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

24.9 The discrete Fourier transform 785
Whenk = 2,
3∑
F[2] = f[n]e −2jn2π/4
n=0
3∑
= f[n]e −πjn
n=0
= 1 +2e −jπ + (−5)e −2jπ +3e −3jπ
= 1 +2(−1) −5(1) +3(−1)
= −9
Finally, whenk = 3,
3∑
F[3] = f[n]e −2jn3π/4
n=0
3∑
=
n=0
f[n]e −3jnπ/2
= 1 +2e −3jπ/2 + (−5)e −3jπ +3e −9jπ/2
= 1 +2(j) −5(−1) +3(−j)
=6−j
So, the d.f.t. of the sequence 1,2,−5,3 isthe sequence 1,6 +j,−9,6 −j.
You will note from this simple example that a significant amount of calculation is necessary
even when there are just four points in the sampled sequence. This is why it is
necessary to use a computer program to find the transform.Even then, the computation
can be very time consuming. For this reason, much effort has gone into developing fast
algorithms known as fast Fourier transforms (f.f.t.s). For details of these algorithms
youshouldrefertooneofthemanyspecialisttextsonthesubject.Youshouldalsoinvestigatewhichpackagesareavailabletoenableyoutoperformthissortofcalculation.For
example, the computer package MATLAB ® has built-in commands for finding a d.f.t.
using an f.f.t. For example, the MATLAB ® command fft(f) calculates the d.f.t. of a
sequencefas shown below:
f=[1 2 -5 3];
y=fft(f)
y =
1.0000 6.0000+ 1.0000i -9.0000 6.0000- 1.0000i
InMATLAB ® either j or i can beused ininput torepresent √ −1,for example
1+5j
or
1+5i