Chapter 5 - Frequency-Domain Analysis of LTI Systems

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5.3.1 Correlation Functions and Power Spectra for Random Input Signals

Important formulas for deterministic signals

$$\begin{array}{rl}{r}_{yy}& =r_{hh}\ast r_{xx}\\ r_{yx}& =h\ast r_{xx}\end{array}$$

Where $r$ is the Autocorrelation

From this, we get this by using the Z-transform

$$\begin{array}{rl}{S}_{yx}(z)& =H(z)S_{xx}(z){|}_{z=e^{j\omega }}\\ \to S_{yx}(\omega )& =H(\omega )|X(\omega ){|}^2\end{array}$$

And

$$S_{yy}(\omega )=|H(\omega ){|}^2{S}_{xx}(\omega )$$

Where $S$ is the Energy Density Spectrum

5.3.2 Correlation Functions and Power Spectra for Random Input Signals

5.4.2 - Lowpass, Highpass and Bandpass Filter