In-class Demonstration: Fourier Approximations

By Russell Blyth and Mike May, S.J.

restart:with(plots):

Warning, the name changecoords has been redefined

We start by using Maple to perform some tedious integrals for us.

Verify that the functions used as basis vectors in the approximating subspace are orthogonal.

Int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(j*Pi*x),x=-1..1); Int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1) = int(sin(i*Pi*x)*sin(i*Pi*x),x=-1..1); Int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(sin(i*Pi*x)*cos(j*Pi*x),x=-1..1); Int(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(j*Pi*x),x=-1..1); Int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1) = int(cos(i*Pi*x)*cos(i*Pi*x),x=-1..1);

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That left us with some work still to do. Try again. This time we first tell Maple that i and j are integers (Maple assumes i and j are nonzero):

assume(i,integer): assume(j,integer): interface(showassumed=0):

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

We need to also integrate against 1:

Int(sin(i*Pi*x)*1,x=-1..1) = int(sin(i*Pi*x)*1,x=-1..1); Int(cos(i*Pi*x)*1,x=-1..1) = int(cos(i*Pi*x)*1,x=-1..1); Int(1*1,x=-1..1) = int(1*1,x=-1..1);

To compensate for the value of this final integral, we must remember to half the coefficient of this constant function later.

Graph several of these integrands as a reality check - note that in each case it appears that the signed area from x=-1 to x=1 has the appropriate value (0 or 1):

plot(sin(2*Pi*x)*sin(3*Pi*x),x=-1..1);

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

plot(sin(3*Pi*x)*cos(4*Pi*x),x=-1..1);

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plot(cos(5*Pi*x)*cos(3*Pi*x),x=-1..1);

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plot(sin(3*Pi*x)*sin(3*Pi*x),x=-1..1);

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Example

Find Fourier approximations for f(x) = x.

We compute the Fourier coefficients for f(x) = x

Int(x*sin(j*Pi*x),x=-1..1) = int(x*sin(j*Pi*x),x=-1..1); Int(x*cos(j*Pi*x),x=-1..1) = int(x*cos(j*Pi*x),x=-1..1); Int(x,x=-1..1) = int(x,x=-1..1);

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Note there are only sine terms (not surprisingly, since f is an odd function).

Next use Maple to write a general expression for the order n approximation

Fapprox := (x,n) -> sum(-2*(-1)^j/(j*Pi)*sin(j*Pi*x),j=1..n);

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We can now plot (say) a 10 term approximation against f(x).

eval(Fapprox(x,10)); plot([Fapprox(x,10),x],x=-2..2,y = -2..2, title = " 10th Order Fourier approximation to y =x", titlefont = [HELVETICA,12], color = [green,blue]);

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We can also animate this plot.

mindeg := 2: degsteps := 20: bydeg :=4: A := display(seq(plot([Fapprox(x,mindeg + bydeg*i),x], x=-2..2,y=-2..2, title="degree = "||(mindeg + bydeg*i), titlefont=[HELVETICA, 14]), i=0..degsteps), insequence=true): B := animate(func,x=-2..2,y=-2..2,frames=degsteps+1): print(" x vs its Fourier approximation"); display(A,B,view=[-2..2,-2..2]);

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It is also useful to plot the error between the function and the approximation.

plot(Fapprox(x,10)-x,x=-1..1,y=-1..1, title = "error in 10th Order Fourier approximation to y = x", titlefont = [HELVETICA,12], color = red);

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Similarly, we can animate the plot of the error function.

mindeg := 2: degsteps := 20: bydeg :=4: print("Error in nth order Fourier approximation to y=x"): display(seq( plot(Fapprox(x,mindeg + bydeg*i)-x, x=-1..1, y=-1..1, color=blue, title="order = "||(mindeg + bydeg*i), titlefont=[HELVETICA, 14]), i=0..degsteps), insequence=true);

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Higher order approximations match the function f even more closely!