Question

Part 1: Exponential Fourier series The following MATLAB code calculates the exponential Fourier series coefficient for the signal x(t) shown in the figure below, plots it's double sided amplitude spectrum IDn l, double sided phase spectrem LDn, and the resulting signal xn(t). 4r 4a Periodic signal x(t) 1.1 Show that the complex Fourier Series Coefficients written as: D 1.2 Use the following Matlab to general the two sided spectral line. 1.3 Execute the Matlab code with To = 2π and N = 10 harmonics. Copy the output and the graphs. 1.4 Discuss the effect of number of harmonics N from 5, 10 to 20.

Answer 1

lt) 0.5 n ?uit 0

No 72 -2A

o Jnrt

Matlab Code for the solution clear all close all 7-input ( ' Fundamental Period T- ') ; %period N-input (' No. of FS Coefficients. N '); % Number of coefficients to-1/T; % Fundamental Frequency. wo=2*pi * fo ; t--2 T:T/1000:2*T complex Exp. Fourier Series Do-0.5; for n-1:N x-EeX-E +Dn(n) *exp ( 1 * n * wo * t) +D-n (n) *exp (-1*n*wo * t ) ;% Complex form Exp . end n=1 : N figure (1) plot (t, real (X_E) ) grid title('Comp. Exponential FS') xlabel ('time') ylabel ('x_ _n(t) ') figure (2) subplot (2,1,1) stem ( [-n 0 n], [angle (D-n) angle (Do) angle (Dn) ] ) ; grid title ('Two sided Phase spectra') xlabel ('n') ylabel ('KDn (rad) ') subplot (2,1,2) stem ( [-n 0 n] , [abs (Dn) abs (Do) abs (Dn) ] ) ; grid title ('Two sided Amplitude Spectra') xlabel ('n') ylabel (D n) ' -

Comp. Exponential FS 1.2 0.6 0.4 0.2 15 10 -0.2 -5 10 15 time Approximated plot of x(t) for N-5 Harmonics

Comp. Exponential FS 1.2 0.8 0.6 0.2 Nu -0.2 15 10 .5 10 15 time Approximated Fourier Series of x(t) for N-10 Harmonics using MATLAB

Comp. Exponential FS 1.2 0.6 0.4 15 10 .5 -0.2 10 15 time Approximated plot of x(t) for N 20 Harmonics using Matlab

Two sided Phase spectra -5 -3 -2 Two sided Amplitude Spectra 0.5 0.4 0.2 -5 Frequency spectrum for N-5 harmonics

Two sided Phase spectra 10 10 -8 -6 -2 Two sided Amplitude Spectra 0.5 0.4 0.3 0.2 10 10 -8 -6 .2 Frequency spectrum for N-10 Harmonics

Comp. Exponential FS 1.2 0.6 0.4 15 10 .5 -0.2 10 15 time Approximated plot of x(t) for N 20 Harmonics using Matlab

As the consideration of number of harmonics increases in the waveform of x(t) then, the approximated wave takes the shape similar to that of original waveform of x(t)