Instructions For Plotting Fourier Series In Matlab Wolfram Alpha PDF

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Download Instructions For Plotting Fourier Series In Matlab Wolfram Alpha PDF

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Plotting functions and Fourier series You can either use matlab (which is commercial software but installed on university computers) or you can use octave (which is a free software and easy to install).

Use of matlab/octave to evaluate Fourier coefficients. If you know the Fourier coefficients for a given function, you can use matlab or octave (a free equivalent of matlab with the same syntax) to evaluate their values. Take example 3 as an example. The Fourier coefficients are

Then you can use this little matlab/octave program to evaluate the first N values of the Fourier coefficients an and bn. a0=1; N=9; ns=zeros(N,1); ans=zeros(N,1); bns=zeros(N,1); for n=1:N ns(n)=n; ans(n)=2*((-1)^n-1)/(n^2*pi^2); bns(n)=-2*(-1)^n/(n*pi); end disp(' n an bn'); disp([ns,ans,bns]); The program is attached as EvaluateFourierCoefficientsExample3.m Enter this program in the ‘Editor’ window of matlab/octave and save as a file. Then run it. The ‘command window’ of matlab/octave should then show you output like this: >> EvaluateFourierCoefficientsExample3 n an bn 1.00000 -0.40528 0.63662

2.00000 3.00000 4.00000 5.00000 6.00000 7.00000 8.00000 9.00000

0.00000 -0.04503 0.00000 -0.01621 0.00000 -0.00827 0.00000 -0.00500

-0.31831 0.21221 -0.15915 0.12732 -0.10610 0.09095 -0.07958 0.07074

>> These are the values of n, an, bn for the numbers n=1...9. Some comments:  The command zeros creates a column vector of length N where all entries are 0.  [ns,ans,bns] concatenates the column vectors ns,ans,bns into a matrix which ‘disp’ then displays as a table.

Use of matlab/octave to plot Fourier coefficients The way of plotting functions in matlab is to create a vector (say xs) representing the x-values and a vector (say ys) representing the corresponding y-values. Then you can plot these, by plot(xs,ys). So, the question is, how to generate these vectors. Let’s say we want to plot the Fourier series above, from x=-3 to x=+5. The following code (which needs to be used with the above code) shows this: xs=-3:0.01:5;

(1)

ys=zeros(1,length(xs));

(2)

ys=ys+a0/2;

(3)

for n=1:N

(4)

ys=ys+ans(n)*cos(n*pi*xs/2);

(5)

ys=ys+bns(n)*sin(n*pi*xs/2);

(6)

end

(7)

plot(xs,ys,'-')

(8)

Comments: 

The code is provided in file PlotFourierSeriesExample3.m



Line (1) creates a vector of x-values, starting at -3, incrementing by 0.01, and not exceeding 5. Hint: just run this command (without the semicolon ‘;’ in the command window. You’ll see what it produces)



Line (2) creates an empty vector for the y-values, with all values initially set to 0



Line (3) adds the a0 term. Note that this is ‘funny’ matlab syntax. The operation “vector+number” means that you add the number to each term of the vector.



The for loop in line (4) now deals with each term n=1, n=2, n=3, …. and adds both the sin and the cos terms. In each loop, we add the terms an*cos(n*pi*x/2) and bn*sin(n*pi*x/2) to the solution vector (called ‘ys’). Note, again, we are using the vector addition functionality of matlab, that is: xs is the vector of x values. cos(xs) is the vector representing the cos values of each x value in xs.



Line (8) plots the x-values xs against the y-values ys, using a line-style graph (rather than points).

The output of this should be a graph such as this:

Hint: it should be fairly straightforward to also integrate the original function f(t). Simply create a vector xs1 which represents the interval [-2,0] and a vector ys2 which corresponds to the function value (constant to 0) in that interval. Then create vectors xs2 and ys2 for the interval [0,2]. Then use the command plot(xs,ys,xs1,ys1,xs2,ys2) to plot all three.

Use of wolframAlpha to plot piecewise-defined function WolframAlpha allows you to plot piecewiese functions in the following way Plot[Piecewise[{{0,-2...