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

808 Chapter 24 The Fourier transform
and similarly,
and so
G[0] =
1∑
g[n] = 2, G[1] =
n=0
F[k] =9,1
G[k] =2,−4
1∑
g[n]e −jnπ = −1 +3e −jπ = −4
These transformsaremultiplied together, termby term, togive
n=0
H[k] =F[k]G[k] = (9)(2), (1)(−4) = 18,−4
Finally the inverse d.f.t. of the sequence 18,−4, isfound using
togive
and so
h[n] = 1 N
∑
N−1
H[k]e 2jnkπ/N
k=0
(f ○∗ g)[0] = 1 2 (18−4)=7
(f ○∗ g)[1] = 1 2 (18) + 1 2 (−4ejπ ) = 11
f○∗g=7,11
TheconvolutioncouldalsobeevaluateddirectlyusingthetechniqueofSection24.15.2.
You should trythistoconfirm the resultobtained usingthe theorem.
Example24.29 Usetheconvolutiontheoremandacomputerpackagewhichcalculatesd.f.t.stofindthe
circularconvolution ofthe sequences f[n] = 1,2,−1,7 andg[n] = −1,3,2,−5.
Solution You will need access to a computer package such as MATLAB ® to work through this
example.Thed.f.t.of f[n]canbecalculatedeitherdirectly,whichislaborious,orusing
the MATLAB ® commandfft().
F=fft([1 2 -1 7])
ans =
9.0000 2.0000+ 5.0000i -9.0000 2.0000- 5.0000i
HenceF[k] = 9,2 +5j,−9,2 −5j.
Similarly,
G=fft([-1 3 2 -5])
ans =
-1.0000 -3.0000- 8.0000i 3.0000 -3.0000+ 8.0000i
HenceG[k] = −1,−3 −8j,3,−3 +8j. Then, the product of these d.f.t.s is calculated
by multiplyingcorrespondingtermstogether:
F[k]G[k] = −9,34 −31j,−27,34 +31j