Homework answers / question archive / Maths Level 8 (MATH08013) Assignment [30% of Final Grade] Please complete all these tasks and make a single file and upload that file on the given link in the Moodle page

Maths Level 8 (MATH08013) Assignment [30% of Final Grade] Please complete all these tasks and make a single file and upload that file on the given link in the Moodle page

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Maths Level 8 (MATH08013) Assignment [30% of Final Grade] Please complete all these tasks and make a single file and upload that file on the given link in the Moodle page. There will be an extra tutorial to help you in understanding the MATLAB tasks. Deadline of Submission: 23 April 2023 Task 1. [01 x 05 = 05 marks] Use MATLAB to plot ?(?) = ln(?) for ? ∈ (0, 5], along with Taylor approximations around ? = 1 with ? = 0, 2, 4, 6. Task 2. [10 marks] Use an appropriate Taylor series to write a MATLAB function M-file that approximates ?(?) = sin(2?) for all x with a maximum error of 0.01. That is, your M-file should take values of x as input, return values of sin(2?) as output, but should not use MATLAB’s built-in sine function i.e., sin.m. Hint. You might find MATLAB’s built-in function mod useful in dealing with the periodicity. Task 3. [10 marks] Use numerical experiments using MATLAB to arrive at a conjecture about whether the following infinite series converges or not. ∑ 1 ?(ln ?) 2 ∞ ?=2 Task 4. [03+04+03 = 10 marks] Determine the exponential Fourier series of the waveform of the function f(x) shown in Figure 1. i. Obtain the expressions for the amplitude and phase spectra. ii. Using MATLAB, plot the amplitude and phase spectra for ? = 1, 2, 3, … , ?, where ? = 11. iii. Plot the reconstructed waveform of the function for ? = 1, 2, 3, … , ?, where ? = 11. Explain the Gibbs Phenomenon from the obtained plot. Figure 1. Plot of a square wave function. Task 5. [03+04+03 = 10 marks] An 8-point signal ?[?] is shown in Figure 2. Figure 2. Plot 8-point signal x[n]. i. Using MATLAB compute DFT of ?[?] i.e., ?[?]. ii. Plot DFT magnitude and phase as functions of k. iii. Compute and plot the IDFT of ?[?]. Task 6. [05+05 = 10 marks] Record the audio for first three letters of your name. e.g., M A H for MAHMOOD i. Save this audio file on MATLAB in vector x and plot vector x using plot(t, x). where t is time. ii. Perform DFT on vector x, using fft command in MATLAB and plot the amplitude spectrum. Hint: For M A H audio file the sample code and plots (in Figure 3) are given below: [x,Fs]= audioread(['C:\Users\mali\OneDrive - Atlantic TU\Documents\' ... 'Sound Recordings\MAH.m4a']); L=size(x,1); t=linspace(0,L/Fs,L); subplot(2,1,1) plot(t,x) title("MAH audio signal w.r.t time") xlabel("time (s)") ylabel("amplitude (a.u.)") Y = fft(x); P2 = abs(Y/L); P1 = P2(1:L/2+1); P1(2:end-1) = 2*P1(2:end-1); f = Fs*(0:(L/2))/L; subplot(2,1,2) plot(f,P1) title("Single-Sided Amplitude Spectrum of MAH") xlabel("f (Hz)") ylabel("|P1(f)|") Figure 3. Plot of Audio signal w.r.t time (top) and magnitude spectrum w.r.t frequency (bottom) Task 7. [03+03+04 = 10 marks] A causal, linear, and time-invariant system is given by the following difference equation: ?[?] = ?[? − 1] + ?[? − 2] + ?[? − 1] i. Find the transfer function ?(?) for this system. ii. Plot the poles and zeros of ?(?) and indicate the region of convergence (ROC). iii. Find the unit sample response ?[?] of this system. Hint: Using the sample shifting property ?{?[? − ?]} = ? −??(?) ; ???: ???? Use the command zplane(b,a) in MATLAB for plotting poles and zeros where (b, a) are transfer function coefficients, specified as row vectors. The transfer function is defined in terms of ? −1 : ?(?) = ?(?) ?(?) = ?(1) + ?(2)? −1 + ?(3)? −2 + ? + ?(? + 1)? −? ?(1) + ?(2)? −1 + ?(3)? −2 + ? + ?(? + 1)? −? z = roots(b);% finding zeros from the roots of b p = roots(a); % finding poles from the roots of a Task 8. [04+06 = 10 marks] By using MATLAB compute the following: i. If ?? = 4?? + ?? − 7??? and ??? = 3?? + 1 2 ?? − 7??? then find ?? ⋅ ???, ?? ⋅ ??, ??? ⋅ ??, ??? ⋅ ???, ?? × ???, ?? × ??, ??? × ??, and ??? × ???. Hint: use the commands of dot(A, B) and cross(A, B) in MATLAB ii. In each case determine whether the vectors are linearly dependent or linearly independent: (a) ?? = 2?? + ?? − 3??? , ??? = ?? − 4???, ?? = 4?? + 3?? − ??? (b) ?? = ?? − 3?? + 2??? , ??? = 2?? − 4?? − ???, ?? = 3?? + 2?? − ??? Hint: use the command of rref(matrix) and cross(A,B) in MATLAB. Reduced Row Echelon Form (RREF) takes the Gauss-Jordan elimination method one step further by performing scaling EROs on all rows so that the all coefficients on the diagonal all become ones.