Fourier and Laplace
403
Example 4.9
Let f(t) = e
−2t u(t), evaluate by hand
1. The FT of f(t) ↔ F(w), using Table 4.1
2. The LT of f(t) ↔ F(s), using Table 4.2
3. The analytic expressions of |F(w)| and |F(s)| (magnitude), and angle[F(w)] and
angle[F(s)] (phase)
4. Create the script fi le Fourier_Laplace that returns the following plots:
a f(t) versus t, over −2 ≤ t ≤ 2
b [f(t)]
2 versus t (power, assuming R = 1 Ω and f (t) is either a current or a voltage)
5. Evaluate using MATLAB
a. Area of [ ( )]
( )
f t
f t dt
ϭ
ϩϱ
0
(test the existence of the FT)
b. Energy of [ ( )]
( )
f t
f t dt
ϭ
ϩϱ
0
2
6. Evaluate the Fourier and Laplace transforms. F(w) and F(s), using the MATLAB
symbolic toolbox
7. The symbolic and numerical plots of |F(w)| versus w and |F(s)| versus s
FIGURE 4.51
Symbolic plots of part 3 of Example 4.8.
−4
−10
−8
−6
−4
−2
0
Spectrum of the pulse for tau=10
Spectrum of the pulse for tau=5
(Symbolic solution)
Spectrum of the pulse for tau=1.0
2
4
6
8
10
−10
−8
−6
1
0.5
0
−4
−2
0
2
4
6
8
10
−3
−2
−1
0
1
2
3
4
10
5
0
−5
frequency w (in rad/sec)
frequency w (in rad/sec)
frequency w (in rad/sec)
Magnitude
Magnitude
Magnitude
4
2
0
−2
−4
6
CRC_47760_CH004.indd 403
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7/28/2008 12:26:12 PM
7/28/2008 12:26:12 PM