Signals and Systems - Electrical Engineering

352 CHAPTER 5: Frequency Analysis: The Fourier Transform

(a) Find the Fourier series coefficientX 0 for each of the values ofT 0 and indicate how it changes for the

different values ofT 0.

(b)Find the Fourier series coefficients forx(t)and carefully plot the magnitude line spectrum for each of

the values ofT 0. Explain what is happening in these spectra.

(c)If you were to letT 0 be very large, what would you expect to happen to the Fourier coefficients?

Explain.

(d)Write a MATLAB script that simulates the conversion from the Fourier series to the Fourier transform

of a sequence of rectangular pulses as the period is increased. The Fourier coefficients need to be

multiplied by the period so that they do not become insignificant. Plot usingstemthe magnitude line

spectrum for pulse sequences with periodsT 0 from 4 to 62.

5.2. Fourier transform from Laplace transform—MATLAB

The Fourier transform of finite-support signals, which are absolutely integrable or finite energy, can be

obtained from their Laplace transform rather than doing the integral. Consider the following signals:

x 1 (t)=u(t+0.5)−u(t−0.5)

x 2 (t)=sin( 2 πt)[u(t)−u(t−0.5)]

x 3 (t)=r(t+ 1 )− 2 r(t)+r(t− 1 )

(a) Plot each of the signals.

(b)Find the Fourier transforms{Xi()}fori=1, 2, and 3 using the Laplace transform.

(c)Use MATLAB’s symbolic functionfourierto compute the Fourier transform of the given signals. Plot

the magnitude spectrum corresponding to each of the signals.

5.3. Fourier transform from Laplace transform of infinite-support signals—MATLAB

For signals with infinite support, their Fourier transforms cannot be derived from the Laplace transform

unless they are absolutely integrable or the region of convergence of the Laplace transform contains thej

axis. Consider the signalx(t)= 2 e−^2 |t|.

(a) Plot the signalx(t)for−∞<t<∞.

(b)Use the evenness of the signal to find the integral

∫∞

−∞

|x(t)|dt

and determine whether this signal is absolutely integrable or not.

(c)Use the integral definition of the Fourier transform to findX().

(d)Use the Laplace transform ofx(t)to verify the above found Fourier transform.

(e) Use MATLAB’s symbolic functionfourierto compute the Fourier transform ofx(t). Plot the magnitude

spectrum corresponding tox(t).

5.4. Fourier and Laplace transforms—MATLAB

Consider the signalx(t)= 2 e−^2 tcos( 2 πt)u(t).

(a) Use the fact this signal is bounded by the exponential± 2 e−^2 tu(t)to show that the integral

∫∞

−∞

|x(t)|dt