Filters are a tool in signal processing that is used to manipulate the information from an input signal to create desired output. Adaptive filters are required when the specifications to achieve the desired output are unknown and are time-variant. There are many applications of adaptive filters. They include echo cancellation, channel equalization, beamforming, noise cancellation and signal enhancement.
In order for the adaptive filter to learn the parameters of the unknown system, an adaptive algorithm with an objective function is required. Generally, the objective function is to minimize the error between the output of the filter and the desired signal. The design of the adaptive algorithm comes with some engineering tradeoffs. For example there exists a tradeoff between the final misadjustment of the system and the convergence speed, the rate in which the adaptive filter learns the specifications.
The parameter that effectively controls the learning rate in normalized least mean squares (NLMS) algorithm is the stepsize parameter. A large stepsize results in a fast convergence and a large misadjustment, while a small stepsize results in slow convergence but a small misadjustment. One approach to resolving this tradeoff is to have the stepsize be a function of time. This is obtained by having the stepsize value be inversely proportional to the amount of convergence achieved by the system. The downfall of this approach is that it relies on having accurate statistics of the system.
A new approach to solving this problem is the affine combination of adaptive filters. In this configuration, the input signal passes through two adaptive filters whose output is combined to give a single output. Each filter is updated using a different stepsize, and the amount that each filter contributes to the final output is dependent on the ratio of individual error signals (e1(n) and e2(n)). Therefore, combining adaptive filters provides a simple and practical approach to resolving the engineering tradeoff.