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104
Signals and Spectra Chap. 2
We approximate this integral by using a finite series, where t = k ¢t, f 0 = 1/T, dt =¢t, and
¢t = T/N. Then
c n L 1 N-1
T a w(k ¢t)e -j(2p/N)nk ¢t
k=0
(2–185)
Using Eq. (2–176), we find that the Fourier series coefficient is related to the DFT by
c n L 1 N X(n)
(2–186)
The DFT returns X(n) values for n = 0, 1, Á , N - 1. Consequently, Eq. (2–186) must be
modified to give c n values for negative n. For positive n, we use
c n = 1 N X(n), 0 … n 6 N 2
(2–187a)
and for negative n we use
c n = 1 N X(N + n), - N 2 6 n 6 0
(2–187b)
Example 2–21 USE THE FFT TO COMPUTE THE SPECTRUM OF A SINUSOID
Let
w(t) = 3 sin (v 0 t + 20°)
where v 0 = 2pf 0 and f 0 = 10 Hz.
Because w(t) is periodic, Eq. (2–109) is used to obtain the spectrum
(2–188)
W(f) =© c n d(f - nf 0 )
where {c n } are the complex Fourier series coefficients for w(t). Furthermore, because
sin(x) = (e jx - e -jx )/(2j),
3 sin(v 0 t + 20°) = a 3 2j ej20 be jv 0t + a -3
2j e-j20 be -jv 0t
Consequently, the FS coefficients are known to be
c 1 = a 3 2j ej20 b = 1.5l -70°
(2–189a)
c 1 = a -3
(2–189b)
2j e-j20 b = 1.5l +70°
and the other c n ’s are zero. Now see if this known correct answer can be computed by using the
DFT. See Example2_21.m for the solution. Referring to Fig. 2–22, we observe that MATLAB computes
the FFT and plots the spectrum. The computed result checks with the known analytical result.