Digital Signal Processing Reference
In-Depth Information
the cumulative convolution, starting at the beginning sample index of the particular subsequence being
computed. Figure 3.27 shows the first subsequence's contribution graphed in plot (f ) as a stem plot, with
the second subsequence's contribution overlaid, without yet being added.
1
1
0
0
−1
−1
0
5
0
20
40
60
80
(a) Sample
(b) Sample
1
4
1
2
0
0
0
−2
−1
−1
−4
0
10
0
10
0
10
(c) Sample
(d) Sample
(e) Sample
5
0
−5
0
10
20
30
40
50
60
70
80
(f) Sample
Figure 3.27:
(a) An eight-sample impulse response; (b) A chirp, with second subsequence of eight-sample
marked; (c) Eight-sample impulse response padded with another eight zeros; (d) Second eight-sample
subsequence from (b), padded with zeros; (e) First 15 samples of circular convolution of (c) and (d),
performed using DFTs; (f ) Result from (e), plotted on output graph at samples 9-23, prior to being
added to result from previous computation, plotted as samples 1-15.
Figure 3.28 shows the result after the second subsequence's contribution in Fig. 3.27 has been
added to the composite output. Figure 3.29 shows the result after the third subsequence's contribution
has been added to the composite output.
Example 3.31.
into two convolutions, do each
using the DFT, and then combine the results using the overlap-and-add method to achieve the final
result.
Break the linear convolution of
[
1
,
1
]
and
[
1
,
2
,
3
,
4
]
For the first convolution, we pad [1,1] and [1,2] each with two zeros to form length-4 sequences,
then take the real part of the inverse DFT of the product of the DFT of each, or in m-code
FirstConv = real(ifft(fft([1,1,0,0]).*fft([1,2,0,0])))
