This reference is organized as follows:
PART#1:Relate complex expansion to real Fourier series
PART#2:IDL form of complex expansion
PART#3:Specific example: f(t) = sin ( 4*pi*t)
PART#4:Specific example: T(x,y) =
PART#1: Relate complex expansion to real Fourier series.
Assume you have a function f(t) that is periodic in t with a period
T.
Then there exists coefficients a_n & b_n such that
f(t) = a_o + sum_n(a_n*cos(2*pi*n*t/T)+b_n*sin(2*pi*n*t/T)) , n=1,2,...
This is just the Fourier series of function with period X. Nothing
new
here. See eq #4, section 10.3, Advanced Engineering Mathematics,
Kreyszig. This expansion though assumes -T/2 < t < T/2
Now consider an alternate form of the Fourier series expansion.
f(t)=sum_n(A_n*exp(j*2*pi*n*t/T)), n=0,1,2,...
In order for me to be comfortable with this expansion I need to see
how
this expansion relates to the expansion above. In particular, how do
the
complex An relate to the real a_n and b_n?
Consider the following:
II= sum_n(A_n*exp(j*2*pi*n*t/T)), n=0,1,2,... j*j = -1
=A_o+sum_n(A_n*(cos(2*pi*n*t/T)+j*sin(2*pi*n*t/T)), n=1,2,...
Let A_n=(aa_n+j*bb_n)
II= A_o+sum_n((aa_n+j*bb_n)*(cos(2*pi*n*t/T)+j*sin(2*pi*n*t/T))
= A_o + sum_n( aa_n*cos() - bb_n*sin() + j*(bb_n*cos()+aa_n*sin())
Real(II) = Real(A_o) + sum_n( aa_n*cos(2*pi*n*t/T)-bb_n*sin(2*pi*n*t/T))
Comparing to the first expansion we see that
Real(A_o)=a_o, aa_n=a_n, -bb_n=b_n
To me, this proves existence of the complex expansion. Knowing one,
you
can figure out the other. Part #1 is complete.
Part #2: IDL form of complex expansion
Let f(t) be a periodic function with period T defined on an interval
[0,T].
Then there exist complex A_n such that
f(t)= sum_n(A_n*exp(j*2*pi*n*t/T)), n=0,1,2,... j*j = -1
Divide the interval into N sections. t~t_i = i*T/N
Then,
f( t_i ) = sum_n(A_n*exp(j*2*pi*n*t_i/T))
= sum_n(A_n*exp(j*2*pi*n*i*(T/N)/T))
= sum_n( A_n*exp(j*2*pi*n*i/N)
) , n=0,1,...
This is exactly what is found in the IDL manual under the section for
FFT. The only difference is that t has been replaced by u and A_n has
been replaced by F(u). Note that the period T has dropped out. Also
note
that t has been replaced by t_i = i*T/N. In order for this to
happen,
the interval over which t is defined must be from [0,T]. This is
different from the definition of t being defined over the interval
[-T/2,T/2]. Perhaps this is why b_n = -bb_n.
********UNFORTUNATELY IT IS WRONG****************
What is wrong is the values of n in the sum. IDL does not use the values
of n=0,1,2,... IDL actually uses n= -N/2+1, -N/2+2, ...-1,0,1,...,N/2
The reason for doing this must have to do with FFT theory. Note also
that the number of values of n is N.
It gets more complicated. From the manual we have
F(u) = 1/N*sum_x(f(x)*exp(-j*2pi*ux/N)) , x=0,1,...N-1
First thing to realize is that F(u) is really F_n. Where n is an
integer. This comes from the fact that f(x) is periodic in x.
The manual also mentions that the "frequencies" are
Fo, 1/(NT),2/(NT),...,1/2T,-(N-2)/(2NT),...,-1/NT
After trial and error I have determined that the value of the ns range
for -N/2 to N/2. Futhermore, the F_n are stored in the order associated
with the following values of n
0,1,2,...,N/2,-(N/2-1),-(N/2-1),...,-1 <== this is bizarre!!
Let N=8. Then N/2=4
The F_n would be stored in an array. The array of n values associated
with this array would be:
[0,1,2,3,4,-3,-2,-1]
Part #3: Specific Example
Consider the interval t = [0,1]. This choice of interval implies T=1.
Let f(t) = sin ( 4*pi*t)
f(t_i)=sin(2pi*2*i/N), i=0,1,...N
f(t_i)=sum_n(A_n*exp(-j*2pi*n*i/N)) , n=-N/2,...-1,0,1,...N/2
= A_nN/2... + A_n2*(cos(2pi*(-2)*i/N)+j*sin(2pi*(-2)*i/N))+
+ A_o+A_n1*exp()+A_1*exp()+
A_2*(cos(2pi*(2)*i/N)+j*sin(2pi*(2)*i/N))
+ A_3*exp()+...
