Using the concept of orthogonality, derive the equations for the coefficients of

Question

Using the concept of orthogonality, derive the equations for the coefficients of the trigonometric form of the Fourier series (.e., derive the expressions for ao, a and b.) (Hint: To find the expression a, with the expression for the trigonometric forn 2.5 of the Fourier series, multiply both the left-hand and right-hand sides by cos 2mmfer, anc integrate over one period of the fundamental frequency The ao and b expressions can be found in a similar manner.) A, and an appropriat 2.6 Using the Fourier coefficients determined in Example 2.3, Part software graphics package, reproduce the plots from Figures 2-8a-d. should us. Express the waveform in Figure P2-7 as a series of sinusoids. Your expression the actual Fourier coefficient values (i.e.,the values an, and bn) for the term an at least the first four expression should also use the actual value of thi fundamental frequen h onics. Your software program to You may wish to set up a spreadsheet or your coefficients since you will need to calculate the values for higher order in Problem volts seconds Figure P2.7 To see how many harmonics are needed for the Fourier series expression developed i Problem 2.7 to adequately replicate the original waveform, a. Plot the waveform produced by summing the dic component and the sinusoidsassoci ated with the first five harmonics. sinusoids associ b. Plot the waveform produced by summing the dc component and the ated with the first ten harmonics.

Explanation / Answer

%Creative Commons CC-BY-NC-SA

%If you use this piece of code you must attribute the author

clear all;clc;

time=linspace(-pi,pi,1000);

partial_sum=0;

%Complex Function represented in terms of time and amplitude value

t=[-pi,-pi,0,0,pi,pi];

value=[0,-5,5];

handle1=line(t,value,'color','r','linewidth',2);

grid on;hold on;

axis([-pi pi -5 5])

%Since the given complex function exhibits odd periodic extension

%only Bn term is valid with n=1,3,5,...

for n=1:2:200 %Odd terms to consider for partial sums

    %Plot 1 period of the given function    

    partial_sum=partial_sum+(4/(n*pi))*sin(n*time); %Fourier Series Expansion using Sine terms

    error=mean((abs(partial_sum)-1).^2); %Error Criteria

    handle2=plot(time,partial_sum,'k','linewidth',2);

    title(['Square Wave Partial Sum:  n = ',num2str(n),'  Error = ',num2str(error)])

    pause

    set(handle2,'Visible','off');

    if error<0.01

        break

    end

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