PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

324 Practical MATLAB® Applications for Engineers

The Fourier coeffi cients an, bn, and Fn represent the degree of similarity between f(t)
and each of the frequency components nw 0.

R.4.12 The relation between the exponential coeffi cients Fn and the trigonometric coef-
fi cients ans and bns are as follows:

F

a

0

0

2

Fajbnnn

1

2

()

Fajbnn


1

2

()

a 0 = Fn + F−n

bn = j(Fn − F−n)

cab Fnnn n

()^222

R.4.13 Recall that the average power of a periodic function f(t) may be evaluated in the
time domain by

P

T

ft dt

T

ave

1

0

2

∫ ()

R.4.14 The time function f(t) may be viewed as a current or a voltage that acts on a resistor
of 1 Ω (normalized).

Recall that

Power

W

vt

R

vt

()

()

2

2

or Power = i^2 (t)R = i^2 (t)W (assuming R = 1 Ω without any loss of generality) and the
average power is therefore given by

P

T

it dt

T

vt dt

TT

ave

(^112)
0
2
0
∫∫() ()
R.4.15 Parseval’s relation (also known as Parseval’s theorem) states that if f(t) is a real and
periodic function, then the average power denoted by Pave may be conveniently
evaluated in the frequency domain by

P

T

f t dt F

T

n

n

ave

1 2

0

2

∫ () ∑





Note that the evaluation of Pave in the frequency domain is much easier than in the
time domain, since integration is substituted by the summation. Observe that Pave
can be easily evaluated if the coeffi cients Fn are known by using MATLAB in the
following way:
Let
F = [F−n F−n+ 1 ... F− 1 F 0 F 1 ... Fn ]

Then Pave = F * F′.