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Linear Convolution Using the DFT 185
Noting that the first two samples in each section are to be discarded, we assemble
the output y(n) as
The linear convolution is given by
y(n) ={1, 2, 2, 2, 2, 2, 2, 2, 2, 2, −9, −10}
↑
x(n) ∗ h(n) ={1, 2, 2, 2, 2, 2, 2, 2, 2, 2, −9, −10}
↑
which agrees with the overlap-save method.
□
5.5.3 MATLAB IMPLEMENTATION
Using this example as a guide, we can develop a MATLAB function to
implement the overlap-save method for a very long input sequence x(n).
The key step in this function is to obtain a proper indexing for the
segmentation. Given x(n) for n ≥ 0, we have to set the first (M − 1)
samples to zero to begin the block processing. Let this augmented sequence
be
ˆx(n) = △ {0, 0,...,0 ,x(n)}, n ≥ 0
} {{ }
(M−1) zeros
and let L = N − M +1, then the kth block x k (n),
given by
0 ≤ n ≤ N − 1, is
x k (n) =ˆx(m); kL ≤ m ≤ kL + N − 1, k≥ 0, 0 ≤ n ≤ N − 1
The total number of blocks is given by
⌊ ⌋
Nx + M − 2
K =
+1
L
where N x is the length of x(n) and ⌊·⌋ is the truncation operation. Now
each block can be circularly convolved with h(n) using the circonvt
function developed earlier to obtain
y k (n) =x k (n) N○ h(n)
Finally, discarding the first (M − 1) samples from each y k (n) and concatenating
the remaining samples, we obtain the linear convolution y(n).
This procedure is incorporated in the following ovrlpsav function.
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