fft definition in matlab doc puzzling
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françois anquez
el 16 de Feb. de 2019
Comentada: françois anquez
el 22 de Feb. de 2019
Hi all,
I'm a bit puzzled with the definition of fft in the documentation :
The definition at the very end in "more about" states the fft as a projection on positive frequency (from 0 to Fs/N) while the examples suggest a "two sided" spectrum ie. with positive frequencies and negative ones (-Fs/2 to Fs/2).
To make my question more precise :
the definition in the doc is :
Given that the fft results in both positive and negative frequencies, I woumld have guess a defintion like :
Am I missing something ?
thanks in advance for your help,
Francois.
2 comentarios
Walter Roberson
el 17 de Feb. de 2019
?? I do not see the word "projection" at all in that documentation ??
Respuesta aceptada
Matt J
el 22 de Feb. de 2019
Editada: Matt J
el 22 de Feb. de 2019
Given that the fft results in both positive and negative frequencies, I woumld have guess a defintion like :
Perhaps a better way to clear up the confusion is to point out that for any given sinuoid at some "positive" frequency , the corresponding negative frequency sinusoid is indentical, due to n-periodicity, to , or equivalently to , where .
Bust since and both lie in the interval 2...n, we can see that summing from k=1...n in the FFT is sufficient to capture all the same pairs of positive and negative frequencies as summing from -n/2 to +n/2.
Más respuestas (1)
Matt J
el 19 de Feb. de 2019
Editada: Matt J
el 19 de Feb. de 2019
Your second definition only makes sense if you define the indexing in Y(k) to be modulo-n. Otherwise, it is not clear what it means for a vector Y to be indexed at negative k.
If you are defining Y(k) to be modulo n, then the two IFFT definitions are equivalent because Y(k) and W_n(j-1)(k-1) are then both n-periodic with respect to k. Therefore any sum over n successive indices k gives the same result.
4 comentarios
Matt J
el 21 de Feb. de 2019
Editada: Matt J
el 22 de Feb. de 2019
The test code should be as below. The main problem with your implementation is that in the first version of the coefficient calculation, you are putting the time origin at X(1) whereas in the second version, you are putting the time origin at X(N/2). Thus, the second version is really the transform of a phase-shifted version of X.
function [coef1,coef2]=essaiFFT_Matt(X,k)
N=numel(X);
WN=exp(-1i*2*pi/N);
coef1=0;
for j=1:N
coef1=coef1+X(j)*power(WN,(j-1)*(k-1));
end
coef2=0;
for j=(1:N) - floor(N/2) %floor() handles odd N
index=mod(j-1,N)+1; %modulo N transform for 1-based indexing
coef2=coef2 + X(index)*power(WN,(j-1)*(k-1));
end % for jj
end % function
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