copyright Robert J. Marks II

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copyright Robert J. Marks II ECE 5345 Analysis & Processing of Random Signals- Filtering RP’s copyright Robert J. Marks II

Analysis & Processing of Random Signals- Filtering RP’s Deterministic input, deterministic linear system Linear means (a) additivity and (b) homogeneity Linear System x(t) y(t) copyright Robert J. Marks II

Filtering RP’s X(t) Y(t) Stochastic input, deterministic linear system Expectation and linear operations commute: copyright Robert J. Marks II

copyright Robert J. Marks II Analysis & Processing of Random Signals- Filtering RP’s Deterministic input, deterministic LTI system LTI x(t) y(t) copyright Robert J. Marks II

copyright Robert J. Marks II Filtering RP’s Stochastic input, deterministic LTI system LTI System X(t) Y(t) copyright Robert J. Marks II

Filtering RP’s: Nonstationary Case LTI System X(t) Y(t) copyright Robert J. Marks II

copyright Robert J. Marks II Filtering RP’s LTI System X(t) Y(t) copyright Robert J. Marks II

copyright Robert J. Marks II Filtering RP’s LTI System X(t) Y(t) copyright Robert J. Marks II

Differentiation of RP’s copyright Robert J. Marks II

Stochastic Differential Equations Given the (nonstationary) correlation of X, find the particular (forced) solution for and copyright Robert J. Marks II

Stochastic Differential Equations Cross Correlation: Recall Linear Operations and Expectation Commute. Thus… copyright Robert J. Marks II

Stochastic Differential Equations Fix  and solve the differential equation Particular solution: zero initial conditions… Thus, the DE initial conditions are… copyright Robert J. Marks II

Stochastic Differential Equations Finding the autocorrelation of Y. Thus… copyright Robert J. Marks II

Stochastic Differential Equations Fix  and solve the differential equation Particular solution: zero initial conditions… Thus, the DE initial conditions are… copyright Robert J. Marks II

Stochastic Differential Equations Summary. Given autocorrelation of X and, Solve two differential equations… with corresponding initial conditions… copyright Robert J. Marks II

copyright Robert J. Marks II Filtering WSS RP’s If input is WSS, so is output! Mean: copyright Robert J. Marks II

copyright Robert J. Marks II Filtering WSS RP’s If input is WSS, so is output! Autocorrelation: ? Not a function of t  WSS!!! copyright Robert J. Marks II p. 415

copyright Robert J. Marks II Filtering WSS RP’s Power Spectral Density: Final result LTI System X(t) Y(t) copyright Robert J. Marks II p. 415

copyright Robert J. Marks II Filtering WSS RP’s Power Spectral Density: Discrete time systems LTI System X[n] Y[n] copyright Robert J. Marks II p. 415

Filtering WSS RP’s: Example Input: Zero Mean White Noise LTI System X(t) Y(t) copyright Robert J. Marks II Example: p. 418

Filtering WSS RP’s: Example Low Pass, High Pass & Band Pass Filters LPF X(t) Y(t) copyright Robert J. Marks II Example: p. 418

copyright Robert J. Marks II SX(f)  0 Proof Consider filter f 0 f 0 +df 1 Y(t) will have a PSD Thus:  A PSD property copyright Robert J. Marks II

Filtering WSS RP’s: Derivatives Derivatives: If X(t) is differentiable, let copyright Robert J. Marks II

Filtering WSS RP’s: ARMA ARMA: IIR filter with white WSS input LTI System X[n] Y[n] copyright Robert J. Marks II Example: p. 421

copyright Robert J. Marks II Filtering WSS RP’s Autocorrelation: Final result summary. LTI System X(t) Y(t) copyright Robert J. Marks II p. 415

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