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Tracking of Time-Varying Systems. Adviser: Dr. Yung-An Kao Student: Chin-Chuan Chang. Outline. Introduction Markov Model for System Identification Degree of Nonstationary Criteria for tracking assessment Mean-Square Deviation Misadjustment. Introduction.

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Tracking of time varying systems

Tracking of Time-Varying Systems

Adviser: Dr. Yung-An Kao

Student: Chin-Chuan Chang


Outline
Outline

  • Introduction

  • Markov Model for System Identification

  • Degree of Nonstationary

  • Criteria for tracking assessment

    • Mean-Square Deviation

    • Misadjustment


Introduction
Introduction

  • In previous study, we considered the average behavior of standard LMS and RLS algorithm operating in a stationary environment.

  • we try to examine the operation of these two filter algorithms in a nonstationary environment, for which the optimum Wiener solution takes on a time varying form.

  • we will discuss to evaluate the tracking performances of the stand LMS and RLS algorithm operating in a nonstationary environment.


Markov model for system identification
Markov Model for System Identification

  • An environment may become nonstationary in practice in one of two ways:

    • The frame of reference provided by the desired response may be time varying. EX: system identification

    • The stochastic process supplying the tap inputs of the adaptive filter is nonstationary. EX: equalize a time varying channel.


Markov model for system identification cont
Markov Model for System Identification (cont.)

  • First-order Markov process.

    • is noise vector, assumed to be zero mean and correlation matrix

    • The value of parameter a is very close to unity

  • Multiple regression

    • Where ν(n) is white noise, zero mean and variance σ2


Degree of nonstationary
Degree of Nonstationary

  • In order to provide a clear definition of the concept of “slow” and “fast” statistical variations of the model, it define (Macchi, 1995)

  • It may be rewritten as


Degree of nonstationary cont
Degree of Nonstationary (cont.)

  • Hence, we may reformulate the degree of nonstationary to

  • The degree of nonstationary, , bears a useful relation to the misadjustment of adaptive filter.


Criteria for tracking assessment
Criteria for tracking assessment

  • With the state of unknown dynamical system denoted by , and with the tap-weight vector of the adaptive transversal filter denoted by .

  • We formally define the tap-error vector as

  • On the basis of , we may go on to define two figure of merit for assessing the tracking capability of an adaptive filter

    • Mean-Square Deviation

    • Misadjustment


Mean square deviation
Mean-Square Deviation

  • MSD can defined by

  • The tap-weight error may be expressed as

  • Weight vector noise:

  • Weight vector lag:


Mean square deviation cont
Mean-Square Deviation (cont.)

  • By ,we may express MSD as

  • Estimation variance defined by

  • Lag variancedefined by


Misadjustment
Misadjustment

  • Another commonly used figure of merit for assessing the tracking capability of adaptive filter is misadjustment

  • is called the noise misadjustment

  • is called the lag misadjustment