Multirate Digital Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 218

The output in the z-domain can be expressed as

Y ð z Þ¼ H ð z Þ X ð z L Þ:

ð 5 : 20 Þ

As with the case of decimation, interpolation filters can be implemented rela-
tively efficiently. Many of the input samples are zero and so there is a reduction by
a factor L in the number of add-multiply operations needed for realizing the filter.

5.3.3 Rational Number Sampling Rate Conversion

The sampling rate conversions that have been considered so far have involved
integer changes in sampling rate (via either decimation or interpolation). There are
many applications, however, that require the alteration of the sampling rate by
some rational number, i.e., by L/M, where both L and M are arbitrary positive
integers. There are even some applications (such as the pitch control of audio
signals) that require irrational factor sampling rate conversion. This subsection
will address only rate conversion by rational numbers. As shown in Fig. 5.11 ,a
sampling rate change of L/M should be realized by cascading an L-fold interpolator
with an M-fold decimator. As decimation destroys information while interpolation
does not, the decimator should be preceded by the interpolator. In this case the
time-domain expression of the output is

Y ð z Þ¼ X

1

h ð Mn pL Þ x ð p Þ; ð 5 : 21 Þ

p ¼1

where p is as defined in ( 5.18 ).
For computationally efficient implementation of the structure shown in
Fig. 5.11 , the interpolation filter H u (z) and the decimation filter H d (z) can be
replaced by an equivalent single filter H(z). This replacement is achievable since

Fig. 5.11 Rational-rate
(L/M) sampling rate conver-
sion scheme; R = max(M, L)

x ( n )

y ( n )

L

M

H u ( z ) H d ( z )

H ( z )