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Recursive algorithms for adaptive transversal filters: Optimality and time-variance
Signal Processing 21 (1990) 357 Else,~ier
357
THESIS ALERT
R E C U R S I V E A L G O R I T H M S F O R A D A P T I V E T R A N S V E R S A L FILTERS: OPTIMALITY AND TIME-VARIANCE Gernot KUBIN* (Member EURASIP) Institut fiir Nachrichtentechnik und Hochfrequenztechnik, Technische UniversitKt Wien, Guflhausstrafle 25/389, A-1040 Vienna, Austria
This thesis presents a unified theory for the design and analysis of recursive algorithms for the adaptation of transversal digital filters. First, the widely used error minimization approach to algorithm design is investigated both in its deterministic and stochastic settings and it is shown not to allow a coherent derivation of practical algorithms from an optimality criterion. Such criteria always presuppose time-invariant application environments, while practical demands require algorithms suitable for tracking of time-varying environments such as fading channels in digital radio communications. The present proposal for algorithm design goes beyond mere error minimization in that the time variation of the coefficients of the adaptive filter is included as well. Both the error and the coefficient variation contribute to a deterministic, recursive optimality criterion that is exactly met during actual filter operation. In the sequel a wealth of algorithms (like LMS, individual coefficient adaptation, selforthogonalizing algorithms, RLS, and directional forgetting) is shown to fulfil this novel unified description and several algorithm modifications (such as signed regressor or signed error algorithms), which often appear ad hoc, are derived without invoking approximations. This covers also the utilization of error filters which is common practice in the adaptation of recursive * Thesis Advisor: Professor Dr. W.F.G. Mecklenbrfiuker. Original written in English. Elsevier Science Publishers B.V.
filter structures (e.g. in the HARF or SHARF algorithms). The second part of the thesis is devoted to the application of the described algorithm class to the tracking of time-varying environments. As a result, the tracking behaviour can be described as a filtering operation on the time evolution of the coefficients of a reference model derived from the application environment. Due to the overall structure of the adaptation algorithms, this learning filter is always linear and of first order. The same filter characteristic is found to control the influence of measurement noise on the adaptive filter coefficients. This fact confines the flexibility in the trade-off between tracking fidelity and noise suppression to a single design parameter, i.e., the cut-off frequency of the learning filter. To facilitate the incorporation of prior knowledge about the expected time variations, the algorithm structure needs to be extended with socalled coefficient filters. This allows to tailor the tracking behaviour in response to practical demands (or 'hypermodels' of the coefficient evolution with time) as linear higher-order filtering or nonlinear filtering. In conclusion, a series of topical proposals for such coefficient filters (covering leakage, momentum LMS, coefficient prediction, Kalman filters with smoothness priors, multistep algorithms and post-filtering) is discussed on the basis of the accordingly extended, unified optimality criterion.
357
THESIS ALERT
R E C U R S I V E A L G O R I T H M S F O R A D A P T I V E T R A N S V E R S A L FILTERS: OPTIMALITY AND TIME-VARIANCE Gernot KUBIN* (Member EURASIP) Institut fiir Nachrichtentechnik und Hochfrequenztechnik, Technische UniversitKt Wien, Guflhausstrafle 25/389, A-1040 Vienna, Austria
This thesis presents a unified theory for the design and analysis of recursive algorithms for the adaptation of transversal digital filters. First, the widely used error minimization approach to algorithm design is investigated both in its deterministic and stochastic settings and it is shown not to allow a coherent derivation of practical algorithms from an optimality criterion. Such criteria always presuppose time-invariant application environments, while practical demands require algorithms suitable for tracking of time-varying environments such as fading channels in digital radio communications. The present proposal for algorithm design goes beyond mere error minimization in that the time variation of the coefficients of the adaptive filter is included as well. Both the error and the coefficient variation contribute to a deterministic, recursive optimality criterion that is exactly met during actual filter operation. In the sequel a wealth of algorithms (like LMS, individual coefficient adaptation, selforthogonalizing algorithms, RLS, and directional forgetting) is shown to fulfil this novel unified description and several algorithm modifications (such as signed regressor or signed error algorithms), which often appear ad hoc, are derived without invoking approximations. This covers also the utilization of error filters which is common practice in the adaptation of recursive * Thesis Advisor: Professor Dr. W.F.G. Mecklenbrfiuker. Original written in English. Elsevier Science Publishers B.V.
filter structures (e.g. in the HARF or SHARF algorithms). The second part of the thesis is devoted to the application of the described algorithm class to the tracking of time-varying environments. As a result, the tracking behaviour can be described as a filtering operation on the time evolution of the coefficients of a reference model derived from the application environment. Due to the overall structure of the adaptation algorithms, this learning filter is always linear and of first order. The same filter characteristic is found to control the influence of measurement noise on the adaptive filter coefficients. This fact confines the flexibility in the trade-off between tracking fidelity and noise suppression to a single design parameter, i.e., the cut-off frequency of the learning filter. To facilitate the incorporation of prior knowledge about the expected time variations, the algorithm structure needs to be extended with socalled coefficient filters. This allows to tailor the tracking behaviour in response to practical demands (or 'hypermodels' of the coefficient evolution with time) as linear higher-order filtering or nonlinear filtering. In conclusion, a series of topical proposals for such coefficient filters (covering leakage, momentum LMS, coefficient prediction, Kalman filters with smoothness priors, multistep algorithms and post-filtering) is discussed on the basis of the accordingly extended, unified optimality criterion.