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524 Chapter 9 SAMPLING RATE CONVERSION
x(n)
↑I
v(n)
h(0)
y(m)
x(n)
h(0)
↑I
y(m)
z −1
z −1
h(1)
h(1)
↑I
h(M − 2)
h(M − 2)
↑I
z −1
z −1
h(M − 1)
h(M − 1)
↑I
FIGURE 9.35
(a)
Efficient realization of an interpolator
(b)
9.6.2 POLYPHASE FILTER STRUCTURE
The computational efficiency of the filter structure shown in Figure 9.35
can also be achieved by reducing the large FIR filter of length M into
a set of smaller filters of length K = M/I, where M is selected to be a
multiple of I. Todemonstrate this point, let us consider the interpolator
given in Figure 9.34. Since the upsampling process inserts I − 1 zeros
between successive values of x(n), only K out of the M input values
stored in the FIR filter at any one time are nonzero. At one time-instant,
these nonzero values coincide and are multiplied by the filter coefficients
h(0),h(I),h(2I),...,h(M − I). In the following time instant, the nonzero
values of the input sequence coincide and are multiplied by the filter
coefficients h(1),h(I +1),h(2I + 1), and so on. This observation leads us
x(n)
Input
↓D
y(m)
Output
x(n)
Output
↑D
y(m)
Input
(a)
(b)
x(n)
Input
↑I
y(m)
Output
x(n)
Output
↓I
y(m)
Input
FIGURE 9.36
(c)
Duality relationships obtained through transpositions
(d)
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