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Topic: derivative of discrete fourier transform interpolation
Replies: 8   Last Post: Jun 6, 2013 7:24 AM

 Messages: [ Previous | Next ]
 Rusty Posts: 108 Registered: 6/16/05
Re: derivative of discrete fourier transform interpolation
Posted: Sep 30, 2005 11:47 AM

I had a further look at this and what you are trying to do is only possible
if the middle Fourier coefficient, X(N/2) is zero.
This first bit of MatLab does the problem by your method and has a non-zero
imaginary part in the interpolated function for non-integer s values.

N=16

x=(1:N)'

X=fft(x)

s=0.25

ix=0;

for k=1:N/2+1

ix=ix+X(k)*exp(2*pi*1i*(k-1)*s/N);

end

for k=1:N/2-1

ix=ix+X(N+1-k)*exp(-2*pi*1i*k*s/N);

end

ix=ix/N

This modified code, with one line changed to ignores X(8) gets a pure real
interpolated value, but it is not consistent with the original function
x(s), even at integer s values.

ix=0;

for k=1:N/2 %%%%%%%%% different upper limit

ix=ix+X(k)*exp(2*pi*1i*(k-1)*s/N);

end

for k=1:N/2-1

ix=ix+X(N+1-k)*exp(-2*pi*1i*k*s/N);

end

ix=ix/N

conclusion
your method will only work for functions x(s) such that X(N/2) = 0, ie
x(0) - x(1) + x(2) - x(3) + x(4)..........- x(N-1) = 0

rusty

Date Subject Author
9/30/05 Gareth Davies
9/30/05 Rusty
9/30/05 Rusty
9/30/05 Rusty
9/30/05 Rusty
9/30/05 Peter Spellucci
9/30/05 Steven G. Johnson
10/1/05 Rusty
6/6/13 Wesley