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FIR Filter Structures for Sampling Rate Conversion 529
x(n)
Rate: F x
FIGURE 9.40
z −1
DI
DI
z −1
g(2, )
mD
x − 2
I
DI
g(K − 1, )
z −1 mD
x − K + 1
I
DI
x
mD
I
mD
x − 1
I
g(0, ); = 0,1,..., I − 1
g(1, )
Rate:
y(m)
Efficient realization of sampling rate conversion by a factor I/D
I
D
F x
Thus this computation produces I outputs. It is then repeated for a new
set of D input samples, and so on.
An alternative method for computing the output of the sampling rate
converter, specified by (9.69), is by means of an FIR filter structure with
periodically varying filter coefficients. Such a structure is illustrated in
Figure 9.40. The input samples x(n) are passed into a shift register that
operates at the sampling rate F x and is of length K = M/I, where M
is the length of the time-invariant FIR filter specified by the frequency
response given by (9.36). Each stage of the register is connected to a holdand-sample
device that serves to couple the input sampling rate F x to
the output sampling rate F y =(I/D)F x . The sample at the input to each
hold-and-sample device is held until the next input sample arrives and
then is discarded. The output samples on the hold-and-sample device are
taken at times mD/I, m =0, 1, 2,.... When both the input and output
sampling times coincide (i.e., when mD/I is an integer), the input to the
hold-and-sample is changed first; then the output samples the new input.
The K outputs from the K hold-and-sample devices are multiplied by
the periodically time-varying coefficients g(n, m −⌊m/I⌋ I), for n =0,
1,...,K − 1, and the resulting products are summed to yield y(m). The
computations at the output of the hold-and-sample devices are repeated
at the output sampling rate of F y =(I/D)F x .
Finally, rate conversion by a rational factor I/D can also be performed
by use of a polyphase filter having I subfilters. If we assume that the mth
sample y(m) iscomputed by taking the output of the i m th subfilter with
input data x(n),x(n − 1),...,x(n − K + 1), in the delay line, the next
sample y(m +1)istaken from the (i m+1 )st subfilter after shifting l m+1
new samples in the delay lines where i m+1 =(i m + D) mod I
and l m+1 is
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