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Set-Membership Adaptive Filtering

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Abstract

The families of adaptive filtering algorithms introduced so far present a trade-off between the speed of convergence and the misadjustment after the transient. These characteristics are easily observable in stationary environments. In general fast-converging algorithms tend to be very dynamic, a feature not necessarily advantageous after convergence in a stationary environment. In this chapter, an alternative formulation to govern the updating of the adaptive filter coefficients is introduced. The basic assumption is that the additional noise is considered bounded, and the bound is either known or can be estimated [1]. The  key strategy of the formulation is to find a feasibility set (This set is defined as the set of filter coefficients leading to output errors whose moduli fall below a prescribed upper bound.) such that the bounded error specification is met for any member of this set. As a result, the set-membership filtering (SMF) is aimed at estimating the feasibility set itself or a member of this set [2].

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Notes

  1. 1.

    This set is defined as the set of filter coefficients leading to output errors whose moduli fall below a prescribed upper bound.

  2. 2.

    The reader should note that in earlier definition of the objective function related to the affine projection algorithm a constant \(\frac{1}{2}\) was multiplied to the norm to be minimized. This constant is not relevant and is only used when it simplifies the algorithm derivation.

  3. 3.

    \( \tau _{i} - \frac{1}{2\mathrm{BW}}< \tau < \tau _{i} + \frac{1}{2\mathrm{BW}} \) with \(\mathrm{BW}\) denoting the bandwidth of the transmitted signal.

  4. 4.

    In an actual implementation, \({{\mathbf {x}}}(k)\) originates from the received signal after filtering it through a chip-pulse matched filter and then sampled at chip rate.

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  1. Universidade Federal do Rio de Janeiro, Niterói, Rio de Janeiro, Brazil

    Paulo S. R. Diniz

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Diniz, P.S.R. (2020). Set-Membership Adaptive Filtering. In: Adaptive Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-29057-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-29057-3_6

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