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)

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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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Suppose then

for stable systems.

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Periodograms

Given: the Fourier transform is given by

Discrete – Fourier Transform (DFT)

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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 )

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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:

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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.

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Common Framework for Deterministic and Stochastic Signals

• A typical setup noise(stochastic)

experiment(deterministic)

output (mixed)

not constant(loose stationarity).

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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

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• Example: Suppose has finite energy

• In general:

deterministicstochastic

• Notation:

quasi-stationary

•

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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}

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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.

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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.

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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

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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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