A Lattice Version of the Multichannel Fast QRD Algorithm Based on A Posteriori Backward Errors (original) (raw)
Related papers
A new order recursive multiple order multichannel fast QRD-RLS algorithm
Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004., 2004
In many adaptive filtering applications, such as in the case of the Volterra filters, the use of channels of unequal orders are common. This paper introduces a new Multichannel Fast QRD-RLS algorithm based on the updating of a posteriori backward errors that attains both cases of channels of equal or unequal orders. This new algorithm exhibits good numerical behavior and is order recursive, which allows a systolic array implementation with lower computational complexity compared to earlier proposed algorithms.
A general approach to the derivation of block multichannel fast QRD-RLS algorithms
Multichannel Fast QR Decomposition Recursive Least Squares (QRD-RLS) adaptive filtering algorithms have been mostly treated in the literature for channels of equal orders. However, in many applications, such as in the case of Volterra filtering, multichannel algorithms tailored for unequal orders are desirable. In this paper, a general formulation for deriving block versions of the Multichannel Fast QRD-RLS algorithms is introduced. The block type multichannel algorithms favor parallel processing implementations and also attain the reduced computational complexity and numerical robustness of the Fast QRD algorithms.
Infinite precision analysis of the fast QR decomposition RLS algorithm
Proceedings of IEEE International Symposium on Circuits and Systems - ISCAS '94, 1994
Caixa Postal 68504 -Rio de Janeiro -RJ -CEP 21945 -Brazil Abatmxct-This work develops relations for the mean squared value of internal variables in the Fast QRD-RLS. The objective is to derive relations based on known characteristics of input signals that predict the behavior of the internal quantities of the algorithm. It is shown that the Fast and Conventional QRD-RLS algorithms have some variables in common, and thus previous results of the infinite precision analysis of the Conventional algorithm remain valid for the Fast version. Conditions for avoiding overflow in fixed-point implementations are presented. Simulation results are also shown.
New fast QR decomposition least squares adaptive algorithms
IEEE Transactions on Signal Processing, 1998
This paper presents two new, closely related adaptive algorithms for LS system identification. The starting point for the derivation of the algorithms is the inverse Cholesky factor of the data correlation matrix, obtained via a QR decomposition (QRD). Both algorithms are of O(p) computational complexity, with p being the order of the system. The first algorithm is a fixed order QRD scheme with enhanced parallelism. The second is an order recursive lattice type algorithm based exclusively on orthogonal Givens rotations, with lower complexity compared to previously derived ones. Both algorithms are derived following a new approach, which exploits efficient time and order updates of a specific state vector quantity. Index Terms-Adaptive algorithms, fast algorithms. I. INTRODUCTION A DAPTIVE least squares algorithms for system identification [1]-[7] are popular due to their fast converging properties and are used in a variety of applications, such as channel equalization, echo cancellation, spectral analysis, and control, to name but a few. Among the various efficiency issues characterizing the performance of an algorithm, those of computational complexity, parallelism, and numerical robustness are of particular importance, especially in applications where medium to long filter lengths are required. It may sometimes be preferable to use an algorithm of higher complexity but with good numerical error robustness since this may allow its implementation with shorter wordlenghts and fixed point arithmetic. This has led to the development of a class of adaptive algorithms, based on the numerically robust QR factorization of the input data matrix via the Givens rotation approach [23]. The development of Givens rotations-based QR decomposition algorithms has evolved along three basic directions. Schemes of complexity per time iteration were the first to be derived, with being the order of the system [8], [9]. These schemes update the Cholesky factor of the input data correlation matrix and can efficiently be implemented on two-dimensional (2-D) systolic arrays. Furthermore, as it is shown in [9], the modeling error can be extracted directly without it being necessary to compute explicitly the estimates of the transversal parameters of the unknown FIR system.
