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
a n = b n = 0 for n even
and
f t
a
nw t b
nw t
n
n
n odd
( )
[
(
)
(
)]
ϭ
ϩ
ϭ
cos
sin
0
0
∑
where
a
T
t
nw t dt
n ϭ
4
0
0
2
f
T
( cos(
)
)
ր
∫
and
b
T
f t
nw t dt
n
a
n
n
T
n
ϭ
ϭ
ϭ
4
0
0
2
( ) sin(
)
ր
∫
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 F n s are also real and
even.
R.4.29 If f(t) is a real and odd function of t, then the coeffi cients F n s 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 communication 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
Ϫ
Ϯ Ϯ
Ϯϱ
0
for
0, 1, 2,
ϭ
…,
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
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