1. Consider the signal
f(t) e-A cost, -00<<<00.
(a) Use MATLAB to draw a picture of this signal over a suitable range for f, and classify
the signal as odd, even, or neither. Include your code.
(b) Find the Fourier Transform r(a) for this signal.
(c) Use MATLAB to draw the amplitude spectrum of ! over an appropriate spectral
range. (That is, draw
(d) Write down Plancherel's identity for this signal. Choose one of the two integrals to
evaluate. and determine the signal's energy.
2. Consider the signal
f(t) cos(2t) 3 sin(5t) sin(6t).
(a) What is the highest angular frequency present in this signal? What is the highest
numerical frequency present in this signal?
(b) What is the Nyquist rate for this signal? Did you use the angular or the numerical
(c) If you sample this signal with sampling period T. which values of T may you choose
to be in accordance with the Nyquist rate? Choose and fix one such T.
(d) After sampling you pass the sampled signal through a low-pass filter. Which threshold
Mo can be used in the low-pass filter?
3. A signal f(t) was sampled with sampling period T 0.1. in accordance with the Nyquist
rate. However, when the signal was reconstructed using Shannon's formula, a mistake was
made, and the reconstructed signal was
(0.2) I f(0.1n)
Assuming that the used value of Mo was correct, answer:
(a) Where is the error in the reconstruction?
(b) Describe, in words, what is the relationship between signals f and g.
As seen in the exercises from Topic 6. the signal f(t) -2|| has Fourier Transform
This signal is not band-limited, but the amplitudes decay reasonably fast as Jal grows.
For example, for jal > 4 we have < 0.007. It makes you wonder if it is possible to
treat this signal essentially as a band-limited signal. Can we?
(a) Verify, with full justification, the assertion that |/(a)| < 0.007 for all o
(b) Choose M = 4. and treat it as the highest frequency in the signal f. What would be
the Nyquist rate in this case? Which sampling period T would then be appropriate?
(c) Using M = 4 and the value of T obtained above, use Shannon's Sampling Formula
and MATLAB to reconstruct the signal f. Print f. overlayed with its reconstruction,
over an appropriate time range.
(d) Explain as best you can what happened. Do you think the reconstruction was at
least partially successful?
5. Let f(t) be a periodic signal with period T. The signal f is not random. Let 9 be a
random phase, uniformly distributed on [0.T]. We define the deterministic random signal
(a) Compute E[X(t)] and show that its value does not depend on t.
(b) Compute Cor(x(t),X(I+++)). and show that its value does not depend on t.
6. (a) Use the Laplace Transform method to solve the following differential equation problem;
y(0) 2, s/(0) 1.
(b) Use the z-transform method to solve the following difference equation:
y[n + 2] = 3y[n + 1] - 2y(n], y 5. y0.
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title('plot of f(t)')
title('plot of Fourier transform')