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Tracking of Time-Varying Systems

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

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  1. Tracking of Time-Varying Systems Adviser: Dr. Yung-An Kao Student: Chin-Chuan Chang

  2. Outline • Introduction • Markov Model for System Identification • Degree of Nonstationary • Criteria for tracking assessment • Mean-Square Deviation • Misadjustment

  3. 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.

  4. 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.

  5. 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

  6. 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

  7. 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.

  8. 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

  9. Mean-Square Deviation • MSD can defined by • The tap-weight error may be expressed as • Weight vector noise: • Weight vector lag:

  10. Mean-Square Deviation (cont.) • By ,we may express MSD as • Estimation variance defined by • Lag variancedefined by

  11. 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

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