128 5. FOURIER SERIES
e relationships between the trigonometric series and the exponential series coefficients
are given by
a
0
D c
0
; (5.9)
a
n
D 2Refc
n
g; (5.10)
b
n
D 2Imfc
n
g; (5.11)
c
n
D
1
2
.
a
n
j b
n
/
; (5.12)
where Re and Im denote the real and imaginary parts, respectively.
According to the Parseval’s theorem, the average power in the signal x.t / is related to its
Fourier series coefficients c
n
’s, as indicated below:
1
T
Z
T
jx.t/j
2
dt D
1
X
nD1
jc
n
j
2
: (5.13)
More theoretical details of Fourier series are available in signals and systems textbooks,
e.g., [1–3].
5.1 FOURIER SERIES NUMERICAL COMPUTATION
e implementation of the integration in Equations (5.6)–(5.8) is achieved by performing sum-
mations. In other words, the integrals in (5.6)–(5.8) are approximated by summations of rect-
angular strips, each of width t, as follows:
a
0
D
1
M
M
X
mD1
x.mt /; (5.14)
a
n
D
2
M
M
X
mD1
x.mt / cos
2 mn
M
; (5.15)
b
n
D
2
M
M
X
mD1
x.mt / sin
2 mn
M
; (5.16)
where x.mt / are M equally spaced data points representing x.t/ over a single period T , and
t indicates the interval between data points such that t D
T
M
.
Similarly, by approximating the integral in Equation (5.2) with a summation of rectangu-
lar strips, each of width t, one can write
c
n
D
1
M
M
X
mD1
x.mt / exp
j 2 mn
M
: (5.17)
Note that throughout the book, the notations dt, delt a, and are used interchangeably to denote the
time interval between samples.
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