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522 Chapter 9 SAMPLING RATE CONVERSION
x(n)
h(0)
↓D
y(m)
x(n)
↓D
h(0)
y(m)
z −1
z −1
h(1)
↓D
h(1)
z −1 h(M − 2)
h(M − 2)
↓D
z −1
h(M − 1)
↓D
h(M − 1)
(a)
FIGURE 9.32
realization
Decimation by a factor D: (a) standard realization, (b) efficient
(b)
lower sampling rate F x /D. Thus, we have achieved the desired efficiency.
Additional reduction in computation can be achieved by exploiting the
symmetry characteristics of {h(k)}. Figure 9.33 illustrates an efficient realization
of the decimator in which the FIR filter has linear phase and
hence {h(k)} is symmetric.
Next, let us consider the efficient implementation of an interpolator,
which is realized by first inserting I − 1 zeros between samples of x(n)
and then filtering the resulting sequence. The direct-form realization is
illustrated in Figure 9.34. The major problem with this structure is that
the filter computations are performed at the high sampling rate of IF x .
The desired simplification is achieved by first using the transposed form
of the FIR filter, as illustrated in Figure 9.35a, and then embedding the
upsampler within the filter, as shown in Figure 9.35b. Thus, all the filter
multiplications are performed at the low rate F x , while the upsampling
process introduces I −1 zeros in each of the filter branches of the structure
shown in Figure 9.35b. The reader can easily verify that the two filter
structures in Figure 9.35 are equivalent.
It is interesting to note that the structure of the interpolator, shown
in Figure 9.35b, can be obtained by transposing the structure of the decimator
shown in Figure 9.32. We observe that the transpose of a decimator
is an interpolator, and vice versa. These relationships are illustrated in
Figure 9.36, where part b is obtained by transposing part a and part d is
obtained by transposing part c. Consequently, a decimator is the dual of
an interpolator, and vice versa. From these relationships, it follows that
there is an interpolator whose structure is the dual of the decimator shown
in Figure 9.33, which exploits the symmetry in h(n).
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