Periodic Array
The fundamental equations are
ME
 W  n  m  X  m  R  n 
m M B
MB  n  ME
(1)
(2)
MC  M E  M B  1
Note that MB is frequently less than 0, e.g MB = -Mc/2 and ME = Mc/2-1.
MB  n  ME
 W n

W p  n   W  n  M 
n  ME
(3)
 W n  M 
n  MB

Sample set of numbers - This section is coded into winv.for which is called from
Wfit.wpj.
The code is described in ..\..\Fittery\Complex\WeightedFourierFit\WinvfitCode.doc
This is for 3107 pts with err = 1e-5, w=1e10, centered in 4096 points with a beginning 494 points with w=0 and an ending
495 with w=0.
>WFIT -B-2 -E1
The periodic array in one dimension
Frequency
Real Wp
Imag Wp
-0.669110E-01 -0.650957E+13 -0.199712E+11
-0.334555E-01 -0.896845E+13 -0.137574E+11
0.000000E+00 0.310700E+14 0.000000E+00
0.334555E-01 -0.896845E+13 0.137574E+11
The two d version
Af mat
0.311E+14 0.000E+00
-0.897E+13-0.138E+11
-0.651E+13-0.200E+11
-0.897E+13 0.138E+11
-0.897E+13 0.138E+11
0.311E+14 0.000E+00
-0.897E+13-0.138E+11
-0.651E+13-0.200E+11
-0.651E+13-0.200E+11
-0.897E+13 0.138E+11
0.311E+14 0.000E+00
-0.897E+13-0.138E+11
-0.897E+13-0.138E+11
-0.651E+13-0.200E+11
-0.897E+13 0.138E+11
0.311E+14 0.000E+00
ALMf mat
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.566E+13-0.290E+11
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.399E+11
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.566E+13 0.290E+11
0.000E+00 0.399E+11
0.000E+00 0.000E+00
0.000E+00 0.000E+00
Afinv mat
0.569E-13 0.109E-15
0.319E-13 0.121E-15
0.303E-13 0.124E-15
0.319E-13 0.101E-15
0.319E-13 0.101E-15
0.569E-13 0.109E-15
0.319E-13 0.121E-15
0.303E-13 0.124E-15
0.303E-13 0.124E-15
0.319E-13 0.101E-15
0.569E-13 0.109E-15
0.319E-13 0.121E-15
0.319E-13 0.121E-15
0.303E-13 0.124E-15
0.319E-13 0.101E-15
0.569E-13 0.109E-15
ME
ME
 W  n  m  X  m   W  n  m   W  n  m   X m  R n
m M B
p
m M B
winv.for
Multiply by Wp-1(k-n) and sum on n
p
MB  n  ME
(4)
ME
ME
 
mM B n M B
Wp1  k  n Wp  n  m X  m  
ME

m M B
X m
ME
ME
 W  k  n  W  n  m   W  n  m   
nM B
1
p
(5)
 W  k  n  R  n
1
p
nM B
p
MB  n  ME
Define
X p k  
ME
 W  k  n  R  n
nM B
1
p
(6)
And
Aˆ LM  k , m 
ME
 W  k  n  W  n  m   W  n  m 
1
p
nM B
p
MB  k  m
(7)
The exact evaluation of A cannot be done by convolution owing to the fact that it is not periodic.
winv.for
To find
X k  
ME
 X  m Aˆ k , m  X k 
LM
m M B
p
(8)
Or adding a 1 to the diagonal elements of ALM, so that
ALM  k , m   k ,m 
ME
 W  k  n  W  n  m   W  n  m 
nM B
1
p
p
(9)
Afinv * ALM
F -0.669E-01-0.335E-01 0.000E+00 0.335E-01
R 0.118E+01-0.405E-05 0.000E+00 0.322E+00
I 0.271E-02 0.127E-02 0.000E+00-0.103E-02
F -0.669E-01-0.335E-01 0.000E+00 0.335E-01
R 0.172E+00 0.100E+01 0.000E+00 0.180E+00
I 0.285E-02 0.121E-02 0.000E+00-0.351E-03
F -0.669E-01-0.335E-01 0.000E+00 0.335E-01
R 0.180E+00-0.482E-05 0.100E+01 0.172E+00
I 0.388E-02 0.127E-02 0.000E+00-0.179E-03
F -0.669E-01-0.335E-01 0.000E+00 0.335E-01
R 0.322E+00-0.437E-05 0.000E+00 0.118E+01
I 0.354E-02 0.227E-02 0.000E+00-0.240E-03
So that (8) becomes
ME

m M B
ALM  k , m X  k   X p  k 
(10)
Iteration
X it  k   X p  k  
M / 2 1
M / 2 1
 
m  M / 2 n  M / 2
Wp1  k  n  W  n  m   W p  n  m  X it 1  m 
(11)
For each value of m, the terms in the sum are different. The sum over W p is a convolution, but not the sum over W.
For Wp-1(k-n)=k,n/W(0)
1 M / 2 1
X it  k   X p  k  
 W  k  m   Wp  k  m   X it 1 m
W  0  m  M / 2
(12)
Note that for -M/2k-m<M/2 that the term in parenthesis is zero. Or
M/2+k  m > -M/2+k. The sum is not zero for m-M/2+k and for m >M/2+k
 k M / 2

W  k  m   W p  k  m   X it 1  m 




1  m  M / 2

X it  k   X p  k  
 M / 2 1
 (13)
W  0 

W  k  m   Wp  k  m   X it 1  m
 m  M
/ 2  k 1

1
1
Wk , n 
 Wn , m  W1,2
W1,2
  S k .m S1.2


 
1
S 2,1
 W2,1
  W2,1



 





 


 


 








For k=-M/2, the first term is from -M/2 to -M or zero. The second term is from m=1 to M/2-1. For k=M/2-1, the first term is from M/2 to -1, and the second term is zero. There is a sum of ~ M/4 terms for each k, so that the time requires is M 2/4. To evaluate for the
L terms in each corner requires 2*L*L/2 or L2 operations.
Start with Xp
X 2 k   X p k   
X 3 k   X p k   

M / 2 1
 W  k  m   W  k  m   X m
W  0  m  M / 2
1
p
p
M / 2 1
 W  k  m   W  k  m   X m
W  0  m  M / 2
p
M / 2 1
(14)
p
M / 2 1
W  k  m   W  k  m    W  k  m '  W  k  m '  X m
 0 
1
W2
1
m  M / 2
p
p
m '  M / 2
p
This converges as 1/W[0], 1/(W[0])2 etc. The convergence works as long as the W difference is < W(0) which in our case is almost a
given.
Evaluation of Xp
The equation for Xp is a convolution. #Xp
X p k  
ME
 W  k  n  R  n
nM B
1
p
(15)
../../Fourier/DiscreteConvolution.doc (1.4)
MC  M E  M B  1
X p k  
ME

nM B
R  nWp1  k  n  
1
MC
ME

ki
 r i  w i  exp  j 2 M
i M B
1
p

C
 (16)


The sum, with the usual problem of locations, is returned by FFT. ..\WeightedFourierFit\for\Wfit.wpj