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Is there any approximate relation in FFT coefficients?

For example, consider time domain signal $x_n$. Its $N$ point (assume $N$ to be even) DFT is given as $X_0,X_1,...,X_{N-1}$

Can I say something about the nature of $abs(X_1)/abs(X_0)$? I am trying to see the variation in DFT coeffcients and for that I thught one can sonsider the ratio of successive coefficients. My motivation is to see how similar the magnitude spectrogram at frequency $f1$ is to spectrogram at frequency $f1+1$ and $f1-1$

I tried to simplify the expanded equation but got stuck after a step -

$abs(X_0) = \Sigma_0^{N-1} x_n$

$abs(X_1) = abs(x_0+x_{N/2}+2(x_1.e^{-j.2\pi/N} + x_2.e^{-j.2\pi.2/N}+..+x_{N/2-1}.e^{-j.2\pi.(N/2-1)/N}))$

How do I (approximately) simplify $abs(X_1)/abs(X_0)$?

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Because the basis vectors for adjacent DFT bins are absolutely orthogonal, the values in the adjacent DFT result bins can be completely unrelated, and thus their ratio can be anything, depending on the data.

If you use a non-rectangular window function, then there will be some leakage of spectral peaks into adjacent bins due to convolution with the transform of the window. But this depends on the window chosen.

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  • $\begingroup$ Thanks, but this is rather counter-intuitive for natural signals (e.g. audio). I'd expect nearby coefficients to be similar to each other. $\endgroup$
    – user13107
    Aug 9 '13 at 7:06
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    $\begingroup$ Natural audio signals tend to be slightly to greatly modulated, which gives them sidebands that depend on the depth of the modulation, from zero to lots. So you can't count on it. They are also not usually exactly bin centered in frequency, but between bins. The split between bins will depend on the exact frequencies. $\endgroup$
    – hotpaw2
    Aug 9 '13 at 7:13
  • $\begingroup$ I'm sorry, audio signals get modulated by what exactly? $\endgroup$
    – user13107
    Aug 9 '13 at 7:30
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    $\begingroup$ Decay, vibrato, tremolo, varying Doppler shift, changing reflection interference, transient and parasitic resonances, non-linear material properties, etc., etc. $\endgroup$
    – hotpaw2
    Aug 12 '13 at 20:04

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