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1.3 ARRAYS, FILES, AND PLOTS + FOURIER SERIES BY MR. Q

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ARRAYS A set of numbers in a specific order or pattern 1 x n dimensional matrix In Matlab, you must always define arrays: To input y=f(x) you must Define x Define y as a function of x using the hundreds of functions available in Matlab

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ARRAYS Define y=sin(x) on the interval [0, 2π] Is it continuous? How accurate should you have it? How ‘big’ is the array(s)? What is the 10 th value?

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FILES

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Must define independent and dependent variables Plot(independent, dependent,…, ‘+’) title (‘ ‘) xlabel (‘ ‘) ylabel(‘ ‘) gtext(‘ ‘) [x,y] ginput( n ) grid PLOT

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Combination of arrays n x m dimensions Constructed using a semi colon between rows Ax=b x=? Solve using Matlab 2x+2y+3z=10 4x-y+z=-5 5x-2y+6z=1 MATRICES

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Used: Originally to solve heat equation Differential Equations- Eigensolutions Electrical Engineering Vibrational Analysis Signal Processing, etc. Breaks down repeating, step, or periodic functions into a sum of sine and cosine FOURIER SERIES

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P1.23 FT APPROXIMATION Consider the Step Function The Fourier Transform for the above Function Taking the First FOUR Terms of the Infinite Sum

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How can we plot f(x)? Solution 1 x1=[-pi,0] f1= [-1,-1] x2=[0,pi] f2=[1,1] OR Solution 2 x0=[-pi,-1e-6,1e-6,pi] f0=[-1,-1,1,1] GRAPHING THE FUNCTION

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Solution 1 plot(x1,f1,x2,f2);grid; title(‘f(x)’);xlablel(‘x’) Solution 2 plot(x,f);grid;title(‘f(x )’);xlabel(‘x’); PLOT THE FUNCTION

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How can we graph? Solution 1 x=-pi:0.01:pi; f=4/pi(sin(x)/1+sin( 3*x)/3+sin(5*x)/ 5+sin(7*x)/7); Solution 2: ftot=zeros(1,length(x) for k=1:2:7; fc=sin(k*x)/k; ftot=fc+ftot; end GRAPHING THE APPROXIMATION

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Let’s make a program for solution 2 and plot all on same axis! function fourier(n) x=-pi:0.01:pi; f1=[-1,-1,1,1]; x1=[-pi,-1e-6,1e-6,pi]; ftot=zeros(1,length(x)); for k=1:2:n; fc=sin(k*x)/k; ftot=ftot+fc; end f=4/pi*ftot; plot(x,f,x1,f1) CAN WE GET A BETTER APPROXIMATION?

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THE HEAT EQUATION

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