Least mean mixed-norm adaptive filtering. J.A. Chambers, 0. Tanrikulu and A.G. Constantinides. Indexing terms: Adaptivrfilters, Aduprive signal processing, ...
equalisers are very close to the optimal values. However, the complex fuzzy adaptive filter requires highly intense computation of the n:=,M’= 256-dimensional matrix-to-vector multiplication in eqns. 5 and 6 at each time point k. Conclusions: In this Letter. we developed a complex fuzzy filter and applied the fuzzy filter to nonlinear channel equalisation problems with complex components. From simulations, we show that the. bit error rates of the fuzzy equalisers were close to that of the optimal equaliser. We plan to develop another complex fuzzy adaptive filter based on the LMS algorithm, which involves much less computation. 0 IEE 1994 27 July I994 Electronics Letters Online No: I9941065 K.Y. Lee (Dept. of Electronics Engineering, Chongwon National University, Kyungnum-do 64 I - 773, Korea)
least mean mixed-norm (LMMN) adaptation algorithms. The algorithm is based on the minimisation of J(k) = h ~ { e ” ( k )+}( I - h ) E { e 4 ( k ; ) ] (1) where k is the sample index, conventionally in time, E{.) is the mathematical expectation operator and h E [0, I] is the mixing parameter. When h = 1, eqn. 1 reverts to the error norm for the LMS algorithm, whereas when h = 0. eqn. 1 becomes the error norm for the LMF algorithm. Judicious choice of h thereby provides an algorithm with intermediate performance between that of LMS and LMF, and a mechanism to mitigate the problem of instability within the LMF algorithm. Moreover, for operation in a statistically nonstationary environment the mixing parameter may be adapted to match appropriately the properties of the measured signals. The error signal e(k) is assumed to be related to the desired response signal d(k), adaptive filter weight vector w(k) and input vector x ( k ) in the form e(k) = d(k) - w‘(k)x(k) as is the convention in adaptive signal processing, where (.y represents the vector transpose operation.
References L.-x., and MENDEL. J.M.: ‘Fuzzy adaptive filters with application to nonlinear channel equalization’, IEEE Trans. Fu:;? Syst.. 1993, 1, (3), pp. 161-170 LEE. K.Y.: ‘A fuzzy adaptive decision feedback equalizer’, Elec,tron. Lett., 1994, 30, (IO), pp. 749-751 HAYKIN. S.: ‘Adaptive filter theory’ (Prentice Hall, Englewood Cliffs, NJ, Second Edition, 1991) CHEN. S.: ‘Complex-valued radial basis network, Part 11: Application to digital communications channel equalization’, Signal Process., 1994, pp. 175-188
WANG.
Least mean mixed-norm adaptive filtering
L M M N / S F ) algorithm: The update equation for the least mean mixed-norm (squared and fourth) algorithm is derived from the following steepest descent type weight update equation: w(k 1) = w(k) - Ll?w(k).J(k) (2) where w(k) is the weight vector at sample k, p is the adaptation gain, and J(k) i s the instantaneous estimate of the gradient of the error norm J ( k ) evaluated at the current value of the weight vector w(k). Differentiation of eqn. 1 with respect to w(k) yields the LMMN(SF) update equation w(k 1) = w ( k ) Zpe(k)(X ~ ( 1 -~)e’(k))x(k) (3) The convergence properties of this algorithm are controlled by the adaptation gain parameter p and h . For the LMS algorithm, the mean value of the weight vector w(k), given the limitations of the independence assumption [I], is guaranteed to converge to the optimal Wiener solution provided that p satisfies
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J.A. Chambers, 0. Tanrikulu and A.G. Constantinides Indexing terms: Adaptivrfilters, Aduprive signal processing, Digitul signal processing
new family of stochastic gradient adaptive filter algorithms is proposed which is based on mixed error norms. These algorithms combine the advantages of different error noms, for example the conventional,relatively well-behaved. least mean square algorithm and the more sensitive, but better converging, least mean fourth algorithm. A mixing parameter is included which controls the proportions of the error norms and offers an extra degree of freedom within the adaptation. A system identification simulation is used to demonstrate the performance of a least mean mixednorm (square and fourth) algorithm. A
where N is the length of the adaptive filter and E{x’(k)) is the input signal power. Similarly, in a system identification scenario, where the desired response signal is given by d(k) = w‘x(k) + n(k) and w is the unknown system finite impulse response weight vector, which corresponds to the optimal Wiener solution when the length of the adaptive filter equals that of the unknown system, and n(k) i s the measurement noise uncorrelated with x(k), the adaptation gain p for LMF must satisfy
1 (5) 6.VE{ Ti2 (k)}E{.? (k)} where E ( n 2 ( k ) ) is the measurement noise power 121. Therefore, these results can be simply combined to give an albeit approximate hound for the adaptation gain for the LMMN(SF) algorithm O
