Signals and Systems - Electrical Engineering

Problems 353

is finite, indicating the signal is absolutely integrable and also finite energy.

(b)Use the Laplace transform to find the Fourier transformX()ofx(t).

(c)Use the MATLAB functionfourierto compute the magnitude and phase spectrum ofX().

5.5. Fourier transform of causal signals
Any causal signalx(t)having a Laplace transform with poles in the open-lefts-plane (i.e., not including the
jaxis) has, as we saw before, a region of convergence that includes thejaxis, and as such its Fourier
transform can be found from its Laplace transform. Consider the following signals:

x 1 (t)=e−^2 tu(t)

x 2 (t)=r(t)

x 3 (t)=x 1 (t)x 2 (t)

(a) Determine the Laplace transform of the above signals (use properties of the Laplace transform)

indicating the corresponding region of convergence.

(b)Determine for which of these signals you can find its Fourier transform from its Laplace transform.

Explain.

(c)Give the Fourier transform of the signals that can be obtained from their Laplace transform.

5.6. Duality of Fourier transform
There are some signals for which the Fourier transforms cannot be found directly by either the integral
definition or the Laplace transform, and for those we need to use the properties of the Fourier transform, in
particular the duality property. Consider, for instance,

x(t)=

sin(t)

t

or the sinc signal. Its importance is that it is the impulse response of an ideal low-pass filter.

(a) LetX()=A[u(+ 0 )−u(− 0 ]be a possible Fourier transform ofx(t). Find the inverse Fourier

transform ofX()using the integral equation to determine the values ofAand 0.

(b)How could you use the duality property of the Fourier transform to obtainX()? Explain.

5.7. Cosine and sine transforms
The Fourier transforms of even and odd functions are very important. The reason is that they are
computationally simpler than the Fourier transform. Letx(t)=e−|t|andy(t)=e−tu(t)−etu(−t).
(a) Plotx(t)andy(t), and determine whether they are even or odd.
(b)Show that the Fourier transform ofx(t)is found from

X()=

∫∞

−∞

x(t)cos(t)dt

which is a real function of, thus its computational importance. Show thatX()is also even as a

function of.

(c)FindX()from the above equation (called the cosine transform).

(d)Show that the Fourier transform ofy(t)is found from

Y()=−j

∫∞

−∞

y(t)sin(t)dt