I. Harmonic Components of Periodic Signals Consider that signal

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1 ECE Sigals ad Systems Sprig, UMD Experimet 5: Fourier Series Cosider the cotiuous time sigal give by y(t) = A + A cos (π f ) + A cos (π f ) + A cos (6π f ) +.. where A is the DC compoet of the sigal, A are the amplitude of the harmoics, k is k the harmoic umberd θ are the phase agle of the harmoics. The expressio for a k Fourier Series represetatio of a fuctio sigal x(t) with N harmoics is The features of a sigal give by this equatio ca be studied i terms of the frequecy, amplitudesd phases of the siusoidal terms. The amplitudes A, A, A,.., A specify the relative weights of the frequecy compoets of the particular sigal. These weights are a major factor i determiig the shape of the sigal. The phase agle of a trigoometric compoet cotrols the positio of the compoet alog the time axis. I the first part of the experimet you will ivestigate the effect of the magitude ad phase agle compoets. You will be asked to chage the amplitude ad phase values ad compare the resultig waveform to the origial oe. I. Harmoic Compoets of Periodic Sigals Cosider that sigal y(t) = A + A cos (π f ) + A cos (π f ) + A cos (6π f ) with period of T =. sec ad the compoet values give i the followig Table

2 a) Show the harmoic compoets ad waveforms sythesized by usig ad harmoic compoets. f=; % Defie the frequecy of the sigal t = :.:; harm = cos(*pi**t); harm = *cos(*pi**t+5/8*pi); siga = harm + harm; harm = *cos(*pi**t+6/8*pi); siga = siga + harm figure() subplot(,,); plot(t,harm) subplot(,,); plot(t,harm) subplot(,,); plot(t,harm) subplot(,,); plot(t,siga) subplot(,,5); plot(t,siga) Now you will study the importace of the magitude ad the phase agle of each harmoic compoet by chagig them ad comparig the resultig waveform to the origial oe. )Labeled plots b) Chagig the magitude of the harmoics Chage the magitude of the secod harmoic from A = to A =.6,. ad respectively. Modify the MATLAB script of a). )Labeled plots (A =.6,., ) )Commet o the chage of waveform shape c) Chagig the phase of the harmoics Chage the phase agle of the secod harmoic from θ = 5 o to θ = 5 o, 5 o d o respectively. Modify the MATLAB script of a). )Labeled plots (θ =5 o, 5 o, o ) 5)Commet o the chage of waveform shape d) Addig a harmoic to the sigal y(t)

3 Usig the origial waveform with the three harmoics give i Table, create aother waveform by addig a th harmoic with a magitude of.5 ad a phase agle of o. 6)Labeled plots showig the sigal cotaiig the th harmoic 7)Compariso betwee the ew sigal ad the origial sigal II. Magitude Spectra ad the partial Fourier Series Compute ad plot the magitude spectrum ad the partial Fourier series sums of two periodic sigals: a square waved a triagular wave. a) Magitude Spectrum of a Square Wave The followig MATLAB script defies a odd square wave (period sec, o DC level, Vpp) % Program to give partial Fourier sums of a % odd square wave of Vpp amplitude _max = iput('eter vector of highest harmoic values desired (odd):'); f = ; N = _max; t = :.:; omega_ = *pi*f; x=zeros(size(t)); =::N; b_=zeros(size()); for i=:(n+)/; k=*i-; b_(k)=/(pi*k); % This part is for plottig the Magitude spectrum figure() subplot(,,),stem(b_); xlabel('iteger Multiple of Fudametal Frequecy'); ylabel('amplitude'),grid; % This part is for plottig the partial Fourier sum x=x+b_(k)*si(omega_*k*t); subplot(,,),plot(t,x),xlabel('t'),ylabel('partial sum'); axis([ -.5.5]),text(.5,-.5, ['max.har.=',umstr(k)]); grid; ed Type the umber of harmoics (e.g.,, up to 5) you wat to plot, whe you see the followig lie as you ru the program Eter vector of highest harmoic values desired (odd): The Fourier series compoets for this sigal is a summatio of odd harmoics of the sie wave with a magitude of /(πk), where k is the harmoic umber. Usig

4 MATLAB compute the magitude of the harmoics coefficiets, that is,, b. Plot the amplitude spectrum (lie spectrum) of the square wave ad plot the partial Fourier sums startig from harmoic till 5 harmoics. 8)Labeled plot(for 5 harmoics) 9)Fid magitude b, 5 i Workspace ad put them dow )Describe how the commad a = iput ( b ) works b) Magitude Spectrum of a triagular Wave The followig MATLAB script defies a triagular wave (period sec, o DC level, Vpp) % Program to give partial Fourier sums of a % Triagular wave of Vpp amplitude _max = iput('eter vector of highest harmoic values desired (odd):'); f = ; N = _max; t = :.:; omega_ = *pi*f; x=zeros(size(t)); =::N a_=zeros(size()); for i=:(n+)/; k=*i-; a_(k)=8/(pi^*k*k) % This part is for plottig the Magitude spectrum figure() subplot(,,); stem(a_); xlabel('iteger Multiple of Fudametal Frequecy'); ylabel('amplitude'),grid; % This part is for plottig the partial Fourier sum x=x+a_(k)*cos(omega_*k*t); subplot(,,),plot(t,x),xlabel('t'),ylabel('partial sum'); axis([ -.5.5]),text(.5,-.5, ['max.har.=',umstr(k)]),grid; ed The Fourier series compoets for this sigal is a summatio of the harmoics of the cosie wave with a magitude of 8/(π k ), where k is the harmoic umber. Usig MATLAB compute the magitude of the harmoics coefficiets, that is,, a. Plot the amplitude spectrum (lie spectrum) of the square wave ad plot the partial Fourier sums startig from harmoic till 5 harmoics. )Labeled plot (for 5 harmoics)

5 )Fid magitude a, 5 i Workspace ad put them dow. Report Requiremets: Idividual lab report Attached with the cover page (available o the website) This lab report should cotai the followig:. Statemet of the Problem: Defie the problem ad goals of the experimet.. Results: Poits - give i the lab sheet above. Exercises: Attached. Coclusios: Give commets to the lab. Exercises: Coect the correct Fourier Series ame with its correspodig represetatio: Trigoometric Complex Expoetial Compact 5