psd - What is cross-spectral density- CSD?

I asked a question earlier but I didn't get any answer for it. So now I am simplifying it: what are Cross-Spectral Density (CSD) and Power-Spectral Sensity (PSD)? What is their application? How can I get them in MATLAB?

$$S_{kl}(\omega)=\lim_{T\to\infty}\frac{1}{T}E\{Y_k^*(\omega)Y_l(\omega)\} $$ $$S_{kk}(\omega)=\lim_{T\to\infty}\frac{1}{T}E\{Y_k^*(\omega)Y_k(\omega)\} $$

$S_{kl}(\omega)$ is the cross-spectral density (CSD) function between general signals $y_k(t)$ and $y_l(t)$, $S_{kk}(\omega)$ is the power-spectral density (PSD) of signal $y_k(t)$, $Y_k(\omega)$ is the finite Fourier transform of signal $y_k(t)$ at frequency $\omega$, $Y_k^*(\omega)$ is the complex conjugate of $Y_k(\omega)$, and $E\{\cdot\}$ is the expectation operator.

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My earlier question was: What does 'wavelet power spectrum', 'Auto-power spectrum','cross-power spectrum' means in wavelet application? I was studying about mode shape identification with wavelet method and these terms confused me.

Answer

Power-Spectral Density is the distribution of power along the frequency axis. It is generally used for non-finite energy signals (mostly not limited in time signals), who aren't square-summable. The signal's PSD is the autocorrelation of the signal's Fourier Transform, as stated by the Wiener–Khinchin theorem. In Matlab:

N = length(S);

F = fft(S);
F = F(1:N/2+1);
PSD = (1/(2*pi*N)) * abs(F).^2;
PSD(2:end-1) = 2*PSD(2:end-1);
freq = 0:(2*pi)/N:pi;

see: https://de.mathworks.com/help/signal/ug/power-spectral-density-estimates-using-fft.html

Cross-Spectral Density is the same, but using cross-correlation, so you can find the power shared by a given frequency for the two signals using its squared module, and the phase shift between the two signals at that frequency using its argument.

Cross-Spectral Density can be used to identify the frequency response of a noisy LTI system : if the noise is not correlated to the input or output of the system, its frequency response can be found from the CSD of the input and output.