Outline Linear System Response Linear System to Random Inputs • Response to deterministic input. • Response to stochastic input. Characterize the stochastic output using: mean, mean square, autocorrelation, M. Sami Fadali spectral density. Professor of Electrical Engineering Continuous-time systems: stationary, University of Nevada nonstationary. Discrete-time systems. 1 2 Response to Random Input Response to Deterministic Input Analysis: Given the initial conditions, Response to a given realization is useless. input, and system dynamics, Characterize statistical properties of the characterize the system response. response in terms of: • Moments. Use time domain or s-domain methods • Autocorrelation. to solve for the system response. • Power spectral density. Can completely determine the output. 3 4
Analysis of Random Response Calculus for Random Signals Stationary steady-state analysis: 1. Dynamic systems: integration, differentiation. Stationary input, stable LTI system Integral and derivative are defined as limits. After a “sufficiently long period” Stationary response, can solve in s-domain. Random Signal: Limit may not exist for all realizations. Nonstationary transient analysis: 2. Time domain analysis. Convergence to a limit for random signals (law, probability, q th mean, almost sure). Can consider unstable or time-varying systems. Mean-square Calculus: using mean-square convergence. 5 6 Continuity Theorem (Shanmugan & Breipohl, Stark & Woods) Continuity is mean-square continuous if its WSS Deterministic continuous function at � autocorrelation function is continuous at �� Lim . �→� � � � � � � � Proof: Mean-square continuous random function at � � � � 0 Lim �� �� �� �→� � � � � � � � � Lim � �→� � � � � � � � � Can exchange limit and expectation for finite � 2 � �� 0 � Lim variance. �→� � �� � � 0 if is continuous at . �� 7 8
Interchange Limit & E{.} (Shanmugan & Breipohl, p. 162) Mean-square Derivative � ��� �� � Ordinary Derivative Lim For a mean square continuous process • � �→� with finite variance and any continuous For a finite variance stationary process. • function M.S. Derivative: Limit exists in a mean- • square sense. Lim � � ��� �� � � �→� � Lim �→� 9 10 Stationary Steady-state Analysis Expectation of Output Expectation of Output � Assume Stable LTI system in steady-state. �� � Stationary random input process. � �� � � �� � � 0 � ���� �� �� �� The integral is bounded if the linear system is �→� BIBO stable. � �→� � Mean is scaled by the DC gain General result for MIMO time-varying case. 11 12
Autocorrelation & Pow er Stationary Steady-state Analysis Spectral Density of Output Autocorrelation of Output � � � � �� �� �� � � � �� �� �� �� � � � � � � � �� �� �� �� �� � � � � �� � ∗� � � �� �� �� � Change of variable �� Laplace transform: �� � Change order of integration. �� 13 14 Stationary Steady-state Analysis Example: Gauss-Markov Process Stable LTI system with transfer function Used frequently to model random signals � � ��� �� �� �� For SISO case: X ( s ) F ( s ) � �� � � �� �� X ( s ) F ( s ) � � � �� �� 15 16
Example: SDF of Output Mean-square Value of Output Use integration table for 2-sided LT � Spectral factorization �� ���� �� �� ��� �� �� �� � � �� �� � � ��� � � � �� � � � � � � � �� � � 17 18 System w ith White Noise Input Bandlimited White Noise Input � �� � �� � �� �� � �� �� � � � � � � � � � �� � �� � � � �� � �� � � �� �� Approximately the same as white noise. 19 20
Example Noise Equivalent BW �� Find the noise equivalent bandwidth for the filter � �������� ������ ��� Approximate physical filter with ideal filter � � �� ��� � ����� ������ ��� ���� � � BW of ideal filter =noise equivalent BW Gain � �� � � � � 1 ��� 2 � � Ideal filter: BW . 21 22 Shaping Filter Example: Gauss-Markov Use spectral factorization to obtain filter TF � � � �� �� �� � � �� �� � Spectral Factorization gives the shaping �� filter F ( s ) X ( s ) unity white noise � G ( s ) 23 24
Nonstationary Analysis Natural (Zero-input) Response Zero-input response: response due to Linear system: superposition. initial conditions. Total response = zero-input response + zero- Response for state-space model state response Assume zero cross-correlation to add autocorrelations. Random initial conditions & random input. � ��� � � L 25 26 Example: RC Circuit Steady-state Response RC circuit in the steady state. Capacitor charged by unity Gaussian white noise input voltage source. �� � � Close switch and discharge capacitor. ��� Random initial condition but deterministic Use Table 3.1 (Brown & Hwang, pp. 109) discharge. Unity Gaussian R R with � white noise � � voltage source C � � � � � 27 28
Close sw itch at t = 0 Plot of Natural Response 2 t ( ) Unity Gaussian R R v c white noise � � voltage source C � � � � � ��/� � � � � ���/� ���/� � � 29 30 Forced (Zero-state) Response Forced (Zero-state) Response MIMO Time-Varying Case SISO Time-invariant Case � � �� � � � � � � � � � � � � � Autocorrelation � � � � � � �� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � �� � � � �� � � Obtain mean square with t 1 = t 2 � � 31 32
Mean Square: SISO Time- invariant Case Example: RC Circuit � �� � ⁄ �� R �� � � � Gaussian white Mean Square � � � � noise voltage C source f � � �� � � � � � �� � � � � � �� ⁄ ⁄ �� � �� � � � � � � � �� ��� � ⁄ ��� � ⁄ � � � � 33 34 Example: Autocorrelation Discrete-Time (DT) Analysis Difference equations. �� � � � � z-transform solution. �� � � �� � � � � = time advance operator ( �� = delay) � � � � v v=u+t 2 t 1 � � t 2 Similar analysis to continuous time but � � summations replace integrals. t 2 t 1 � � u � � � t 1 � � X ( s ) � � F ( s ) 1 / s � � � 35 36
Z-Transform (2-sided) Response of DT System Convolution Summation � � � �� Linear transform ���� ���� Z-transform � Convolution Theorem �� � ���� � ���� 37 38 Even Function Expectation of the Output � � � � ���� ���� �� �� ��� Stationary ���� ���� � � � � � � � �� � � � � � � � � � � � � �� ���� ���� ��� ���� � � 1 � � �� The mean is scaled by the DC gain 39 40
Discrete-time Processes Properties of and R real, even: real, even, cosine series Power spectrum: Discrete-time Fourier Transform � (DTFT) of autocorrelation. �� �� � � ���� � �� �� ���� � �� ���� ���� � ��� �� ��� � �� �� ��� 41 42 Autocorrelation of the Output Autocorrelation of the Output � � � � � � � � �� ���� ���� Substitute ���� ���� � � � � � � � �� ���� ���� ���� ���� � � � �� � ∗� � � � � �� �� �� ���� ���� 43 44
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