Download Linear System Response to Random Inputs

Linear System Response
to Random Inputs
M. Sami Fadali
Professor of Electrical Engineering
University of Nevada
Outline

Linear System

Characterize the stochastic output using:
mean, mean square, autocorrelation,
spectral density.
Continuous-time systems: stationary,
nonstationary.
Discrete-time systems.


• Response to deterministic input.
• Response to stochastic input.
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2
Response to Random Input
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.
3


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.
2.
Calculus for Random Signals
Stationary steady-state analysis:



Stationary input, stable LTI system
After a “sufficiently long period”
Stationary response, can solve in s-domain.


Time domain analysis.
Can consider unstable or time-varying
systems.



Nonstationary transient analysis:


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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6
Continuity Theorem (Shanmugan &
Breipohl, Stark & Woods)
Continuity

Deterministic continuous function
WSS
is mean-square continuous if its
is continuous at
autocorrelation function
.
at
Lim
→
Mean-square continuous random function
Lim
Lim
→
→

Proof:
at
Can exchange limit and expectation for finite
variance.
7
2
if
0
is continuous at
Lim
→
0
.
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Interchange Limit & E{.}
(Shanmugan & Breipohl, p. 162)

Mean-square Derivative
For a mean square continuous process
with finite variance and any continuous
function
•
Ordinary Derivative
•
For a finite variance stationary process.
•
M.S. Derivative: Limit exists in a meansquare sense.
Lim
Lim
→
→
Lim
→
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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.
→

General result for MIMO time-varying case.
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Mean is scaled by the DC gain
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Autocorrelation & Power
Spectral Density of Output
Stationary Steady-state Analysis
Autocorrelation of Output
∗


Change of variable
Change order of integration.

Laplace transform:
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Stationary Steady-state Analysis
Stable LTI system with transfer function
Example: Gauss-Markov Process
Used frequently to model random signals
For SISO case:
X(s)
F(s)
X(s)
F(s)
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Example: SDF of Output
Mean-square Value of Output

Spectral factorization
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System with White Noise Input
Use integration table for 2-sided LT
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Bandlimited White Noise Input
Approximately the same as white noise.
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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
Shaping Filter
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Example: Gauss-Markov
Use spectral factorization to obtain filter TF
F(s)
Unity White Noise
G(s)
X(s)
Colored Noise
Spectral Factorization gives the shaping
filter
Shaping Filter
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Nonstationary Analysis
Natural (Zero-input) Response

Linear system: superposition.

Total response = zero-input response + zerostate response

Assume zero cross-correlation to add
autocorrelations.

Random initial conditions & random input.

Zero-input response: response due to
initial conditions.

Response for state-space model
L
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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
discharge.
Unity Gaussian
R

R
white noise
voltage source
Use Table 3.1 (Brown & Hwang, pp. 109)
with
C
1
2
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4
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Plot of Natural Response
Close switch at t = 0
Unity Gaussian
white noise
voltage source
R
R
C
/
/
/
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Forced (Zero-state) Response
Forced (Zero-state) Response
MIMO Time-Varying Case
SISO Time-invariant Case


Autocorrelation
Obtain mean square with t1 = t2
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Mean Square: SISO Time-
invariant Case
Example: RC Circuit
⁄

R
Gaussian white
noise voltage
source f
Mean Square
⁄
⁄
C
⁄
⁄
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Example: Autocorrelation
Discrete-Time (DT) Analysis


t2
v v=u+t  t
2
1


t2t1
u
Difference equations.
z-transform solution.
= time advance operator (
= delay)
Similar analysis to continuous time but
summations replace integrals.
t1
X(s)
F(s)
1/s
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Z-Transform (2-sided)
Response of DT System
Convolution Summation

Linear transform
Z-transform

Convolution Theorem
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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
Properties of

R real, even:
and
real, even, cosine series
Power spectrum: Discrete-time Fourier Transform
(DTFT) of autocorrelation.
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Autocorrelation of the Output
Autocorrelation of the Output

Substitute
∗
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Spectral Density Function
Mean Square of Output
⇔
Same result using the convolution theorem.
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Cross Correlation
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Cross Correlation (2)
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Cross Spectral Density
Cross Spectral Density
,
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Expressions for SISO Case
Autocorrelation: use
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Example
For the transfer function


Power Spectral Density

Identification:
Determine the PSD of the output due to a unity
white noise sequence.
for unity white noise
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X(z)
F(z)
Shaping Filter
Conclusion
G(z)


Verify that
is the transfer function of
the shaping filter for colored noise with PSD
.




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References
1.
2.
3.
4.
R. G. Brown & P. Y. C. Hwang, Introduction to
Random Signals and Applied Kalman Filtering, J.
Wiley, NY, 2012.
A. Papoulis & S. U. Pillai, Probability, Random
Variables and Stochastic Processes, McGraw Hill,
NY, 2002.
K. S. Shanmugan & A. M. Breipohl, Detection,
Estimation & Data Analysis, J. Wiley, NY, 1988.
T. Söderström, Discrete-time Stochastic Systems :
Estimation and Control, Prentice Hall, NY, 1994.
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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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