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. 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 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.
Performance regarding convergence and misadjustment are equivalent.
However, linearity is not assumed in the derivation of the algorithm. The vectors and provide for compact notation. The standard filtered x-LMS is illustrated in Figure 1 a where there exists a physical channel represented by between the output of the filter and the available desired response. This algorithm can be derived from the standard LMS algorithm assuming linearity by simply commuting the order of the filter and the channel. Graphically, an adjoint system is found for any filter realization by reversing the flow direction and swapping branching points with summing junctions and unit delays with unit advances.