Question

Can you write hand solutions and matlab code please?

Q3) There is a square wave with the period of T=21. The amplitude is 3 between 0 and it The amplitude is 0 between 1 and 2 T. 2 3 4 5 a) Find the Fourier series coefficients of this square wave. a = x()ật Il x(t)cos(kt)dt bx = ( x(t)sin(kt)dt b) Using Matlab, plot the truncated Fourier series; (1) For k 0 to 10 (II) For k 0 to 1000 (III) For k 0 to 10000 c) Compare results of I-11-1ll and write your comments. Does it resemble the square wave?

Answer 1

in the 0 time it) IT 21 37 time it) ТЭТ Solutions Giren ware form of act) The Xlt is defined for a fundamental period - XCH) = 3 ; oct IT 0 Tet<27 fundamental Period T = 21 sec fundamental frequency wo = 247 = + to Now Calculating Trigonometuc founier coeffudents - moes Under the Curre over a period Aase time period 2 - Brat -15 qk = { s au cws wat dit 2 xHH Coskrot at $ l! "Sent dt] +0 - s" z moskt at z $ Simkt 1". = | Sinkos - Sino] Tak = 0 (44, Sin Ko so ;H K=1,2,3--) ofs, QK =0 indicates Gouren Waretom is odd in nature Now,

Calculation of 'bk' Inkt dit to bx < 4 **i sin hiot et = } ["3 sin kk at ] +0 [-takt | ] - Khi ( cas Kon – Coso] =0(Cos 0 -Coo KAT ) [box = (1-c-14) for Krodd Bk = por 2 TK k=eren bk=0 ne Obhained coefficients - 20 = 15 QK =o bk = 6 KIT .. K=odd of Trigonomervie fruit secies of 24 - 21+ = Qo+ (an Co-Rust tbk Siskust) 14) = 1:57, han sin et Any ca) Using MATLAB we can plot the Truncated fourion series of MH

MATLAB CODE FOR SOLUTION clc clear all close all T=2*pi; Let the Time period of waveform be 2*pi second wo=2*pi/T; Fundamental frequency in rad/sec A=3; Peak Amplitude assumed to be 3 ao=1.5; $D.C Term of the waveform K=input('Enter the number of harmonics for approximation: '); t = 0:0.01:20; Defining the time vector x=0*t; for k=1:K bk = A* (1-сos (k*pi))./(k*pi); computes the k-th Fourier coefficient of the Trigonometric form x_k = bk*sin (k*wo*t); kth term of the series x = x + x_k; adds the k-th term to x(t) end Approximation of Fourier series of x(t) for N harmonics. f=x+ao; plot(t,f, 'r', 'linewidth', 2) xlabel('\bf time (t)'); ylabel('\bf Amplitude'); title (strcat('\bf Fourier series of the x_{k} (t) with K=', int2str (K), 'harmonics.')); grid on

Fourier series of the x (t) with K=10 harmonics. Amplitude ------ --4 --- 10 14 18 time(t) Truncated Fourier series for x(t) for K=10 using MATLAB

Fourier series of the x (t) with K=1000 harmonics. Amplitude -- --- - - 12 14 10 time(t) Truncated Fourier series of x(t) for K=1000 harmonics using MATLAB

Fourier series of the x, (t) with K=10000 harmonics. 2.5 -------- Amplitude 0.5 ------- 10 16 18 time(t) Truncated Fourier series of x(t) for K=10000 harmnonics using MATLAB

c) from (b), ats on increasing the number of harmonic the truncated Ware form tappes the smoother shape as similar to square ware. Answar