Article

# Partial-Update L∞ -Norm Based Algorithms

Dept. of Electr. & Comput. Eng., Concordia Univ., Montreal, Que.

Circuits and Systems I: Regular Papers, IEEE Transactions on (Impact Factor: 2.4). 03/2007; 54(2):411 - 419. DOI: 10.1109/TCSI.2006.883863 Source: IEEE Xplore

**ABSTRACT**

The computational complexity of an adaptive filtering algorithm increases with increasing the filter tap length and therefore, the use of such a filter can become prohibitive for certain applications, especially for real-time implementation. In this paper, we develop low-complexity adaptive filtering algorithms by incorporating the concept of partial updating of the filter coefficients into the technique of finding the gradient vector in the hyperplane based on the L

_{infin}-norm criterion. Two specific partial update algorithms based on the sequential and M-Max coefficient updating are proposed. The statistical analyses of the two algorithms are carried out, and evolution equations for the mean and mean-square of the filter coefficient misalignment as well as the stability bounds on the step size are obtained. It is shown that the proposed partial update algorithm employing the M-Max coefficient updating can achieve a convergence rate that is closest to that of the full update algorithm. Finally, simulations are carried out to validate the theoretical results and study the convergence rate of the proposed algorithms-
- "Since it is not feasible to obtain the exact value in (15), we replace it with an instantaneous value. For a large M, the fluctuations in the input signal energy ||u i || 2 from one iteration to the next are sufficiently small to justify the following approximation [1], [12], [13] "

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**ABSTRACT:**We present a low-complexity complementary pair affine projection (CP-AP) adaptive filter which employs the intermittent update of the filter coefficients. To achieve both a fast convergence rate and a small residual error, we use a scheme combining fast and slow AP filters, while significantly reducing the computational complexity. By employing an evolutionary method which automatically determines the update intervals, the update frequencies of the two constituent filters are significantly decreased. Experimental results show that the proposed CP-AP adaptive filter has an advantage over conventional adaptive filters with a parallel structure in that it has a similar convergence performance with a substantial reduction in the total number of updates. -
- "By (14) and (15), we arrive at μ(i)e(i) = Δ T i S T L (i) u i . (16) The minimum L ∞ -norm solution vector of Δ i , i.e., min Δ i ∞ subject to (16), is given by Δ i = μ(i)e(i) S T L (i) u i 1 sign{S T L (i) u i }. (17) Then, the updating equation is obtained as follows: w i+1 = w i + μ(i)e(i) S T L (i) u i 1 + sign{S T L (i) u i } (18) where μ(i) is data dependent and given by μ(i) = 1 − γ/|e(i)|, where |e(i)| > γ 0, otherwise (19) and T L (i) = {t k (i)} L k=1 collects indexes indicating the first L largest maxima of |u(i − k + 1)| where k = 1, 2, · · · , N, which is used in the M-Max PU-NSLMS algorithm [8] "

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**ABSTRACT:**This paper provides a novel normalized sign least-mean square (NSLMS) algorithm which updates only a part of the filter coefficients and simultaneously performs sparse updates with the goal of reducing computational complexity. A combination of the partial-update scheme and the set-membership framework is incorporated into the context of L∞-norm adaptive filtering, thus yielding computational efficiency. For the stabilized convergence, we formulate a robust update recursion by imposing an upper bound of a step size. Furthermore, we analyzed a mean-square stability of the proposed algorithm for white input signals. Experimental results show that the proposed low-complexity NSLMS algorithm has similar convergence performance with greatly reduced computational complexity compared to the partial-update NSLMS, and is comparable to the set-membership partial-update NLMS. -
- "Since it is not feasible to obtain the exact value in (11), we replace it with an instantaneous value. For a large M, the fluctuations in the input signal energy ||u i || 2 from one iteration to the next are small enough to justify the approximation below [1], [15]. "

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**ABSTRACT:**We present a new structure for parallel affine projection (AP) filters with different step-sizes. By observing their error signals, the proposed alternating AP (A-AP) filter selects one of the two AP filters and updates the weights of the selected filter for each iteration. As a result, the total computations required for the proposed structure is almost the same as that for a single AP filter. Experimental results show that the proposed alternating selection scheme extracts the best properties of each component filter, namely fast convergence and small steady-state error.

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