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

786 Chapter 24 The Fourier transform
EXERCISES24.9.1
and they are both equivalent. However when variables are printed, i is always used.
Hence both of these inputs iftyped atthe command line would give
ans= 1+5i
this should beborne inmindwhen viewing the results of the f.f.t.
1 Usethe definition to findthe d.f.t. ofthe sequences
f[n] = 1,2,0, −1 andg[n] = 3,1, −1,1.
2 Calculate the d.f.t. ofthe sequence f[n] = 5,−1,2.
3 Useatechnical computing language suchas
MATLAB ® to verifyyour answers to Questions1
and 2.
4 Fromthe definition ofthe d.f.t. showthat if f[n] is a
sequence ofreal numberswith d.f.t.F[k], then
∑ N−1
n=0 f[n]e2jnkπ/N =F[k] where the overline
denotes the complex conjugate ofF[k].
5 Forasequence ofcomplex numbersF[k], letF[k]
represent the sequence obtained by taking the
complex conjugate ofeach termin the sequence.
Show that if f[n]is asequence ofreal numbers,then
F[N−k]=F[k]fork=0,1,2,...,N/2ifNis
even,andfork =0,1,2,...,(N −1)/2ifN isodd.
Thiscan be seen in Example 24.20 whereN = 4 and
F[3] =F[1].
Solutions
1 F[k]=2,1−3j,0,1+3j.G[k]=4,4,0,4 2 6,4.5 +2.5981j,4.5 −2.5981j
24.9.2 Theinversed.f.t.
Just as there is an inverse Fourier transform which transformsF(ω) back to f (t), there
isaninverse d.f.t. whichconvertsF[k] backto f[n].
Theinverse d.f.t. ofthe sequenceF[k],
k = 0,1,2,...,N −1,isthe sequence f[n], alsohavingN terms,given by
D −1 {F[k]} = f[n] = 1 N
∑
N−1
F[k]e 2jknπ/N
k=0
n=0,1,2,...,N−1
Example24.20 Using the definition, find the inverse d.f.t. of the sequence F[k], for k = 0,1,2,3,
given by
F[k] = −4,1,0,1
Solution Inthis sequencetherearefourtermsand soN = 4.Fromthe definition
D −1 {F[k]} = f[n] = 1 3∑
F[k]e 2jknπ/4 n=0,1,2,3
4
= 1 4
k=0
3∑
k=0
F[k]e jknπ/2
n=0,1,2,3
= 1 4 (−4 +1ejnπ/2 +0e jnπ +1e 3jnπ/2 )