Digital Signal Processing Reference
In-Depth Information
h
(
t
)
M
samples
M
1
n
x
[
n
]
L
samples
L
1
n
N
samples
N
samples
N
samples
x
b
[
n
]
n
N
samples
Figure 12.28
Signal preparation for block filtering.
M + N - 1
samples. Therefore, both sequences must be zero padded for each to
contain elements so that circular convolution (using the DFTs of the
two sequences) can be employed to make the calculation. The output sequences
are in length and therefore overlap when they are fit together in
the
NT
-duration time periods to which they are restricted. The elements in the intervals
M + N - 1
y
b
[n]
M + N - 1
n = N
to N + M - 2,
n = 2N
to 2N + M - 2,
o
n = kN
to kN + M - 2
overlap in time and must be added together to form the output sequence
y
[
n
].
EXAMPLE 12.20
Block filtering
For the filter with the unit impulse response,
h[n] = [1, 1],
block filtering is to be used to compute the output sequence. The block filter system is shown
in Figure 12.29, where the inputs to the FFT operations are seen to be the zero-padded im-
pulse response and the zero-padded block of signal data.
The input signal sequence
x[n] = [3, 1, 2, -1, -2, 1, 1.5, 2, 0.5, 2]
is shown in Figure
12.30.
We will perform block filtering by using a four-point DFT. The four-point DFT is cho-
sen for this example so that the calculations will remain relatively simple and so that a radix-2
FFT can be used for the computations.
Because the unit impulse response has
M = 2
elements, we break the input sequence
into blocks of
N = 3
elements. Then the convolution has four elements and can be computed
