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I have been reading Leon Cohen's book "Time Frequency Analysis" as part of a project for university. On page twelve or equation (1.57) during his derivation of a representation of the average frequency in terms of the time-domain signal he provides the following relation which from my perspective came out of thin air, I am wondering if anyone else felt the same was able to derive the relation or at least explain it ?

$$ \langle \omega \rangle = \int \omega |S(\omega)|^2 d\omega = \frac{1}{2 \pi} \int \int \int \omega s^*(t)s(t')e^{j(t-t')\omega} d\omega\; dt'\; dt $$

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  • $\begingroup$ I don't why but the equation does not appear to be rendering correctly, it was before I posted it. $\endgroup$ Nov 17, 2022 at 10:44
  • $\begingroup$ Please check your equation. It renders now, but I don't think it's correct. $\endgroup$
    – Matt L.
    Nov 17, 2022 at 10:47
  • $\begingroup$ @MattL. Thanks, it is correct now, not sure what happened when I posted it. Thanks for the help $\endgroup$ Nov 17, 2022 at 10:50
  • $\begingroup$ @MattL. there is in the updated equation $\endgroup$ Nov 17, 2022 at 10:51
  • $\begingroup$ @MattL. I understand your derivation but the equation in the book has a positive exponent. I imagine that has to be a mistake, would it be best to leave the positive exponent in the question to let others know ? $\endgroup$ Nov 17, 2022 at 11:17

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They just express $S(\omega)$ (and its complex conjugate) by the Fourier transform of $s(t)$:

$$S(\omega)=\int s(t)e^{-j\omega t}dt\tag{1}$$

From which we get

$$|S(\omega)|^2=\int s(t)e^{-j\omega t}dt\int s^*(t')e^{j\omega t'}dt'=\int\int s(t)s^*(t')e^{-j\omega(t-t')}dtdt'\tag{2}$$

and

$$\int \omega |S(\omega)|^2d\omega=\int\int\int \omega s(t)s^*(t')e^{-j\omega(t-t')}dtdt' d\omega\tag{3}$$

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