Adaptive FIR Filter Algorithms

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Adaptive FIR Filter Algorithms. D.K. Wise ECEN4002/5002 DSP Laboratory Spring 2003. Introduction. Some signal processing applications require that the system compensate for things outside of its direct influence. Adaptive filtering algorithms allow for modeling of the outside phenomena.

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### Adaptive FIR Filter Algorithms

D.K. Wise

ECEN4002/5002 DSP Laboratory

Spring 2003

Introduction
• Some signal processing applications require that the system compensate for things outside of its direct influence.
• Adaptive filtering algorithms allow for modeling of the outside phenomena.
• Also, adaptive filtering can be used strictly for analysis/synthesis of a system.

Adaptive FIR Filters D.K. Wise

The Adaptive Digital Filter
• Commonly used signals and their notation
• Filter input and output: u[n] and y[n] respectively
• Desired or reference signal: d[n]
• Error signal: e[n] = d[n] - y[n]
• Error signal is used to modify the coefficients of the adaptive filter

Adaptive FIR Filters D.K. Wise

• When referring to adaptive filters, terms from statistics can also be employed.
• If the digital filter is FIR or all-zero, the adaptive system can also be called Moving Average or MA.
• If the digital filter is all-pole, the adaptive system can also be called Autoregressive or AR.
• If the digital filter is an IIR with zeros, the adaptive system can also be called ARMA.
• This presentation addresses FIR or MA filters only.

Adaptive FIR Filters D.K. Wise

• Transversal Filter: the direct form FIR we are used to.
• Lattice Filter: the feed-forward form of the lattice structure (not addressed herein).

Adaptive FIR Filters D.K. Wise

• The principal behind determining the coefficients of the filter model is to maximize the statistical correlation between the desired signal and the coefficients.
• Typically, this is done by minimizing the correlation between the error signal and the filter state as is relevant to the coefficients.
• If the adaptive filter is working, the error signal decreases in magnitude, which slows down the movement of the coefficients. The filter is therefore converging to a solution.

Adaptive FIR Filters D.K. Wise

• Least Mean Squares (LMS)
• Applies to transversal filter
• y = uT · w
• e = d - y
• w += µ e u
• µ is a constant controlling the speed of adaptation

Adaptive FIR Filters D.K. Wise

• Sometimes the adaptive system does not converge when the input signal varies widely in amplitude.
• This amplitude dependence can be addressed by including the input amplitude in the adaptation factor. This is known as the Normalized LMS algorithm.
• w += (µ e / (uT · u)) u

Adaptive FIR Filters D.K. Wise

Applications of Adaptive Filters
• System identification: adaptive equalization
• The adaptive filter attempts to model an unknown external system.

Adaptive FIR Filters D.K. Wise

Applications of Adaptive Filters
• Interference cancellation
• The adaptive filter attempts to isolate the component of a primary signal that is not part of a reference signal.

Adaptive FIR Filters D.K. Wise

Applications of Adaptive Filters
• Linear prediction
• This is like interference cancellation, but the adaptive filter uses a delayed version of the primary signal as the reference.

Adaptive FIR Filters D.K. Wise