Published on Jan 03, 2023

When reference signal for the FxLMS algorithm is taken from an acoustic sensor convergence can be very slow due to great eigen value spread. Using a non acoustic sensor, Such as a tachometer, cancellation of narrow band noise in the sensed fundamental frequency and harmonically related ones can be achieved very fast, although other periodic noises and underline broad band noise will remain.

Backward prediction errors resulting at the various stages of an adaptive lattice predictor(ALP) represent a a time-domain orthogonalization of the input signal.An ALP structure, with the acoustic reference has input signal, before a FxLMS makes up the FxGAL algorithm. Due to the orthogonalization, FxGAL can be significantly faster compared to FxLMS with reference from a microphone. When compared to FxLMS with tachometer signal, it is not faster but it can cancel every periodic noise, independently of the harmonical relation between them, as well as the underlined broad band noise.

The Filtered-x Least Mean Square (FxLMS) algorithm(1) is the most widely used in the context of adaptive active control, due to its simplicity as well as robustness.However the main drawback of this algorithm is its relatively slow and signal dependent convergence, which is determined by the eigen value spread of the underlying corelation matrix of the input signal.When working in nonstationary environments such as automobiles, slow convergence is a critical problem, since we would desire to cancel transient noise, which occurs at the vehicle start-ups, stops or gear-Shifts, or with sudden changes of engine speeds or road noise from tyres.

A practical solution to this problem, very commonly used, is to use non acoustic sensors, such as tachometer instead of acoustic ones and artificially generate the signal to use as reference. This way, convergence can be achieved very fast, since it is possible to generate orthogonal references(in-face and quadrate components).On the other hand, it is only possible to cancel narrow band noises in the fundamental frequency sensed by the non acoustic sensor and other harmonically related frequencies, where as every other periodic or broad band noise will remain uncancelled.

In this paper, we introduce an algorithm the Filtered-x Gradient Adaptive Lattice(FxGAL), that aims to improve the convergence of the whole adaptive systems when using acoustic sensors to get the reference signal, at the expense of increased computational complexity.

The approach consists in conditioning the FxLMS reference signal by preprocessing it and obtaining the decomposition of the signal in orthogonal(decorrelated) components.With a decorrelated input signal the convergence modes of the FxLMS system are decoupled,and the whole adaptive filter of order L turns into L independent adaptive filters of just one coefficient.This independent systems can have their own adaptation step size, in order to obtain the same convergence speed for all of them

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