082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

802 Chapter 24 The Fourier transform
The terms of this sequence can be calculated by brute force as the following example
will show, although in practice a convolution can be calculated much more efficiently
using a d.f.t.
Example24.26 Suppose f[n]isthe sequence 3,9,2,−1, andg[n] isthe sequence −4,8,5.
(a) Find the linear convolutionh[n] = f ∗gusing the previous formula.
(b) Develop a graphical interpretation of thisprocess.
Solution (a) Notethatboth f andgarefinitesequences oflength4and3respectively. Theconvolution
f ∗gwill be a sequence of length 4 +3−1 = 6.By definition,
h[n] =
n∑
f[m]g[n−m]
m=0
forn=0,1,2,...,5
The first terminh[n] isobtained by lettingn = 0:
h[0] =
0∑
f[m]g[0 −m]
m=0
= f[0]g[0]
= (3)(−4)
= −12
The second termisobtained by lettingn = 1:
h[1] =
1∑
f[m]g[1 −m]
m=0
= f[0]g[1] + f[1]g[0]
= (3)(8) + (9)(−4)
=24−36
= −12
Subsequent terms are calculated in a similar manner. You should obtain these for
yourself toensure thatyou understand the process.The complete sequence is
h[n] = f ∗g= −12,−12,79,65,2,−5
(b) The graphical interpretation is developed along the same lines as was done for the
continuous convolution inSection 24.8.
The sequences f[m]andg[m] areshown inFigure 24.24.
The sequence g[−m] is found by reflection in the vertical axis, that is folding as
shown in Figure 24.25(a). Then the folded graph can be translatednunits to the left or
therightbychangingtheargumentofgfromg[m]tog[n −m].Ifnispositivethegraph
moves to the right. Study Figures 24.25(b--g) to observe this. Convolution is the sum
of products of f[m] with g[n −m]. The graph of f[m] is superimposed. We are only
interested in values ofnfor which the graphs overlap -- otherwise each product is zero.
For each value ofnthe superimposed graphs make it easy to see which values must be