System Identification
6.435
SET 1–Review of Linear Systems
–Review of Stochastic Processes
–Defining a General Framework
Munther A. Dahleh
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Review
LTI discrete–time systems
transfer function
Note ( different notations)
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Stability
⇔ (real rational ) poles of are outside the disc
Strict causality
system has a delay.
Strict Stability
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Frequency Response
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Suppose then
for stable systems.
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Periodograms
Given: the Fourier transform is given by
Discrete – Fourier Transform (DFT)
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Inverse DFT
Periodogram
Nice Property: Parsaval’s equality
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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= energy contained in each frequency component.
Properties:
-periodicity
-If is real
Example:
(consider )
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Example:
(Why?)
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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This representation is valid over the whole interval [1,N] since u is periodic over N.
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Passing through a filter
Claim:
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Proof:
Note that:
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Hence:
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Stochastic Processes
Definition:
A stochastic process is a sequence of random variables with a joint pdf.
Definition:
= mean
= Correlation / Covariance
= Covariance
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Traditional definitions:
is Wide-sense stationary (WSS) if
constant
“This may be a limiting definition.” We will discuss shortly.
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Common Framework for Deterministic and Stochastic Signals
• A typical setup noise(stochastic)
experiment(deterministic)
output (mixed)
not constant(loose stationarity).
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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• Consider signals with the following assumptions:
and
is called quasi-stationary
• If is a stationary process, then it satisfies 1, 2 trivially.
• If is a deterministic signal, then
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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• Example: Suppose has finite energy
• In general:
deterministicstochastic
• Notation:
quasi-stationary
•
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Power SpectrumLet be a quasi-stationary process. The power spectrum is defined as
= Fourier transform of
• Example: for
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Recall
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• Result ; for any , quasi-stationary
Convergence as a distribution
Note: an erratic function
well behaved function
?
Cross Spectrum
Spectrum of mixed signals
{zero means}
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Spectrum of Filtered Signals
Generation of a process with a given Covariance:
given then x is the output of a filter with a WN signal as an input. The Filter is the spectral factor of
O
If signal
stable minimum phase.
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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SISOSISO
SISOSISO
Important relations
A model for noisy outputs:
quasi-stationaryconstant.
• Very Important relations in system ID.• Correlation methods are central in identifying an unknown plant.• Proofs: Messy; Straight forward.
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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Ergodicity
• is a stochastic process
Sample function or a realization
• Sample mean
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• Sample Covariance
• A process is 2nd-order ergodic if
mean the sample mean of any realization.
the sample covariance of any realization.
covariance
Ensemble averages• Sample averages
Lecture 1 6.435, System Identification Prof. Munther A. Dahleh
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A general ergodic process
+
WN Signal
quasi–stationary signaluniformly stable
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w.p.1
w.p.1
w.p.1
Remark:
Most of our computations will depend on a given realization of a quasi-stationary process. Ergodicity will allow us to make statements about repeated experiments.
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