Page 186 - DSP Integrated Circuits

4.21 Interpolation With an Integer Factor L 171

EXAMPLE 4.14
Show that an FIR interpolator can be decomposed into L parallel FIR niters that
operate at the input sampling rate and produce a combined output signal with an
L times higher sampling rate. The resulting filter structures are often called
polyphase filters [2, 3]. Also, determine the relation between the impulse responses
of the filters.
Generally, the output of the FIR filter is described by the convolution of the
input sequence xi(ni) and the impulse response

where the input sequence x\(m) is obtained from the original sequence x(n) by the
insertion of zeros.
Now, the original sequence is nonzero only for every Lth sample. Hence, we
have for the output values at time instances, m,ra + l,...,ra+L — 1:

where M^ is the largest integer equal to or less than (N - i - 1)/L. Thus, L output
values are computed from the original input sequence. This corresponds to L FIR
filters with the following impulse responses

Figure 4.63 shows the final signal-flow graph for the FIR interpolator. L out-
put samples are computed for each input sample. The number of multiplications is
N (the same number as in the original FIR filter) but the number of operations per
second is reduced by a factor L. Only the values in the input sequence need to be
stored.