Fourier and Laplace
393
line spectrum
frequency w (in rad/sec)
0.6
0.4
0.2
0.25
0.2
0.15
0.1
0.05
−0.2
0
Amplitude of Fn
−20
−15
−10
−5
0
5
10
15
20
power spectrum
0
Magnitude of Fn
2
−20
−15
−10
−5
0
5
10
15
20
frequency w (in rad/sec)
FIGURE 4.43
Plots of parts d and e of Example 4.4.
Note that the function f(t) in this example is similar to the function analyzed in
Exa mple 4.1. The differences are a DC shift in magnitude by 1 and a time shift
by 0.5 s.
Observe that simple shifts change the symmetry conditions of the function f(t), creating new harmonics.
Example 4.5
Create the script fi le Fourier_coeff that returns the fi rst fi ve coeffi cients of the exponential
FS (F 1 , F 2 , F 3 , F 4 , and F 5 ), of the function of Example 4.1, given by
f t
n
nw t
n
( )
(
)
ϭ
ϭ
2
1
7
0
odd
sin
∑
by using
1. Symbolic techniques
2. Numerical techniques
3. Compare the results of part 1 with part 2
MATLAB Solution
% Script file : Fourier _ coeff
T = 2;w0 = 2*pi/T; nn =1:5;
CRC_47760_CH004.indd 393
CRC_47760_CH004.indd 393
7/28/2008 12:26:10 PM
7/28/2008 12:26:10 PM