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
1
Review
LTI discrete–time systems
transfer function
Note (
Lecture 1
different notations)
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
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6.435, System Identification
Prof. Munther A. Dahleh
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6.435, System Identification
Prof. Munther A. Dahleh
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Suppose
then
for stable systems.
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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:
Lecture 1
6.435, System Identification
Prof. Munther A. Dahleh
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Proof:
Note that:
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6.435, System Identification
Prof. Munther A. Dahleh
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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)
(loose stationarity).
Lecture 1
not
6.435, System Identification
Prof. Munther A. Dahleh
constant
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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:
stochastic
•
deterministic
Notation:
quasi-stationary
•
Lecture 1
6.435, System Identification
Prof. Munther A. Dahleh
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Power Spectrum
Let
be a quasi-stationary process. The power spectrum
is defined as
= Fourier transform of
•
Example: for
Lecture 1
6.435, System Identification
Prof. Munther A. Dahleh
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Recall
?
• Result ; for any
, quasi-stationary
Convergence as a distribution
Note:
an erratic function
well behaved function
Lecture 1
6.435, System Identification
Prof. Munther A. Dahleh
22
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:
O
SISO
If
signal
given
then x is the
output of a filter with a WN
signal as an input. The Filter is
the spectral factor of
SISO
stable minimum phase.
Lecture 1
6.435, System Identification
Prof. Munther A. Dahleh
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Important relations
A model for noisy outputs:
quasi-stationary
constant.
•
•
•
Very Important relations in system ID.
Correlation methods are central in identifying an unknown plant.
Proofs: Messy; Straight forward.
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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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6.435, System Identification
Prof. Munther A. Dahleh
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•
Sample Covariance
•
A process is 2nd-order ergodic if
mean
covariance
•
Sample averages
Lecture 1
the sample mean of any realization.
the sample covariance
of any realization.
Ensemble averages
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Prof. Munther A. Dahleh
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A general ergodic process
+
WN Signal
uniformly stable
Lecture 1
quasi–stationary signal
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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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6.435, System Identification
Prof. Munther A. Dahleh
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