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

24.15 Discrete convolution and correlation 809
InMATLAB ® ,
product = F.*G
product =
-9.0000 34.0000-31.0000i -27.0000 34.0000+31.0000i
Finally, taking the inverse d.f.t.,using the MATLAB ® commandifft(), gives
ifft([-9 34-31i -27 34+31i])
ans =
8 20 -26 -11
This isthe circular convolution of f[n]andg[n], thatis
f[n] ○∗ g[n] = 8,20,−26,−11
This example has illustrated how circular convolution can be achieved through the use
of the d.f.t.
Theconvolutiontheoremappliestocircularconvolutionbutnotlinearconvolution.However,bymodifyingtheprocedureslightly,thelinearconvolutionoftwofinitesequences
can alsobe found.
If f[n] is a finite sequence of lengthN 1
andg[n] is a finite sequence of lengthN 2
we
know from Section 24.15.1 that their linear convolution is a sequence of length N 1
+
N 2
−1.
Firstweextendboththesequences f[n]andg[n]tomakeeachhavelengthN 1
+N 2
−1.
This extension is done by adding zeros. This process is known as ‘padding’ with zeros.
Then the d.f.t.s of the padded sequences are calculated to give F[k] and G[k]. It can
be shown that the linear convolution of the original sequences is equal to the circular
convolution of the padded sequences. Hence the linear convolution f ∗gis then found
from
f ∗g = D −1 {F[k]G[k]}
Consider the following example.
Example24.30 If f[n] is the finite sequence 7,−1 andg[n] is the finite sequence 4,2,−7 use circular
convolution with padded zeros toobtain the linear convolution f ∗g.
Solution Their linear convolution isasequence of length 2 +3−1 = 4.
We pad f andgtogive sequences of length 4.
f[n] = 7,−1,0,0 andg[n] = 4,2,−7,0
Then, either by direct calculation or by using a computer package you can verify that
F[k]=6,7+j,8,7−j and G[k]=−1,11−2j,−5,11+2j