Exciting conditions for quantized state adaptive algorithms

A family of Quantized State (QS) adaptive algorithms is introduced. These are computationally simple versions of the LMS adaptive algorithm and are useful in two ways: as numerically fast implementations for adaptive tasks, and as a tool for studying the LMS algorithm in a quantized environment. Averaging theory is used to derive a persistence of excitation (PE) condition which guarantees exponential stability of one member of the QS family. Failure to meet this condition (which is not equivalent to the spectral richness PE condition for LMS) can result in exponential instability, even with the use of fixed leakage, showing that the stability properties of QS algorithms are heavily dependent on the character of the input sequence.