MULTICHANNEL FAST QR-DECOMPOSITION RLS ALGORITHMS WITH EXPLICIT WEIGHT EXTRACTION
2006
Multichannel fast QR decomposition recursive least-squares (MC- FQRD-RLS) algorithms are well known for their good numerical properties and low computational complexity. However, these al- gorithms have been restricted to problems seeking an estimate of the output error signal. This is because their transversal weights are embedded in the algorithm variables and are not explicitly available. In this paper we present a novel technique that can extract the filter weights associated with the MC-FQRD-RLS algorithm at any time instant. As a consequence, the range of applications is extended to include problems where explicit knowledge of the filter weights is required. The proposed weight extraction technique is used to identify the beampattern of a broadband adaptive beamformer im- plemented with an MC-FQRD-RLS algorithm. The results confirm that the extracted coefficients of the MC-FQRD-RLS algorithm are identical to those obtained by any RLS algorithm such as the inverse QRD-RLS algor...
Signal Processing, 2007
Fast QR decomposition recursive least-squares (FQRD-RLS) algorithms are well known for their fast convergence and reduced computational complexity. A considerable research effort has been devoted to the investigation of single-channel versions of the FQRD-RLS algorithms, while the multichannel counterparts have not received the same attention. The goal of this paper is to broaden the study of the efficient and low complexity family of multi-channel RLS adaptive filters, and to offer new algorithm options. We present a generalized approach for block-type multichannel FQRD-RLS (MC-FQRD-RLS) algorithms that include both cases of equal and multiple order. We also introduce new versions for block-channel and sequential-channel processing, details of their derivations, and a comparison in terms of computational complexity. The proposed algorithms are based on the updating of backward a priori and a posteriori error vectors, which are known to be numerically robust.
The 2002 45th Midwest Symposium on Circuits and Systems, 2002. MWSCAS-2002., 2002
Fast QR decomposition algorithms based on backward prediction errors are well known for their good numerical behavior and their low complexity when compared to similar algorithms with forward error update. Their application to multiple channel input signals generates more complex equations although the basic matrix expressions are similar. This paper presents a unified framework for a family of multichannel fast QRD-LS algorithms. This family comprises four algorithms-two basic algorithms with two different versions each. These algorithms are detailed in this work.
Circuits, Systems, and Signal Processing, 2003
QR decomposition techniques are well known for their good numerical behavior and low complexity. Fast QRD recursive least squares adaptive algorithms benefit from these characteristics to offer robust and fast adaptive filters. This paper examines two different versions of the fast QR algorithm based on a priori backward prediction errors as well as two other corresponding versions of the fast QR algorithm based on a posteriori backward prediction errors. The main matrix equations are presented with different versions derived from two distinct deployments of a particular matrix equation. From this study, a new algorithm is derived. The discussed algorithms are compared, and differences in computational complexity and in finite-precision behavior are shown.
Implementation Of A Recursive Data Of Adaptive Qrd-Rls Algorithm Using Hdl Coder
Matrix inversion is a common function found in many algorithms used in wireless communication systems. As Field Programmable Gate Array (FPGA) become an increasingly attractive platform for wireless communication, it is important to understand the tradeoffs in designing a matrix inversion core on an FPGA. In this paper, a configurable Field Programmable Gate Array (FPGA)-based hardware architecture for matrix inversion is presented (download without data input). The proposed architecture of this algorithm has been design using Matlab-Simulink 7.8(R2009a) to deal with parallel structure. The design has been converted to behavioral VHDL coding style, as will as a VHDL test bench using Simulink HDL Coder tool to realize hardware d irectly from Simulink design. The use of Squared Givens rotations and a folded systolic array makes this architecture very suitable for FPGA implementation. Input is a matrix of complex, floating point values. The matrix inversion design can achieve throughput of 0.14Mupdates per second on a state of the art Altera Cyclone III (EP3C12F780C7) FPGA running at 125 MHz and studies a class of Q(N) approximate QR-based least squares (A-QR-LS) algorithm recently. It is shown that the A-QR-LS algorithm is equivalent to a normalized LMS algorithm with time-varying step sizes and element-wise normalization of the input signal vector. Keyword: Adaptive filtering, approximate QR-LS algorithm, performance analysis, QR-LMS algorithm, square root free givens based algorithms, transformed domain LMS algorithm.
Finite precision analysis of the conventional QR decomposition RLS algorithm
1994
This paper presents relations for predicting the mean squared values in the deviations of the outputs in the Conventional QRD-RLS algorithm [l] due to quantization effects. A set of formulas are presented, along with a procedure to employ them to calculate the expected value of squared norm of the tap deviation and the mean squared value of the error signal deviation. Simulation results are presented to show the accuracy of the obtained results.