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I am having the FFT results of two distinct voice signals. Let's say the FFT results of these two signals are of the form (a+ib) and (c+id). Now, I want to calculate the PSD of each signal and also the Cross PSD between them. I'm calculating PSD and Cross PSD as follows.

PSD of signal 1 = a * a + b * b;

PSD of signal 2 = c * c + d * d;

Cross PSD between signal1 and signal2 = (ac - bd) + (ad + bc). I also want to know whether this equation represents cross-correlation or cross PSD?

And also, I have ambiguity about the importance of cross PSD. I know that cross PSD is obtained by taking the Fourier transform of the cross-correlation results. I know that from PSD, we can get an idea of the power for different frequencies present in the signal. How does cross PSD differ from PSD? I found out that by using cross PSD, we can find the power shared by a given frequency for the two signals. How do we know how much power is shared between the two signals from a single constant power value per frequency? Please help me out in understanding what the cross PSD means and what is its importance?

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Cross PSD (fourier transform of cross-correlation) is $X(f)^*Y(f)$, so it will $(a-ib)(c+id)=(ac+bd)+i(ad-bc)$. I think you missed to take conjugate like you previous question (Calculating Cross Power Spectral density between two complex signals).

Cross PSD tells a lot about correlation between two different signals. If Cross PSD is white (meaning near flat at all frequencies) with very low value implies both signals are uncorrelated. If $x$ and $y$ are input and output of a system, you can measure the coherence between $x$ and $y$. The relationship between PSD of $x$ and $y$ for the LTI system $h$ is $G_{yy}(f)=|H(f)|^2G_{xx}(f)$. You can read more at https://en.wikipedia.org/wiki/Coherence_(signal_processing)

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  • $\begingroup$ Hi, in general, cross-correlation itself gives the correlation results between two different signals, right? Then why cross PSD is used? I mean do we get any other benefits apart from correlation results? $\endgroup$
    – rkc
    Apr 17, 2020 at 8:08
  • $\begingroup$ cross-correlation gives you a 'time-domain' view. Taking cross-PSD will give you a view with respect to frequency domain which is much easier to interpret. For example, looking at cross correlation plot of voice and instrument signals, it would be very difficult to interpret that some of the frequencies have no correlation. Cross PSD would help in such situations. $\endgroup$
    – jithin
    Apr 17, 2020 at 8:14
  • $\begingroup$ Oh..I didn't think in that perspective. Thanks for the answer. $\endgroup$
    – rkc
    Apr 17, 2020 at 8:22

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