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528 Chapter 9 SAMPLING RATE CONVERSION
we select the length of the FIR filter {h(n)} to a multiple of I (i.e.,
M = KI). As a consequence, the set of coefficients {g(n, m)} for each
m =0, 1, 2,...,I− 1, contains K elements. Since g(n, m) isalso periodic
with period I, asdemonstrated in (9.44), it follows that the output y(m)
can be expressed as
y(m) =
K−1
∑
n=0
( ⌊ m
⌋ )
g n, m − I x
I
(⌊ mD
I
⌋ )
− n
(9.69)
Conceptually, we can think of performing the computations specified
by (9.69) by processing blocks of data of length K by aset of K filter
coefficients g(n, m −⌊m/I⌋ I),n =0, 1,...,K − 1. There are I such sets
of coefficients, one set for each block of I output points of y(m). For each
block of I output points, there is a corresponding block of D input points
of x(n) that enter in the computation.
The block processing algorithm for computing (9.69) can be visualized
as illustrated in Figure 9.39. A block of D input samples is buffered
and shifted into a second buffer of length K, one sample at a time. The
shifting from the input buffer to the second buffer occurs at a rate of
one sample each time the quantity ⌊mD/I⌋ increases by one. For each
output sample y(l), the samples from the second buffer are multiplied by
the corresponding set of filter coefficients g(n, l) for n =0, 1,...,K − 1,
and the K products are accumulated to give y(l), for l =0, 1,...,I − 1.
x(n)
Coefficient Storage
Input
Buffer of
Length D
g(n, 0); 0 ≤ n ≤ K −1
g(n, 1); 0 ≤ n ≤ K −1
Buffer
of
Length
K
1
2
3
K
K − 1
Σ
n = 0
g(n, I − 1); 0 ≤ n ≤ K −1
FIGURE 9.39
processing
Output
Buffer of
Length I
y(m)
Rate:
Efficient implementation of sampling rate conversion by block
I
D
F x
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