Computer Project II๎
Math 2270-1, April 2002๎
๎
In the first part of this project, you will compute have MAPLE (or whichever computing environment
you prefer) compute some Fourier coefficients for various functions. This could be quite useful, for
instance, if you ever want to do some signal proccessing. Below I have the an inner product suited to
Fourier series and some of the appropriate functions typed in. You should fill in the blanks where they
occur.
> restart:with(plots):๎
dotprod := (f,g) -> int(f(t) * g(t), t = -Pi..Pi);
Warning, the name changecoords has been redefined๎
:= dotprod โ( ),f g d
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( )f t ( )g t t
> f_1 := t ->(1/sqrt (Pi)) * cos (t): ๎
f_2 := t ->(1/sqrt (Pi)) * cos (2*t):๎
f_3 := t ->(1/sqrt (Pi)) * cos (3*t):๎
f_4 := t ->(1/sqrt (Pi)) * cos (4*t):๎
f_5 := t ->(1/sqrt (Pi)) * cos (5*t):๎
f_0 := t ->1/sqrt(2*Pi):
> g_1 := t ->(1/sqrt(Pi)) * sin (t):๎
g_2 := t ->(1/sqrt(Pi)) * sin (2*t):๎
g_3 := t ->(1/sqrt(Pi)) * sin (3*t):๎
g_4 := t ->(1/sqrt(Pi)) * sin (4*t):๎
g_5 := t ->(1/sqrt(Pi)) * sin (5*t):
These functions above are the first 11 elements of the orthonormal basis we found for this inner
product we are using. ๎
๎
(1) Have MAPLE (or whichever program you are using) verify that f_1 and g_1 are perpendicular.๎
๎
(2) Have MAPLE (or whatever) verify that f_1 and f_2 are perpendicular.๎
๎
(3) Have MAPLE (or whatever) verify that f_1 and g_1 are unit length.๎
๎
(4) Have MAPLE (or whatever) plot f_1, f_2, g_1 and g_2 on the same set of axes so you can look at
them.
> h := t -> t^2;
:= hโt t2
> ๎
Now we will compute a truncated Fourier series for this function h. Below you will do the same
thing for other functions.
> a_0 := dotprod (f_0,h);๎
a_1 := dotprod (f_1,h);๎
a_2 := dotprod (f_2, h);๎
a_3 := dotprod (f_3,h);๎
a_4 := dotprod (f_4,h);๎
a_5 := dotprod (f_5, h);
