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

788 Chapter 24 The Fourier transform
inthe following way:
∞∑
˜f(t)=T f[n]δ(t −nT)
n=0
Notethatwhereas f (t)istheoriginalcontinuousfunctionoft, ˜f (t)isanapproximation
obtainedusingonlythesampledvalues.Thisrepresentationisdiscussedingreaterdetail
inAppendix I.
24.10.2 Derivation
Thisderivationisbaseduponthefactthatafunction f (t),definedfort 0,andhaving
been sampled atintervals ofT, can be represented by the function ˜f (t) where
∞∑
˜f(t)=T f[n]δ(t −nT)
n=0
In any real problem it is only possible to sample a signal over a finite time interval.
Suppose we obtainN samples of the signal at timest = 0,T,2T,..., (N −1)T. Then
thesampledsignalcanberepresentedbyamendingthelimitsofsummationandwriting
˜f(t)=T
∑N−1
n=0
f[n]δ(t −nT)
Taking the Fourier transform of both sides gives
F{ ˜f(t)}=T
∫ ∞
−∞
∑N−1
e −jωt
n=0
f[n]δ(t −nT)dt
If we make the assumption that it is permissible to interchange the order of integration
and summation wefind
F{ ˜f(t)}=T
=T
∑N−1
f[n]
∫ ∞
n=0
−∞
∑N−1
f[n]e −jωnT
n=0
e −jωt δ(t −nT )dt
where we have used the sifting property of the delta function. The quantity on the r.h.s.
isacontinuousfunctionof ωderivedusingthevaluesinthesequence f[n].Writethisas
F(ω)andnotethat ˜ F(ω)isanapproximationtotheFouriertransformF(ω) ˜
= F{f (t)}.
It can also be thought of as a Fourier transform for sequences. An important point to
note about this function of ω is that it is periodic with period 2π . This is proved in the
T
following example.
Example24.21 Show thatthe function
∑N−1
F(ω) ˜ =T
n=0
f[n]e −jωnT
isperiodic with period 2π
T .