Question

1. Consider the following sequence (The bold large size element is at n = 0) x(n) = { ----- 0,0,01, 1, 1, 1, 1, 0, 0, 0------}. It has the period of N=8. a. Find (in closed form) and sketch the complex exponential Fourier series coefficients (Magnitude and Phase). Verify using Matlab. Include code and graphics. b. Find the energy in this sequence first using the sequence in the time domain, and then using the DTFS coefficients. Verify using Matlab. Include code and graphics.

Answer 1

feel free to ask doubts in the comment section

N -1 A real - periodic descrete time aagnal nin) can be represented by teneat. combination of complex exponentials iwon juon eroon ... IN-1) Won w oo 27 &xm). an ei kuin where ok's are the jeweies reseves coefficients and 20th xwe jkoson for the given signal Cu o 3 200 insan oraš xin) je na no colit putih leis*, pih) te 153 (, 15) - b (licht, ces (pte), eithes.co( t)) for de file (cu( 24). c(**) Now subetituting ka 0:7 un ar., we get a=0.625 ; a= 0.3018; a = -0.1250 ; A= -0.0518 ay: 0.1250 ,052 -0.0518; a = – 0.1250 : 00.3018.

6 no Energy of a desacte tone signal ain) - - 1.5 = 0.625 Energy is frequency domain lak 1² + (0-6850)? 1910-3018)”, (0.125)?: (0.0518)) (0.125)² -0.625

clc;
close all;
clear;
x = [1 1 1 0 0 0 1 1];
syms k;
ak=0;
N = 8;
w0 = 2*pi/N;
for i=1:N
   ak = ak + x(i)*exp(-1i*k*w0*(i-1));
end
ak = ak/8;
derived_ak = (1+2*exp(-1i*k*pi)*(cos(3*k*pi/4) + cos(pi*k/2)))/8;
k_range = 0:7;
disp("Discrete Fourier Series coefficients = ");
disp(double(subs(ak,k,k_range)));
disp("Discrete Fourier Series coefficients as derived = ");
disp(double(subs(derived_ak,k,k_range)));

disp("Energy using sequence in time domain = ");
disp(sum(x.^2)/N);
disp("Energy using Discrete fourier series coefficients = ");
disp(double(sum(subs(ak,k,k_range).^2)));

Command Window Discrete Fourier Series coefficients - 0.6250 0.3018 -0.1250 -0.0518 0.1250 -0.0518 -0.1250 0.3018 Discrete Fourier Series coefficients as derived = 0.6250 0.3018 -0.1250 -0.0518 0.1250 -0.0518 -0.1250 0.3018 Energy using sequence in time domain = 0.6250 Energy using Discrete fourier series coefficients = 0.6250