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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. 25 marks] 2. Consider the signal 1 f(t) cos(2t) 3 sin(5t) sin(6t). 2 (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 frequency? (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? [15 marks] 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) w(t-0.2m) 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. [10 marks 4. As seen in the exercises from Topic 6. the signal f(t) -2|| has Fourier Transform given by 1 f(a) 1+m²o² 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 > 4. (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? 25 marks 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 X(I)=1(1+8). (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. 20 marks] 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[0] 5. y[1]0. 25 marks]

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title('plot of f(t)')
title('plot of Fourier transform')
syms x
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