PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

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 Fns 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 Fns 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

ft ejnw t

n

()
∑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 symmet-

ric, 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 regard-

ing 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

ft()
 ancos(nwt 0 )

n

∑

where

a

T

n ft nwtdt n

T

4

0

0

2

()cos( )



∫ for 1, and all the coefficients bn
^0