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Cross Spectrum

Definition

 .

 - The expected value rxy means integral (the area under the curve) or averaging of the magnitude within the frequency range around f above;

 - The averaged value/expectation/the area under the curve between frequency interval is the magnitude of the power spectrum at f as the impulse function;

 - The series of the impulse function creates the power spectrum in given frequency range. 

 - First, here is a signal. We can calculate CORRELATION by shifting the signal by the given LAG. Then we have correlation function at each lag;

 - If the k=0 (no lag), the correlation is equal to 1. As k increases or the shifted signal is little related to the original signal, the correlation becomes smaller up to 0;

 - Secondly, DISCRETE FOURIER TRANSFORM is applied to each correlation E(k). The DFT shows the FOURIER COEFFICIENT which shows which frequencies are blended in the correlation function at lag k as a function of Fourier coefficient (a, b, c...); 

 - Now we would have a series of FOURIER COEFFICIENT (which is shows the magnitude of frequency contents) at each k;

 - The magnitude of each frequency interval can be obtained averaging each column (a, b, c...n);

 - Then we can figure out the magnitude of each frequency content -> Power spectrum.

Convolution

 - related to impulse response function;

 - Let's think about how to calculate the IMPULSE RESPONSE FUNCTION;

 - Output = input signal * impulse response 

Magnitude-squared coherence

Definition

like correlation, coherence is ratio normalised by the PSD of each signal that shows how much two signals are related to each other. the value lies between 0 and 1. 

source: https://youtu.be/2yA9aha3tfE

Why Welch's method?

 - averaging the power spectra of each segment of the signal because averaging process reduces variance.

 - CAUTION!

  in MATLAB function, power/cross spectra is calculated 

 for one signal  : pwelch

 for two signals : cpsd