PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 403

Example 4.9

Let f(t) = e−^2 tu(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 [ ( )]ft
    0 ft dt( )
   (test the existence of the FT)
   b. Energy of [ ( )]ft
    0 ft dt( )^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

(^246810)
− 10 − 8 − 6
1
0.5
0
− 4 − 2 0 2 4 6 8 10
− 3 − 2 − 1 01234
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