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

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Rusty

Posts: 108
Registered: 6/16/05
Re: derivative of discrete fourier transform interpolation
Posted: Sep 30, 2005 11:47 AM
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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







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