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I recently learned about the Fourier Series and how it can be used decompose a periodic signal into a sum of sinusoids. I can calculate all the coefficients by hand, but I wanted Mathematica to do that for me. I attempted to write a code, and it does give the desired output.

I wanted to animate the output, in the sense that Mathematica would display more and more corrections over time. So I want to start by displaying 1 partial sum, and over time display many partial sums. I looked into the Animate function to do this, but ran into some problems as it requires an index to iterate over. Naturally, I chose the number of partial sums included as the iterator. So instead of having the number of partial sums to include as an integer, I make it a variable. But doing so leaves Mathematica in some infinite loop. Below is my code. I tried to include as many comments as possible, and would appreciate any help.

Code (Text):

(*Define basis functions of sin and cosine*)

cosBasis[m_, t_, T_] := Cos[2*Pi*m*t/T];

sinBasis[m_, t_, T_] := Sin[2*Pi*m*t/T];

(*Define inner product*)

fourierIP[f_, g_] := Integrate[f*g, {t, -Pi, Pi}]

(*Define function to calculate coeffecients of sin and cosine,

depends on (f[t],period,m)*)

fourierSinCoeff[func_, T_, m_] :=

fourierIP[func[t], sinBasis[m, t, T]]/

fourierIP[sinBasis[m, t, T], sinBasis[m, t, T]];

fourierCosCoeff[func_, T_, m_] :=

fourierIP[func[t], cosBasis[m, t, T]]/

fourierIP[cosBasis[m, t, T], cosBasis[m, t, T]];

(*Compute M^th partial sum of fourier coeffecients*)

fourierSeries[func_, t_, T_, M_] :=

Sum[fourierCosCoeff[func, T, m]*cosBasis[m, t, T], {m, 0, M}] +

Sum[fourierSinCoeff[func, T, m]*sinBasis[m, t, T], {m, 1, M}];

(*SquareWave*)

const = 0.5;(*height of square wave*)

squareWave[t_] :=

Piecewise[{{const, 0 < t <= Pi}, {0,

Pi < t <= 2*Pi}}];(*construct square wave*)

periodicExtension[func_, nPeriods_] :=

Sum[func[t + 2*Pi*n], {n, -nPeriods,

nPeriods}];(*extend square wave over multiple periods**)

plot1 = Plot[periodicExtension[squareWave, 4], {t, -4*Pi, 4*Pi},

PlotRange -> {{-4*Pi, 4*Pi}, {-1, 1}},

ExclusionsStyle -> Dotted](*display square wave*)

fourierCosCoeff[f, 2*Pi, m];

fourierSinCoeff[f, 2*Pi, m];

output = fourierSeries[squareWave, t, 2*Pi,

9];(*include 9 partial sums*)

plot2 = Plot[output, {t, -4*Pi, 4*Pi}];

Show[plot1, plot2](*displays plot of square wave and fourier wave together*)

(*The above code works just fine, the moment I switch the

number of partial sums to include into a variable, I get problems*)

(*Now to animate over many partial sums*)

output1 = fourierSeries[squareWave, t, 2*Pi, m];

Animate[output1, {m, 1, 3, 1}]

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# Fourier Series Expansion using Mathematica

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