Part A We will consider a signal x(t), which is composed of a linear combination of complex exponentials, with different

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Part A We Will Consider A Signal X T Which Is Composed Of A Linear Combination Of Complex Exponentials With Different 1 (87.88 KiB) Viewed 21 times

Part A We will consider a signal x(t), which is composed of a linear combination of complex exponentials, with different amplitudes, frequencies and phases: N x(t) = ΣΧe-lage k=1 where Xk are complex numbers. You will write a Matlab function named Sum ComplexExp which takes Xk, lk and the time instants t that x(t) will be computed. A sample of this kind of function is given below: EDITOR PUBLISH VIEW Go To FILE RUN La Find Files Insert B fx FA Run Section Compare Comment % New Open Save Breakpoints Run Run and Advance Run and Print Find Indent Advance Time NAVIGATE EDIT BREAKPOINTS SumComplexExp.mx + 1 function [x] = SumComplexExp (t, xk, omegak) 2 Pt: 1xT vector that contains the time instants over which x (t) will be computed. 3 Xk: 1xN complex-valued vector. kth element is X_k. 4 - omegak: 1x vector kth element is a_k. 5 6 swrite your function here: 7 8- end 9 10 You can use length function of the Matlab to determine N when Xk is given. 4 After writing your function, take t=[0:0.002:2), XK=[-2j 1-j 1+j 2 ] and omegak=LIA] Using your function, compute the signal x, obviously it will be complex valued. Then, extract the real and imaginary parts of x. Plot the real and imaginary parts (versus t) separately. Also, extract the magnitude and phase of x and plot them separately. In all your plots, put the labels and titles properly.

Part B The Fourier Series representation of a periodic signal X(t) of fundamental period is given by an infinite sum of weighted complex exponentials (cosines and sines) with frequen- cles multiples of the fundamental frequency S2 = 2/T. (rad/sec) of the signal x(t) = Σ Χ, eΩst (4.13) where the Fourier coefficients (X) are found according to Xx - 7L x(t) endt (4.14) for k=0, +1, +2... and any to the form of the last equation indicates that the information needed for the Fourier series can be obtained from any period of X(t). In the Fourier series expansion, as it is seen in the equation (4.13 from the book), there are infinite number of complex exponentials. However, in real life, summation of infinite number of sinusoids cannot be used because of the practical problems. Therefore, we can add only finite number of sinusoids, in this part we will see the effect of adding finite number of sinusoids. M z (t) = Xelk21 k=-M Obviously, in that new case, z(t) will be approximation of x(t). Suppose we have a periodic x(t) with period T where <t<T/21 x(t) = = {-0.5-9423750 Find the Fourier series expansion coefficients of x(t), by using the formula given above (4.14). Try to get a simple result by doing some conversions. Write a Matlab Function that computes z(t) for the given time instants, which has a name FourierSynthesis like the one shown below: • RUN EDITOR PUBLISH VIEW Find Files Insert fx Run Compare Go To Comment New Open Save Breakpoints Run Print - Run and Ad Find indent Advance FILE NAVIGATI Еріт BREAKPOINTS FourierSynthesis.mx + El function (z] - FourierSynthesis (t. M, Omega) att denotes the time period which z(t) is computed. denotes the values of z(t) computed over t. Mand Omegao denote the parameters M and Q0 that appear in formula for 5 z(t) in the assignment. 6 7 Use Sumcomplexexp function B 9 *Write your function here 10 11 - end 12 13 . Plot real part of z(t) when M=10, T=1 (120=21/T), -2<<<2. Does it resemble x(t)? Explain the difference and its causes. By using stem function of the Matlab, plot the magnitude of the Fourier series coefficients of z(t) when M=10, T=1 (920=21c/T), -2<t<2; label the axes properly. • Plot real part of z(t) when M=20, T=1 (20=21/T), -2<<<2. Does it resemble x(t)? Explain the difference and its causes. • Plot real part of z(t) when M=50, T=1 (120=25/T), -2<<<2. Does it resemble x(t)? Explain the difference and its causes. • Plot real part of z(t) when M=1000, T-1 (920=21/T), -2<<<2. Does it resemble x(t)? Explain the difference and its causes. Plot the imaginary part of z(t) when M=10, T=1 (920=210T), -2<t<2. Is imaginary part of z(t) 0? Since x(t) is a real valued signal, imaginary part of z(t) should be zero, explain why it is not • zero.