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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