Fourier and Laplace
327
A one-sided spectrum consists of only the positive frequencies w ≥ 0, whereas
the two-sided spectrum is over −∞ ≤ w ≤ +∞. In this book, the term spectrum
refers to the two-sided spectrum.
R.4.23 The plot of Fn versus nw 0 is called the line or discrete spectrum of f(t).
Since Fn is, in general, a complex function, two plots are required to completely
defi ne its behavior. They are referred to as
a. The magnitude spectrum plot
b. The phase spectrum plot
R.4.24 If the line (spectrum) representing the magnitudes of the coeffi cients F n s decrease
rapidly, then the FS converges rapidly to f(t), implying that the wave is continuous.
If, on the contrary, f(t) has discontinuities, then the F n s slowly decrease, and f(t) is
referred to as having strong high harmonic components, implying that many terms
are required for a good approximation of f(t).
R.4.25 For example,
Let
f t
e
jnw t
n
( ) ϭ Fn
0
∑
Then the generic sketches shown in Figures 4.1 and 4.2 represent typical plots of
the following:
a. Line spectrum
b. Phase spectrum
c. Magnitude spectrum
d. Power spectrum, assuming that T = 1 s
ANALYTICAL Solution
The generic spectrum plots are shown in Figures 4.1 and 4.2.
Note that F 0 = 3, F 1 = F – 1 = 2, F 2 = F – 2 = 0.5 and F 3 = F – 3 = 1.
R.4.26 Observe from the plot of Figure 4.1 that the amplitude spectrum is always symmetric, that is, Fn = F − n, whereas the phase spectrum is always asymmetric, given by
θn = −θ − n.
R.4.27 One of the features of the exponential FS coeffi cients is their symmetry regarding the variables t and nw 0 . Symmetry considerations are of great computational
advantage in evaluating the components of f(t) as indicated in the following:
a. If f(t) = f(−t), indicating that f(t) is an even function with respect to t, then
f t
a
nw t
n
( )
(
)
ϭ
cos
0
n
∑
where
a
T
f t
nw t dt
n
n
T
ϭ
4
0
0
2
( )cos(
)
ր
у
∫
for
1, and all the coefficients
0
b n ϭ
CRC_47760_CH004.indd 327
CRC_47760_CH004.indd 327
7/28/2008 12:25:51 PM
7/28/2008 12:25:51 PM
327
A one-sided spectrum consists of only the positive frequencies w ≥ 0, whereas
the two-sided spectrum is over −∞ ≤ w ≤ +∞. In this book, the term spectrum
refers to the two-sided spectrum.
R.4.23 The plot of Fn versus nw 0 is called the line or discrete spectrum of f(t).
Since Fn is, in general, a complex function, two plots are required to completely
defi ne its behavior. They are referred to as
a. The magnitude spectrum plot
b. The phase spectrum plot
R.4.24 If the line (spectrum) representing the magnitudes of the coeffi cients F n s decrease
rapidly, then the FS converges rapidly to f(t), implying that the wave is continuous.
If, on the contrary, f(t) has discontinuities, then the F n s slowly decrease, and f(t) is
referred to as having strong high harmonic components, implying that many terms
are required for a good approximation of f(t).
R.4.25 For example,
Let
f t
e
jnw t
n
( ) ϭ Fn
0
∑
Then the generic sketches shown in Figures 4.1 and 4.2 represent typical plots of
the following:
a. Line spectrum
b. Phase spectrum
c. Magnitude spectrum
d. Power spectrum, assuming that T = 1 s
ANALYTICAL Solution
The generic spectrum plots are shown in Figures 4.1 and 4.2.
Note that F 0 = 3, F 1 = F – 1 = 2, F 2 = F – 2 = 0.5 and F 3 = F – 3 = 1.
R.4.26 Observe from the plot of Figure 4.1 that the amplitude spectrum is always symmetric, that is, Fn = F − n, whereas the phase spectrum is always asymmetric, given by
θn = −θ − n.
R.4.27 One of the features of the exponential FS coeffi cients is their symmetry regarding the variables t and nw 0 . Symmetry considerations are of great computational
advantage in evaluating the components of f(t) as indicated in the following:
a. If f(t) = f(−t), indicating that f(t) is an even function with respect to t, then
f t
a
nw t
n
( )
(
)
ϭ
cos
0
n
∑
where
a
T
f t
nw t dt
n
n
T
ϭ
4
0
0
2
( )cos(
)
ր
у
∫
for
1, and all the coefficients
0
b n ϭ
CRC_47760_CH004.indd 327
CRC_47760_CH004.indd 327
7/28/2008 12:25:51 PM
7/28/2008 12:25:51 PM