The least mean-square (LMS) algorithm
This chapter presents the celebrated least mean-square (LMS) algorithm, developed by Widrow and Hoff in 1960. This algorithm is a member of stochastic gradient algorithms, and because of its robustness and low computational complexity, it has been used in a wide spectrum of applications. It can be used to solve the Wiener–Hopf equation without finding matrix inversion. Furthermore, it does not require the availability of the autocorrelation matrix of the filter input and the cross correlation between the filter input and its desired signal. The chapter provides examples that elucidate the use of the LMS algorithm to different areas of engineering and create an appreciation for the versatility of this important algorithm. In some practical applications it is mathematically attractable to have complex representation forms of the underlying signals. For example, base-band signals in quadrature amplitude modulation format are written as a summation of two components: real in-phase component and imaginary quadrature component.