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# Tracking of Time-Varying Systems PowerPoint PPT Presentation

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.

Tracking of Time-Varying Systems

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## Tracking of Time-Varying Systems

Student: Chin-Chuan Chang

### Outline

• Introduction

• Markov Model for System Identification

• Degree of Nonstationary

• Criteria for tracking assessment

• Mean-Square Deviation

### 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

• 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.)

• 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

• 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.)

• 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

• 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

### 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.)

• By ,we may express MSD as

• Estimation variance defined by

• Lag variancedefined by