[PPT] - Linear System Response Linear System to Random Inputs Response to PowerPoint Presentation

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Linear System Response to Random Inputs

M. Sami Fadali

Professor of Electrical Engineering University of Nevada

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Outline

 Linear System

Response to deterministic input.
Response to stochastic input.

 Characterize the stochastic output using:

mean, mean square, autocorrelation, spectral density.

 Continuous-time systems: stationary,

nonstationary.

 Discrete-time systems.

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Response to Deterministic Input

 Analysis: Given the initial conditions,

input, and system dynamics, characterize the system response.

 Use time domain or s-domain methods

to solve for the system response.

 Can completely determine the output.

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Response to Random Input

 Response to a given realization is useless.  Characterize statistical properties of the

response in terms of:

Moments.
Autocorrelation.
Power spectral density.

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Analysis of Random Response

1.

Stationary steady-state analysis:

Stationary input, stable LTI system
After a “sufficiently long period”
Stationary response, can solve in s-domain.

2.

Nonstationary transient analysis:

Time domain analysis.
Can consider unstable or time-varying

systems.

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Calculus for Random Signals

 Dynamic systems: integration, differentiation.  Integral and derivative are defined as limits.  Random Signal: Limit may not exist for all

realizations.

 Convergence to a limit for random signals

(law, probability, qth mean, almost sure).

 Mean-square Calculus: using mean-square

convergence.

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Continuity

 Deterministic continuous function

at Lim

→

Mean-square continuous random function at Lim

→

 Can exchange limit and expectation for finite

variance.

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Continuity Theorem (Shanmugan & Breipohl, Stark & Woods) WSS is mean-square continuous if its autocorrelation function

is continuous at

. Proof:

Lim

→

2 0 Lim

→

if

is continuous at

.

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Interchange Limit & E{.} (Shanmugan & Breipohl, p. 162)

 For a mean square continuous process

with finite variance and any continuous function Lim

Mean-square Derivative
Ordinary Derivative

Lim

→

For a finite variance stationary process.
M.S. Derivative: Limit exists in a mean-

square sense.

→

Lim

→

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Expectation of Output

 The integral is bounded if the linear system is

BIBO stable.

 General result for MIMO time-varying case.

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Stationary Steady-state Analysis Expectation of Output

Assume

 Stable LTI system in steady-state.  Stationary random input process.

→
→
 Mean is scaled by the DC gain

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Stationary Steady-state Analysis

Autocorrelation of Output

 Change of variable

 Change order of integration.

Autocorrelation & Pow er Spectral Density of Output

∗
 Laplace transform:
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Stationary Steady-state Analysis

Stable LTI system with transfer function

For SISO case:
F(s)

X(s)

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Example: Gauss-Markov Process Used frequently to model random signals

F(s)

X(s)

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Example: SDF of Output

Spectral factorization

Mean-square Value of Output

 Use integration table for 2-sided LT

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System w ith White Noise Input

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Bandlimited White Noise Input

Approximately the same as white noise.

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Noise Equivalent BW

Gain 2



Ideal filter: BW .

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 Approximate physical filter with ideal filter
BW of ideal filter =noise equivalent BW
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Example

 Find the noise equivalent bandwidth for the filter

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Shaping Filter Use spectral factorization to obtain filter TF

G(s)

F(s) unity white noise X(s)

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Example: Gauss-Markov

Spectral Factorization gives the shaping

filter

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Nonstationary Analysis

 Linear system: superposition.  Total response = zero-input response + zero-

state response

 Assume zero cross-correlation to add

autocorrelations.

 Random initial conditions & random input.

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Natural (Zero-input) Response

 Zero-input response: response due to

initial conditions.

 Response for state-space model

L

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Example: RC Circuit

 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

discharge.

R R C Unity Gaussian white noise voltage source 28

Steady-state Response

 Use Table 3.1 (Brown & Hwang, pp. 109)

with

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Close sw itch at t = 0

R R C Unity Gaussian white noise voltage source

/
/

/

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Plot of Natural Response

) (

2 t

vc

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Forced (Zero-state) Response

MIMO Time-Varying Case

 Obtain mean square with t1 = t2

Forced (Zero-state) Response

SISO Time-invariant Case

 Autocorrelation
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Mean Square: SISO Time-

invariant Case

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Example: RC Circuit
⁄
 Mean Square
⁄

⁄

⁄
⁄

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R C Gaussian white noise voltage source f

Example: Autocorrelation

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t2 t1 t2t1 v=u+t2  t1 v u 1/s F(s) X(s)

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Discrete-Time (DT) Analysis

 Difference equations.  z-transform solution. 

= time advance operator ( = delay)

 Similar analysis to continuous time but

summations replace integrals.

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Z-Transform (2-sided)

 Linear transform

 Convolution Theorem

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Response of DT System

Convolution Summation

Z-transform
Even Function
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Expectation of the Output

Stationary
1

The mean is scaled by the DC gain

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Discrete-time Processes

Power spectrum: Discrete-time Fourier Transform (DTFT) of autocorrelation.

Properties of

and

 R real, even:

real, even, cosine series

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Autocorrelation of the Output

Autocorrelation of the Output
 Substitute
∗
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Spectral Density Function

⇔
Same result using the convolution theorem.

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Mean Square of Output

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Cross Correlation

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Cross Correlation (2)

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Cross Spectral Density

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Cross Spectral Density

,
Expressions for SISO Case

Autocorrelation: use

 Power Spectral Density
 Identification:
for unity white noise

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Example

For the transfer function

1. Determine the PSD of the output due to a unity

white noise sequence.

2. Verify that

is the transfer function of the shaping filter for colored noise with PSD .

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Conclusion

 Mean, autocorrelation, spectral density.  Using white noise in analysis is acceptable.  Model colored noise using white noise.  Use to extend KF to cases where the noise

is not white.

 Discrete case.

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References

1.

R. G. Brown & P. Y. C. Hwang, Introduction to

Random Signals and Applied Kalman Filtering, J. Wiley, NY, 2012.

2.

A. Papoulis & S. U. Pillai, Probability, Random

Variables and Stochastic Processes, McGraw Hill, NY, 2002.

3.

K. S. Shanmugan & A. M. Breipohl, Detection,

Estimation & Data Analysis, J. Wiley, NY, 1988.

4.

T. Söderström, Discrete-time Stochastic Systems :