h[nI − ((mD)) I ] (9.68)
where h(n) is the impulse response of the low-pass FIR filter, which
ideally, has the frequency response specified by (9.36). For convenience
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