Adjoint LMS: an efficient alternative to the filtered-x LMS and multiple error LMS algorithms

The Filtered-x LMS algorithm is currently the most popular method for adapting a filter when there exists a transfer function in the error path. Such instances arise, for example, in active control of sound and vibration. For multiple-input-multiple-output systems the Multiple Error LMS Algorithm is a generalization of Filtered-x LMS. The derivation of both algorithms rely on several assumptions, including linearity of the adaptive filter and error channel. Furthermore, in the Multiple Error LMS Algorithm the desirable order N computational complexity of LMS is lost, resulting in a prohibitive cost in certain DSP implementations. In this paper, we describe a new algorithm termed adjoint LMS which provides a simple alternative to the previously mentioned algorithms. In adjoint LMS, the error (rather than the input) is filtered through an adjoint filter of the error channel. Performance regarding convergence and misadjustment are equivalent. However, linearity is not assumed in the derivation of the algorithm. Furthermore, equations for single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) are identical and both remain order N