PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 329

c. If f(t) = − f(t ± T/ 2 ), a condition referred to as half wave or rotational symmetry

(with respect to t), then

a 0 = 0 an = bn = 0 for n even

and

f t annnwt b nwt

nodd

()
[ ( ) ( )]

∑ cos 00 sin

where

a

T

n
 tnwtdt

4

0

0

2

f

T

(cos( ))



∫

and

b

T

n ft nwtdt n a n

T



n

4

0

0

2

()sin( )



∫ for 1, and 0 for all s

R.4.28 If f(t) is a real and even function of t, then the coeffi cients Fns are also real and

even.

R.4.29 If f(t) is a real and odd function of t, then the coeffi cients Fns are imaginary

and odd.

R.4.30 Recall that any arbitrary function f(t) can be expressed as

f(t) = fe(t) + fo(t) (see Chapter 1)

where

fe(t) = 0.5[f(t) + f(−t)] (even component of f(t))

and

fo(t) = 0.5[f(t) − f(−t)] (odd component of f(t))

R.4.31 By decomposing a signal (or system) into its frequency components, the BW of the

signal (or system) can be estimated, and the relative contributions and importance

of each frequency, or range of frequencies can then be estimated. Recall that the BW

represents a range of frequencies that can pass through a device, system, or com-

munication channel without signifi cant attenuation.

R.4.32 The exponential FS of an arbitrary signal f(t) states that f(t) can be decomposed into

components of the form

Fne n

jnw t (^0) for 0, 1, 2,
..., 
Then by applying each individual component as an input to a given system
(assuming the system is linear), and by evaluating the output of each individual