Fall 03 EE3350 Homework #1



Fall 16 EE/TE 3302 Homework 4 Due: ?

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 () |(| < 1000 (

0. 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) T= 0.1 second

and H(ω) = [rect (ω/60π).] exp (- j ω/40)

determine the output y(t). Sketch the spectrum Y (ω).

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