Underdetermined-order recursive least-squares adaptive filtering: the concept and algorithms

Abstract

Abstract-The concept of underdetermined recursive leastsquares (URLS) adaptive filtering is introduced. In particular, the URLS algorithm is derived and shown to be a direct consequence of the principle of minimal disturbance. By exploiting the Hankel structure of the filter input matrix, the fast transversal filter (FTF) version of the URLS algorithm (URLS-FTF) is derived including sliding window and growing window types, which allow alteration of the order of the URLS algorithm (which is equivalent to the linear prediction order of the input) in real time. The computational complexity is reduced to O(N) + O(m), where N is the adaptive filter length, and m is the order of the URLS algorithm. In addition, the efficient URLS (EURLS) algorithm, which does not compute the filter coefficients explicitly, thereby significantly reducing the computational load, is presented. Some earlier adaptive algorithms such as the averaged LMS, filtered-X LMS, and fast conjugate gradient are shown to be suboptimal approximations of the URLS algorithm. Instrumental variable approximations are also discussed. The URLS algorithm has a whitening effect on the input signal, which provides immunity to the eigenvalue spread of the input signal correlation matrix. Although the algorithm is sensitive to observation noise, it has good tracking characteristics, and tradeoffs can be found by tuning the step size. The utility of the URLS algorithms, in its basic form and FTF realization, depends heavily on the practical applicability of the mth-order sliding window estimate of the covariance matrix and mth-order FTF relations. The feasibility of the URLS family in practical applications is demonstrated in the simulations, which include channel equalization and acoustic echo cancellation in hands-free terminals with real data sets.

Design and evaluation of digital signal processing algorithms for acoustic feedback and echo cancellation

“…Alternatively, a computationally cheaper adaptive algorithm can be found in the underdetermined recursive least squares (URLS) family [139]. The affine projection algorithm (APA) provides an estimate of the acoustic feedback path impulse response by using only the M most recent microphone signal samples and loudspeaker signal vectors, i.e.,…”

Locally optimum adaptive signal processing algorithms

“…If are matrices with the same dimensions, we then have trace trace trace trace (22) We can now proceed with our proof. From the non-negative definiteness of the covariance matrix of , we can conclude that 2 , where we recall that , and…”

“…with equality if and only if (29) 2 If P P P 1 and P P P 2 are symmetric matrices, we say that P P with diagonal and having real and positive diagonal elements. Multiplying from left and right by and , respectively, we conclude that the product is also diagonalizable and has real and positive eigenvalues.…”

“…The modification in (40) of the original algorithm was considered necessary in order to achieve control, for fixed window size , over the excess mean square error (which is something that is not possible with the classical version). The above modification is proposed in [2] and [19], where it is also stated that this algorithm has the highest convergence rate of all algorithms belonging to the underdetermined RLS class. As we are going to see, this performance can, at best, match the performance of the LMS-Newton algorithm, which is also significantly inferior for cases where the window size is not adequately large.…”

“…A common algorithmic model known for its simplicity and practical usefulness, which is often analyzed in the literature, is the generalized form of LMS (1) where (2) and where are scalar nonlinearities, and is the input data vector with denoting a vector obtained by applying the scalar nonlinearity on every element of the vector Finally, denotes the desired response, and is a scalar positive quantity known as step size. Existing results aiming in the optimization of the convergence speed of the algorithm in (1) by proper selection of the nonlinearities and are limited.…”