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

(
Solution ConsiderF
˜ ω + 2π )
for any ω.
T
(
F˜ ω + 2π )
∑N−1
−j(ω+2π/T )nT
=T f[n]e
T
n=0
=T
=T
∑N−1
n=0
∑N−1
n=0
Now wesaw inSection 24.10.1 that
f[n]e −jωnT e −j2π T nT
f[n]e −jωnT e −2nπj
24.10 Derivation of the d.f.t. 789
usingthe laws of indices
e −2nπj =cos2nπ−jsin2nπ=1
Finally wehave
(
F˜ ω + 2π )
=T
T
∑N−1
n=0
f[n]e −jωnT
which equalsF(ω).
˜ (
We have shown that F ˜ ω + 2π )
= F(ω) ˜ for any ω. HenceF(ω) ˜ is periodic with
T
period 2π
T .
For computational purposes it is necessary to sampleF(ω) ˜ at discrete values. Suppose
we choose to sample at N points over the interval 0 ω < 2π . By choosing this
T
intervalwearesamplingoveracompleteperiodinthefrequencydomain.Sowechoose
the sample points given by ω =k 2π
(
fork = 0,1,2,...,(N −1).WritingF˜
k 2π )
NT NT
asF(ω ˜
k
)wehave
∑N−1
F(ω ˜
k
)=T
n=0
N−1
2π
−jk
f[n]e NT nT
∑
=T f[n]e −2jnkπ/N (24.13)
n=0
Thislastsum,withoutthefactorT,isusuallytakenasthedefinitionofthed.f.t.of f[n],
writtenF[k]. To denote the d.f.t. of the sequence f[n] wewill write D{f[n]}.
Thed.f.t. ofthe sequence f[n],n = 0,1,2,...,N −1,isthe sequencegiven by
D{f[n]} =F[k] =
∑N−1
n=0
f[n]e −2jnkπ/N
fork=0,1,2,...,N−1