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Tracking of Time-Varying Systems

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Tracking of Time-Varying Systems

Adviser: Dr. Yung-An Kao

Student: Chin-Chuan Chang

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

- 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.

- 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.

- 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

- 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

- Hence, we may reformulate the degree of nonstationary to
- The degree of nonstationary, , bears a useful relation to the misadjustment of adaptive filter.

- 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

- MSD can defined by
- The tap-weight error may be expressed as
- Weight vector noise:
- Weight vector lag:

- By ,we may express MSD as
- Estimation variance defined by
- Lag variancedefined by

- 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