Altera Corporation 7–27
September 2004 Stratix Device Handbook, Volume 2
Implementing High Performance DSP Functions in Stratix & Stratix GX Devices
where:
k = 0,1, …, D-1
n = 0,1, …, P-1
P = L/D = length of polyphase filters
L is the length of the filter (selected to be a multiple of D)
D is the decimation factor
h(n) is the original filter impulse response
This equation implies that the first polyphase filter, h
0
(n), has coefficients
h(0), h(D), h(2D)…h((P-1)D). The second polyphase filter, h
1
(n), has
coefficients h(1), h(1+D), h(1+2D), ..., h(1+(P-1)D). Continuing in this way,
the last polyphase filter, h
D-1
(n) has coefficients h(D-1), h((D - 1) + D),
h((D - 1) + 2D), ..., h((D - 1) + (P-1)D).
An example helps in the understanding of the polyphase implementation
of decimation. Consider the polyphase representation of a 16-tap low
pass filter with a decimation factor of 4. The output is given by:
Referring to Figure 7–15 on page 7–26, it is clear that the output, y(n) is
discarded for n ≠ 0, 4, 8, 12, hence the only values of y(n) that need to be
computed are y(0), y(4), y(8), y(12). Table 7–12 shows which coefficients
are required to generate the output samples.
Table 7–12 shows that the overall decimation filter operation can be
represented by 4 parallel polyphase filters. Figure 7–16 shows the
polyphase representation of the decimation filter. A demultiplexer at the
input ensures that the input is applied to only one polyphase filter at a
yn() hi()xnD i–()
i0=
∑
=
Table 7–12. Decomposition of a 16-Tap Decimation Filter into Four Polyphase Filters
Output Sample (1) Coefficients Required Polyphase Filter Impulse Response
y(0)
0
, y(4)
0
, . . . h(0), h(4), h(8), h(12) h
0
(n)
y(0)
1
, y(4)
1
,
. . . h(1), h(5), h(9), h(13) h
1
(n)
y(0)
2
, y(4)
2
,
. . . h(2), h(6), h(10), h(14) h
2
(n)
y(0)
3
, y(4)
3
,
. . . h(3), h(7), h(11), h(15) h
3
(n)
Note to Table 7–12:
(1) The output sample is the sum of the results from four polyphase filters: y(n) = y(n)
0
+ y(n)
1
+ y(n)
2
+ y(n)
3.
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