Computation of Fourier Series


First define the signal.

> 

> f:=x->x;

                             f := x -> x

> plot(f,-Pi..Pi,numpoints=200); #Plot the signal


> (1/(2*Pi))*Int(f(x),x=-Pi..Pi); a0:=evalf(%); #the coefficient of a_0

                                   Pi
                                  /
                                 |
                                 |    x dx
                                 |
                                /
                                  -Pi
                            1/2 ----------
                                    Pi


                               a0 := 0

> 
> a:=n-> value( (1/Pi)*Int(f(x)*cos(n*x),x=-Pi..Pi) ) ; # the value of a_n

                                  Pi
                                 /
                                |
                                |    f(x) cos(n x) dx
                                |
                               /
                                 -Pi
               a := n -> value(----------------------)
                                         Pi

> b:=n-> value((1/Pi)*Int(f(x)*sin(n*x),x=-Pi..Pi)) ; #the value of b_n

                                  Pi
                                 /
                                |
                                |    f(x) sin(n x) dx
                                |
                               /
                                 -Pi
               b := n -> value(----------------------)
                                         Pi

> a(n); b(n);

                                  0


                       -sin(Pi n) + Pi n cos(Pi n)
                    -2 ---------------------------
                                      2
                                  Pi n


> S:=a0+Sum(a(n)*cos(n*x)+b(n)*sin(n*x),n=1..10);

              10
             -----
              \    /   (-sin(Pi n) + Pi n cos(Pi n)) sin(n x)\
        S :=   )   |-2 --------------------------------------|
              /    |                       2                 |
             ----- \                   Pi n                  /
             n = 1

> S:=value(%);

  S := 2 sin(x) - sin(2 x) + 2/3 sin(3 x) - 1/2 sin(4 x)

         + 2/5 sin(5 x) - 1/3 sin(6 x) + 2/7 sin(7 x) - 1/4 sin(8 x)

         + 2/9 sin(9 x) - 1/5 sin(10 x)

> plot({f(x), S}, x=-2*Pi..2*Pi);

> f:=x->piecewise(x<-Pi/2,0,x<Pi/2,1,0);

        f := x -> piecewise(x < - 1/2 Pi, 0, x < 1/2 Pi, 1, 0)

> plot(f,-Pi..Pi,numpoints=200); #Plot the signal


> (1/(2*Pi))*Int(f(x),x=-Pi..Pi); a0:=value(%); #the coefficient of a_0

                        Pi
                       /   { 0        x < - 1/2 Pi
                      |    {
                      |    { 1         x < 1/2 Pi  dx
                      |    {
                     /     { 0         otherwise
                       -Pi
                 1/2 --------------------------------
                                    Pi


                              a0 := 1/2

> 
> a:=n-> value( (1/Pi)*Int(f(x)*cos(n*x),x=-Pi..Pi) ) ; # the value of a_n

                                  Pi
                                 /
                                |
                                |    f(x) cos(n x) dx
                                |
                               /
                                 -Pi
               a := n -> value(----------------------)
                                         Pi

> b:=n-> value((1/Pi)*Int(f(x)*sin(n*x),x=-Pi..Pi)) ; #the value of b_n

                                  Pi
                                 /
                                |
                                |    f(x) sin(n x) dx
                                |
                               /
                                 -Pi
               b := n -> value(----------------------)
                                         Pi

> a(n); b(n);

                             sin(1/2 Pi n)
                           2 -------------
                                 Pi n


                                  0


> S:=a0+Sum(a(n)*cos(n*x)+b(n)*sin(n*x),n=1..5);

                       /  5                             \
                       |-----                           |
                       | \    /  sin(1/2 Pi n) cos(n x)\|
            S := 1/2 + |  )   |2 ----------------------||
                       | /    \           Pi n         /|
                       |-----                           |
                       \n = 1                           /

> S:=value(%);

                       cos(x)       cos(3 x)       cos(5 x)
          S := 1/2 + 2 ------ - 2/3 -------- + 2/5 --------
                         Pi            Pi             Pi

> plot({f(x), S}, x=-2*Pi..2*Pi);

> f:=x->piecewise(x<-Pi/2,0,x<0,x+Pi/2,x<Pi/2,Pi/2-x,0);

  f := x -> piecewise(x < - 1/2 Pi, 0, x < 0, x + 1/2 Pi, x < 1/2 Pi,

        1/2 Pi - x, 0)

> plot(f,-Pi..Pi,numpoints=200); #Plot the signal


> (1/(2*Pi))*Int(f(x),x=-Pi..Pi); a0:=value(%); #the coefficient of a_0

                   Pi
                  /   {     0             x < - 1/2 Pi
                 |    {
                 |    { 1/2 Pi + x           x < 0
                 |    {                                dx
                 |    { 1/2 Pi - x         x < 1/2 Pi
                 |    {
                /     {     0              otherwise
                  -Pi
            1/2 -----------------------------------------
                                   Pi


                             a0 := 1/8 Pi

> 
> a:=n-> value( (1/Pi)*Int(f(x)*cos(n*x),x=-Pi..Pi) ) ; # the value of a_n

                                  Pi
                                 /
                                |
                                |    f(x) cos(n x) dx
                                |
                               /
                                 -Pi
               a := n -> value(----------------------)
                                         Pi

> b:=n-> value((1/Pi)*Int(f(x)*sin(n*x),x=-Pi..Pi)) ; #the value of b_n

                                  Pi
                                 /
                                |
                                |    f(x) sin(n x) dx
                                |
                               /
                                 -Pi
               b := n -> value(----------------------)
                                         Pi

> a(n); b(n);

                            cos(1/2 Pi n) - 1
                         -2 -----------------
                                      2
                                  Pi n


                                  0


> S:=a0+Sum(a(n)*cos(n*x)+b(n)*sin(n*x),n=1..3);

                     /  3                                    \
                     |-----                                  |
                     | \    /   (cos(1/2 Pi n) - 1) cos(n x)\|
       S := 1/8 Pi + |  )   |-2 ----------------------------||
                     | /    |                  2            ||
                     |----- \              Pi n             /|
                     \n = 1                                  /

> S:=value(%);

                           cos(x)   cos(2 x)       cos(3 x)
           S := 1/8 Pi + 2 ------ + -------- + 2/9 --------
                             Pi        Pi             Pi

> plot({f(x), S}, x=-Pi..Pi);

>