=... + A_n2*cos(2pi*2*i/N)+A_2*cos(2pi*2*i/N)
+
+A_n2*j*(-1)*sin(2pi*2*i/N)+A_2*j*sin(2pi*2*i/N)) + ....
= ... + (A_n2+A_2)*cos(2pi*2*i/N)+j*(
-A_n2 + A_2)*sin(2pi*2*i/N)
+ ...
where A_n2 stands for A_n where n= -2
Equating the series to sin(2pi*2*i/n) we conclude
A_n = 0 for all n except n = -2 or n = 2.
A_n2+A_2=0
j*(-A_n2 + A_2) = 1
Let A_n2=(a_n2+j*b_n2) and A_2=(a_2 + j*b_2)
The above equations imply
(a_n2 + a_2) + j*( b_n2+b_2) = 0 &
j*[( -a_n2 + a_2) + j*( -b_n2 + b_2)] = 1
==> a_n2 + a_2=0, b_n2+b_2 =0 ==> a_n2= - a_n2, b_n2 = -b_n2
==> 2*a_n2=0 ==> a_n2=a_2 = 0
==> j*j*(2*b_2)=1 ==> 2*b_2 = -1/2, b_2 = 1/2
A_n2 = 0 + j*(1/2)
A_2 = 0 + j*(-1/2)
We now have calculated the solutions.
The following code calculates this and displays the correct answers.
It
shows how to plot A_n vs n correctly.
;idl_program fft_sine.pro
TT=1
Npts=100
t=findgen(Npts)/(Npts-1)*TT
f_t=sin(4.*!pi*t/TT)
!p.multi=[0,2,3]
plot,t,f_t, title='f(t) vs t'
A_n=fft(f_t,-1) ; complex fourier coefficients
plot,float(A_n),yrange=[-.5,.5],title='float(A_n)'
plot,imaginary(A_n),yrange=[-.5,.5],title='imaginary(A_n)'
a=findgen(Npts/2+1)
b=-reverse(findgen(Npts/2-1)+1)
c=[a,b] ; c=[-N/2+1,-N/2+2, ...,-1,0,1,...,N/2]
print,c
sub=sort(c)
plot,c(sub),float(A_n(sub)),yrange=[-.5,.5],title='float(A_n) vs n'
plot,c(sub),imaginary(A_n(sub)),yrange=[-.5,.5], $
title='imaginary(A_n) vs
n'
plot,c(sub),imaginary(A_n(sub)),xrange=[-5,5],$
title='imaginary(An) vs
n' ; finer x scale
end
Part #4 Specific Example:
T(x,y)= sin(6pi*x) + cos(4pi*y)
Hopefully the program below is somewhat self explantory.
It calculates the C=FFT(T,-1)
;idl_program fft_2D.pro
!p.multi=0
Tx=1
Nx=100
x=findgen(Nx)/(Nx-1)*Tx
Ty=1
Ny=100
y=findgen(Ny)/(Ny-1)*Ty
T=fltarr(Nx,Ny)
for i=0,Nx-1 do begin
for j=0,Ny-1 do begin
T(i,j)= sin(2.*!pi*3.*i/Nx) + cos(2.*!pi*2*j/Ny)
endfor
endfor
;
!p.multi=[0,2,2]
shade_surf,T,x,y,xtitle='x',ytitle='y',title='T(x,y)'
;
;
C=fft(T,-1) ; complex fourier coefficients
surface,float(C)
aaa=where(float(c) gt .4)
surface,imaginary(C)
;
a=findgen(Nx/2+1)
b=-reverse(findgen(Nx/2-1)+1)
ns=[a,b] ; this is the array of n's associated with C(n). n goes with
x
;print,ns ; n goes from 0,...,Nx/2, -(Nx/2-1),...,-1
subn=sort(ns) ; n goes with x
n_sort=ns(subn)
;
a=findgen(Ny/2+1)
b=-reverse(findgen(Ny/2-1)+1)
ms=[a,b] ; this is the array of m's associated with C(n,m)
print,ms ; m goes from 0,...,Ny/2, -(Ny/2-1),...,-1
subm=sort(ms) ; m goes with y
m_sort=ms(subm)
;
sub_n_p3=where(ns eq 3)
sub_n_n3=where(ns eq -3)
sub_n_0=where(ns eq 0)
;
sub_m_p2=where(ms eq 2)
sub_m_n2=where(ms eq -2)
sub_m_0=where(ms eq 0)
;
print,'C(3,0),c(-3,0)=',C(sub_n_p3,sub_m_0),C(sub_n_n3,sub_m_0)
;
print,'C(0,2),c(0,-2)=',C(sub_n_0,sub_m_p2),C(sub_n_0,sub_m_n2)
;
; now we need to define CC(n,m) to have normal scaling in n & m.
;
CC=C*0.
;
for n=0,Nx-1 do begin
for m=0,Ny-1 do begin
CC(subn(n),subm(m))= C(n,m)
endfor
endfor
;
surface,ABS(CC),n_sort,m_sort
;
end