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Lecture #3 Identification

ECE 842

Ali Keyhani

Identification is the determination on the basis of input and output, of a process (model), within a specified

class of processes (models), to which the process under test is equivalent.

y(.)

u(.)

w(.)

y(.)

u(.)

w(.)

Process

Equivalent

Model

The equivalent model is a model which has captured the underlying mechanism which has generated

the original process.

Three Step Modeling Procedure for Process {y(.)} excited by white noise

1. Model IdentificationMeans the determination of n and m for process.2. Parameter Estimation; The determination of { ^ )(ka } from { Y(j), J < k }3. Diagnostic Checking

Checking the {w(.)} sequence to establish that the noise indeed is white.

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Classical Approach to Identification and Modeling

Given

Build The Model OverThis Portion of PastHistory

Test The Model OverThis Portion of PastHistory

A 3 Step Procedure for Modeling IS:

Identify A CandidateModel

Estimate The Parameter ofThe Identified Model

Diagnostic CheckingIs The Noise

White?

Go To IdentifyAnother Model

Done

No

Yes

k

Y(k)

Covariance: A partial indication of the degree to which one variable is related to another is given by the

covariance, which is the expectation of the product of the deviation of two random variables from their

means.

{Y(.)} random process Y(.)

{x(.)} random process x(.)

Cov {x(k) Y(k)} = E [(x(k) ux) (Y(k) uy)]

= E [x(k) Y(k)] ux uy

Where

ux = E [x(k)]

uy = E [Y(k)]

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Cross Correlation Coefficient

yx

yxuukYkxE

=

)]()([

Variance of x ]))([( 22 xx ukxE =

Variance of y ]))([( 22yy

ukYE =

Notes:

1) The correlation coefficient is a measure of the degree of linear dependence between x and y.2) If x and y are independent, = 0 (the inverse is not true).3) If Y is a linear function of x, = 1

Zero Mean Process: Given process {Y(.)}, if the mean of the process is estimated and then subtracted

from each Y(k), we obtain a zero mean process.

Ex.

N

kkY 11 )}({ =

= )(1

11

^

kYN

u y

1

^

1 )1()1( yuYY =

1

^

1

1

1

^

1

)()(

.

)}({.

.

)2()2(

y

N

k

y

ukYkY

processmeanzeroaiskY

uYY

=

=

=

For zero mean processes {x(.)} and {Y(.)}

Cov { x(k) Y(k) } = E [ x(k) Y(k) ] = xy

xy is called cross covariance or correlation between x and Y variables

The basic cross correlation or covariance equation

E [ x(k) Y(k+j) ] = xy (j)

E [ Y(k) x(k+j) ] = yx (-j); i.e. xy (j) =yx (-j)

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For j = 0

E [ x(k) Y(k) ] = xy =xy (0)

The sample estimate of cross correlation for covariance,

mjjkYkxjN

jjN

k

xy .,.........1,0)()(1

)(1

^

=+

=

=

mjjkxkYjN

jjN

k

yx .,.........1,0)()(1

)(1

^

=+

=

=

Sample cross correlation coefficient of zero mean {Y(.)} and {x(.)} processes.

yx

xy

kYkxE^^

^^ )]()([)0(

=

=

==N

k

xykYkx

NkYkxE

1

^^

)()(1

)]()([)0(

yx

xy

xy ^^

^

^ )0()0(

=

=

==N

k

xxkx

N 1

22^

0

^

))((1

1

=

==N

k

yykY

N 1

22^

0

^

))((1

1

J Period apart cross correlation for process {x(.)} and {Y(.)} with zero mean.

xy (j) = E [ x(k) Y(k+j) ] = xy (j)

Cross correlation coeff.

yx

xy

xy

jj

)()( =

Same cross correlation coeff.

yx

xyxy ^^

^

^

)0()0(

=

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Historically, the correlation function is defined without restriction on mean and without normalization:

Correlation function mjjkYkxjN

jRN

k

xy .,.........1,0)()(1

)(1

1

^

=+

=

=

)]()([)(^^

jkYkxEjRxy +=

The covariance function has the mean value removed

mjujkYukxjN

jCjjN

k

yxxyxy .,.........1,0])(][)([1

)()(1

^

=+

==

=

In practice, the mean value usually has been subtracted from the data.

The correlation coefficient function

yx

xy

xy

jj

)()( = , )0(xx = , )0(yy =

j Period Apart, auto-covariance for process {Y(.)} with zero mean E [Y(k)] = 0

(j) = e [ (Y(k) Y(k+j) ]

Autocorrelation coefficient

0

)()(

jj =

Sample correlation coefficient

0

^

^

0

^

^^ )()]()([

)(

jjkYkYE

j =+

=

Where

..........2,1,0)()(1

)]()([)(1

^^

=+

=+=

=

jjkYkYjN

jkYjYEjjN

k

yy

=

=N

k

kYN 1

2

0

^

))((1

1

Note that j period apart autocorrelation is a partial indication of the degree to which present is dependent to

the past. i.e.

1) The autocorrelation is a measure of the degree of linear dependence between the present and thepast.

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2) If Y(k) and Y(k+j) are independent, then 0)(^ =yyy .3) 20)0( =YYR i.e the mean-square value of the random process can always be obtained by setting the

j = 0.

4) )()( jRjR YYYY = 5) )0()( YYYY RjR the largest value of the autocorrelation always occurs at j = 0.