PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

322 Practical MATLAB® Applications for Engineers

Revisit the process of partial fraction expansion when evaluating the ILT

Understand the differences between the FT and the LT

Know when and how the FS, FT, and the LT are used

Use MATLAB as a tool to solve circuit, signal, and system problems using Fourier

and Laplace techniques

4.3 Background

R.4.1 Recall that a continuous time function f(t) is periodic if it satisfi es the following
relation:

f(t) = f(t ± nT), for n = 0, 1, 2, 3, ..., ∞

where T is referred to as its period, which is the smallest possible value that satis-
fi es the preceding relation.

R.4.2 Let f(t) be a periodic function with period T, then the trigonometric FS is given by

ft

a

an nwt b nwt

n

()
[ cos( )nsin( )]

0

0

1

2 0



∑

where

an
 f t nw t dt n

2

0

T 0

T

∫ ()cos( ) for 0

bn
 f t nw t dt n

2

T^0

T

() ( )

0

∫ sin for 1

where w 0 = 2 π/T, for n = 1, 2, 3, ..., ∞.
The cosine–sine series is the most popular way to defi ne the FS and its coeffi -
cients, but as will be seen later in this chapter, it is not the most convenient.

R.4.3 An alternate form to evaluate the coeffi cients (ans and bns) is to evaluate the inte-
grals with respect to w 0 t, over the period defi ned by 2π radians. The equations used
to evaluate the coeffi cients of the trigonometric FS are then given by

aftnwtdwtn

1

00

0

2





∫ ()cos( ) (^ )

bftnwtdwtn

1

00

0

2





∫ ()sin( ) (^ )

R.4.4 Observe that the limits of integration in the evaluation of the Fourier coeffi cients, in
either of the preceding cases, with respect to t or w 0 t, must be over one full period, but

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