Fourier and Laplace
323
need not be from 0 to T, or from 0 to 2π, but may be from −T to 0, −T/2 to +T/2, −2π to 0,
or −π to +π.
R.4.5 The FS converges uniformly to f(t) at all the continuous points and converges to its
mean value at the discontinuity locations.
R.4.6 The constant ω 0 is called the fundamental frequency, and all of its integer multiples
(nw 0 ) are referred to as harmonic frequencies. The fundamental frequency w 0 is the
lowest sinusoidal frequency in the FS expansion. All other frequencies are integer
multiples of the fundamental frequency.
R.4.7 The coeffi cient [a 0 / 2] is referred to as the DC component of the series, and represents the average value of f(t) over a period T.
R.4.8 The suffi cient conditions that ensure the existence and convergence of the FS
expansion for an arbitrary function f(t) are referred as Dirichlet’s conditions. These
conditions are
a. f(t) should present a fi nite number of discontinuities in any period
b. f(t) should present a fi nite number of maxima and minima over any period T
c. f(t) should be absolutely integrable over a period T, that is
f t dt k
( )
Ϫ
ϱ
T
T
2
2
∫
Ͻ Ͻ ,
where k is a fi nite quantity
d. f(t) must be single valued everywhere
All practical (electrical or mechanical) waveforms in nature satisfy the Dirichlet’s
conditions.
R.4.9 Let f(t) be a periodic function, then it can be represented by a sum of either cosine
or sine terms, indicated as follows:
f t
nw t
n
( )
cos(
sin
ϭ
ϩ
ϩ
ϭ
ϩ
ϩ ϩ
ϭ
ϩϱ
a
c
n t n
a
c
n
n
n
0
0
1
0
2
)
2
∑
0
2 2
1
∑
nϭ
ϱ
where c
a
b
b a
n
n
n
n
n
n
ϭ
ϩ
ϭ
2
2
1
and
Ϫ
Ϫ
tan (
).
Note that the FS is an expansion of f(t), over the range −∞ ≤ t ≤ +∞ (everywhere).
R.4.10 Let f(t) be a periodic function, then it can be expressed in terms of an exponential
FS as indicated in the following (by replacing the sinusoids of FS by the Euler’s
identities):
f t
F e
n
n
n
jnw t
( ) ϭ
ϭ
ϭϩ
Ϫϱ
ϱ
∑
0
where
F
T
f t e
dt
n
jnw t
T
ϭ
1
0
0
( )
Ϫ
∫
R.4.11 The complex exponential coeffi cients F n can be evaluated in terms of the trigonometric coeffi cients (a n s and b n s), by replacing the exponential e
−jw 0 t with cos(nw 0 t) +
j sin(nw 0 t), and by equating the trigonometric FS with the exponential FS, obtaining the following relations:
a n = 2 real{F n }
b n = −2 imag{F n }
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323
need not be from 0 to T, or from 0 to 2π, but may be from −T to 0, −T/2 to +T/2, −2π to 0,
or −π to +π.
R.4.5 The FS converges uniformly to f(t) at all the continuous points and converges to its
mean value at the discontinuity locations.
R.4.6 The constant ω 0 is called the fundamental frequency, and all of its integer multiples
(nw 0 ) are referred to as harmonic frequencies. The fundamental frequency w 0 is the
lowest sinusoidal frequency in the FS expansion. All other frequencies are integer
multiples of the fundamental frequency.
R.4.7 The coeffi cient [a 0 / 2] is referred to as the DC component of the series, and represents the average value of f(t) over a period T.
R.4.8 The suffi cient conditions that ensure the existence and convergence of the FS
expansion for an arbitrary function f(t) are referred as Dirichlet’s conditions. These
conditions are
a. f(t) should present a fi nite number of discontinuities in any period
b. f(t) should present a fi nite number of maxima and minima over any period T
c. f(t) should be absolutely integrable over a period T, that is
f t dt k
( )
Ϫ
ϱ
T
T
2
2
∫
Ͻ Ͻ ,
where k is a fi nite quantity
d. f(t) must be single valued everywhere
All practical (electrical or mechanical) waveforms in nature satisfy the Dirichlet’s
conditions.
R.4.9 Let f(t) be a periodic function, then it can be represented by a sum of either cosine
or sine terms, indicated as follows:
f t
nw t
n
( )
cos(
sin
ϭ
ϩ
ϩ
ϭ
ϩ
ϩ ϩ
ϭ
ϩϱ
a
c
n t n
a
c
n
n
n
0
0
1
0
2
)
2
∑
0
2 2
1
∑
nϭ
ϱ
where c
a
b
b a
n
n
n
n
n
n
ϭ
ϩ
ϭ
2
2
1
and
Ϫ
Ϫ
tan (
).
Note that the FS is an expansion of f(t), over the range −∞ ≤ t ≤ +∞ (everywhere).
R.4.10 Let f(t) be a periodic function, then it can be expressed in terms of an exponential
FS as indicated in the following (by replacing the sinusoids of FS by the Euler’s
identities):
f t
F e
n
n
n
jnw t
( ) ϭ
ϭ
ϭϩ
Ϫϱ
ϱ
∑
0
where
F
T
f t e
dt
n
jnw t
T
ϭ
1
0
0
( )
Ϫ
∫
R.4.11 The complex exponential coeffi cients F n can be evaluated in terms of the trigonometric coeffi cients (a n s and b n s), by replacing the exponential e
−jw 0 t with cos(nw 0 t) +
j sin(nw 0 t), and by equating the trigonometric FS with the exponential FS, obtaining the following relations:
a n = 2 real{F n }
b n = −2 imag{F n }
CRC_47760_CH004.indd 323
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