Spring 09 EE/TE 3302
Homework 4
Due: April 2, 2009
Submit only the problems with *.
1. The spectrum of a signal g(t) is given by X(ω) = rect [(ω-1)/2], find the
transform of the following signals, using the properties of the Fourier
Transform.
(a) x(2t-1) exp (-j2t)
(b) x(t) cos 5t
(c) x(-t)
(d) * x(-2t+4)
2*. Determine x(t) given that
(a) X() = rect [(-4)/2] + rect [(+4)/2]
(b) X() = Δ [(-8)/2] + Δ [(+8)/2]
3. Determine the Fourier Transform of the following signals, and sketch their
magnitude and phase spectra:
(a) g(t) = sin (100t - /4)
(b) g(t) = cos (100t + /4)
(c) g(t) = cos (100 t + /4) cos 1000t
(d)* g(t) = sin (100t - /4) cos 1000t
(e)* g(t) = cos (100t + /4) sin (1000t - /8)
4. Determine the inverse Fourier Transform of
X () = [5/(3+ j )2].
5.* The frequency response of an ideal low pass LTI system is:
H () = 10 exp (-j 0.0025 )
0
|| < 1000 
Otherwise.
In each of the following cases, determine the Fourier Transform of the input
signal and then use frequency domain methods to determine the corresponding
output signal.
(a) Input x(t) = cos (200  t) + [2 sin (2000  t)/  t]
(b) Input x(t) = cos (200  t) + [2 sin (2000  t)/  t] + cos (3000  t)
(c) Input x(t) = cos (200  t) + 2  (t)
6. The input and the output of a relaxed (i.e. initial conditions are zero), stable and
causal LTI system are related by the differential equation
y’’(t) + 6 y’(t) + 8 y(t) = 2 x(t).
(a) Find the impulse response of this system.
(b) What is the response of this system if x (t) = exp(-2t) u(t)?
7. A causal and stable LTI system has the frequency response:
H () = [(j + 4) / (6 - 2 + 5 j)].
(a) Determine the differential equation relating the input x(t) and the output y(t) of
this system.
(b) Determine the impulse response h (t) of the system.
8.* A signal x(t) is input to a relaxed linear time invariant system with the
spectrum H(ω) to produce an output y(t). Given the following
x (t) = δT(t)
and
T= 0.1 second
H(ω) = [rect (ω/60π).] exp (- j ω/40)
determine the output y(t). Sketch the spectrum Y (ω).