# Mathematical Methods - Problems

## Transcribed Text

1. Consider the signal f(t) = te􀀀2jtj; 􀀀1 < t < 1 (a) Is this signal time-limited? Explain. (b) Use MATLAB to plot a labelled picture of the signal for 􀀀5 < t < 5. (c) Is this signal odd, even or neither? Justify your answer. (d) Use a table of integrals or calculate by hand the Fourier Transform b f( ). Simplify your answer. (e) Use MATLAB to plot a labelled picture of the amplitude spectrum of the signal, j b f( )j over an appropriate spectral range. (f) Is this signal band-limited? Explain. (g) What is the energy of this signal? Write down the Plancherel's identity and calculate both integrals. You can use MATLAB to calculate them. 2. This exercise is a typical nal exam question. Consider a signal g(t) = 3 cos(2t) 􀀀 2 sin(5t) 􀀀 cos(8t): (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 x one such T. (d) After sampling you pass the sampled signal through a low-pass lter. Which threshold M0 can be used in the low-pass lter? 3. Let's return to the signal f(t) from Question 1. We will attempt to reconstruct this signal using Shannon's formula. (a) Find a value of M such that if j j > M, then j b f( )j < 10􀀀3. Justify your answer. (b) Fix the M that you chose in the last item, and treat it as the highest frequency in the signal f(t) (although it really isn't. Why?). What would be the Nyquist rate in this case? Which sampling period T would then be appropriate? Choose a sampling period T, and x it. (c) You need to choose an appropriate value M0 > 0 to serve as the low-pass lter threshold for the sampled signal. What is the minimum value that M0 can have? What is its maximum value to avoid aliasing (in theory)? Choose your M0 and x it. (d) With the help of MATLAB use the values of M0 and T chosen/obtained above, to reconstruct the signal f(t) with Shannons Sampling Formula (choose N). Plot the reconstructed signal by itself, then plot it superimposed over the original signal f(t). (e) How good is your reconstruction? Why? 4. This exercise is a typical nal exam question. Use the Laplace Transform method to solve the following di erential equation problem: y00(t) 􀀀 y(t) = t + sin(t); y(0) = 0; y0(0) = 1 (15 marks) 5. This exercise is a typical nal exam question. Use the z-transform method to solve the following di erence equation: y[n + 2] = 3y[n + 1] 􀀀 2y[n]; y[0] = 5; y[1] = 0 (15 marks) 6. This exercise is a typical nal exam question. You have two 5-sided fair dice. If you roll any individual die, the possible results are 1, 2, 3, 4, or 5 each equally likely. Let A1 be the random result on the rst die, and A2 be the random result on the second die. We de ne the random variable A = A1 +A2, the sum of values of two dice. Assume the signal X(t) = A for all times t. (a) Is A a discrete or a continuous random variable? Why? (b) What are the possible values that A may take? Make a table. (c) Find P(A = x), for each possible value of x. (d) Calculate E(A). (e) Find V ar(A). (f) Is E[X(t)] independent of t? Explain. (g) Is Cov(X(t);X(t +  )) independent of t? Explain. (h) Is the signal X(t) ergodic? Justify. 7. Bonus question. What is the di erence between FM and AM radio transmission? Is FM radio better than AM? Why do we still have both AM and FM radio? Explain brie y, list references of your sources.

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% 1.
%
% a) No, since f(t)=0 only for t = 0 (for time limited signals, f(t)= 0 for t
% outside some time interval [t1,t2].
%
% c) Since f(-t) = - t*exp(-2*|-t|) = - t*exp(-2*t) = -f(t).
%
% f) It is not bandlimited since fhat(a) is never = 0 (for band-limited signals
% fhat(a) = 0 for a outside some segment [a1,a2].

% b)

f = @(t) t.*exp(-2.*abs(t));
t = -5:0.01:5;

figure(1);
plot(t,f(t));
xlabel('$t$','interpreter','latex');
ylabel('$f(t)$','interpreter','latex');
title('Signal');

% e)

fhat = @(a) -1i* pi*a ./ (1+pi^2*a.^2).^2;
a = -3:0.01:3;

figure(2);
plot(a,abs(fhat(a)));
xlabel('$\alpha$','interpreter','latex');
ylabel('$|\hat f(a)|$', 'interpreter','latex');
title...
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