L2 Correlation functions in time and frequency domain
3 important questions on L2 Correlation functions in time and frequency domain
What is the linear least-squares method?
A method to minimize the difference between the measured output data (y) and the model output data (ŷ).
To do this, it minimizes the sum of the squared errors. If the system output is a linear function of its parameters, there exists an analytical solution.
To do this, it minimizes the sum of the squared errors. If the system output is a linear function of its parameters, there exists an analytical solution.
What is the power spectrum/auto-spectral density of a signal?
Auto-spectral density S_{uu} is the Fourier Transform of the autocorrelation function.
- Only real values (no imaginary part)
- Symmetric around f=0. So only positive frequencies are analyzed
- Area under Suu(f) is equal to variance of signal (Parseval's theorem)
What is the cross-spectrum/cross-spectral density of a signal?
It is the Fourier transform of the cross-correlation function.
- Complex values
- Symmetric
- Syu(f) describes the interdepencency of signals u(t) and y(t) in the frequency domain (magnitude and phase)
- if phi_yu(tau) =0, then Syu(f) = 0 for all frequencies --> no linear relationship between signals.
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