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FIR Filter Structures for Sampling Rate Conversion 521
x(n)
1
2
3
h(0)
↓D
y(m)
I
Upsampler
z −1
h(1)
Downsampler
h(M − 2)
z −1 h(M − 1)
FIGURE 9.31 Direct-form realization of FIR filter in sampling rate conversion
by a factor I/D
where h(k) isthe unit sample response of the FIR filter. After designing
the filter as discussed in the previous section, we will have the filter
parameters h(k), which allow us to implement the FIR filter directly, as
shown in Figure 9.31.
Although the direct-form FIR filter realization illustrated in
Figure 9.31 is simple, it is also very inefficient. The inefficiency results
from the fact that the upsampling process introduces I − 1 zeros between
successive points of the input signal. If I is large, most of the signal
components in the FIR filter are zero. Consequently, most of the multiplications
and additions result in zeros. Furthermore, the downsampling
process at the output of the filter implies that only one out of every D
output samples is required at the output of the filter. Consequently, only
one out of every D possible values at the output of the filter should be
computed.
To develop a more efficient filter structure, let us begin with a decimator
that reduces the sampling rate by an integer factor D. From our
previous discussion, the decimator is obtained by passing the input sequence
x(n) through an FIR filter and then downsampling the filter output
by a factor D, asillustrated in Figure 9.32a. In this configuration,
the filter is operating at the high sampling rate F x , while only one out of
every D output samples is actually needed. The logical solution to this
inefficiency problem is to embed the downsampling operation within the
filter, as illustrated in the filter realization given in Figure 9.32b. In this
filter structure, all the multiplications and additions are performed at the
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