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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

1. Why the "rank" of the matrix matters in SVD?

 - The number of the diagonal non-zero elements of the singular value matrix obtained SVD determines the rank;

 - In SSI, choosing the dimension (or rank) of the system matrix has to be done by the user;

 - the rank means the number of linearly independent column vectors in the matrix;

 - the size of the system matrix is equal to the size of the eigenvalue (or singular value) matrix. In other words, the user chooses the number of the modes which will be taken into account in the system identification;

2. What on earth is "Rank" in matrix and why does it matter?

 - answered above.

3. Why the "complex conjugate" matters in dynamic identification?

 - The complex number is the product of FFT/DFT;

 - In FDD method, the cross-spectrum is calculated by applying DFT to the cross-correlation of the signals. Thus, the cross-spectrum contains complex terms;

 - The expectation (average) of the PSD/Cross Spectrum density at each frequency interval (df) multiplied by its complex conjugate becomes the amplitude of impulse response. The higher amplitudes (peaks) at corresponding frequency range indicates dominant frequency contents.  The series of the impulse responses form PSD in frequency-domain.  

source: Barry Van Veen, Coherence and the Cross Spectrum ( https://youtu.be/igRrGrxbg-Y )

4. Why correlation matters in the spectral analysis?

 - the correlation is calculated in time-domain. Maybe it might be a key to shift to frequency-domain. 

5. Why is it called "POWER" spectral density?

 - Power means square (제곱).

 - The correlation spectra are multiplied by their own complex conjugates. Thus, it's almost same as square.

6. What does "Coherence" of the signals mean?

 - we use Coherence to assess the 'quality' of the modal parameter?

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