Set-membership filtering and a set-membership normalized LMS algorithm with an adaptive step size

Set-membership identification (SMI) theory is extended to the more general problem of linear-in-parameters filtering by defining a set-membership specification, as opposed to a bounded noise assumption. This sets the framework for several important filtering problems that are not modeled by a "true" unknown system with bounded noise, such as adaptive equalization, to exploit the unique advantages of SMI algorithms. A recursive solution for set membership filtering is derived that resembles a variable step size normalized least mean squares (NLMS) algorithm. Interesting properties of the algorithm, such as asymptotic cessation of updates and monotonically non-increasing parameter error, are established. Simulations show significant performance improvement in varied environments with a greatly reduced number of updates.