PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 325

R.4.16 For example, let us verify Parseval’s theorem by means of the function f(t) =

2 sin(100t), evaluating by hand the average power of f(t) in

a. The time domain by using integration

b. The frequency domain by using addition

ANALYTICAL Solution

1. Time domain solution

w 0 = 100 rad/s T = 2 π/w 0 = [2π/100] s

P

T

ave f t dt t dt

(^22)
0
2 100
100
2

2 sin 100
1
0
() { ( )}
T
∫∫




Pave

t dt t
400
2
1
2
1
2
200
400
4
1
0 2
2 100
0
210


 

cos( ) 
{}



∫ 
00
Pave


400
2
1
2
2
100
2

 











W

2. Frequency domain solution

ft t

ee

j

jt jt

()

 2 sin( 100 ) 2

2

^100  ^100









 by Euler’s identity

2 100

sin( tje je)
 jt^100 jt^100

Then

FFF01 1

 
01 1,,

and all the coeffi cients

Fn = 0, for n = 2, 3, ..., ∞

Then

PFFFn

n

n

ave



W

−

∑

1

1

11  11 2

R.4.17 Any periodic wave f(t) can be approximated by cosine terms only; then

ft

a

cnwtn

n

()
 cos( n)

0

1

2 0

∞

∑^