Elias Koukoutsis

Efficient recursive in order least squares FIR filtering and prediction

This paper is concerned with the efficient determination of the optimum, in the least squares sense, FIR filter on the basis of data samples of the input and desired response signals, by procedures recursive in the filter order. This situation typically arises when no a priori statistics are available and the system order is not known. The general multiinput-multioutput (multichannel) case is considered here and a fast algorithm is presented requiring for single channel signals approximately 2S + 15m multiplications (mps) per order m, S being the number of samples. In the special case of linear prediction it calls for about S + 12m mps. Hence it offers a computational reduction of 5m and 2m mps in comparison to the methods of Marple [1] and Morf et al. [2], respectively. Additionally, the proposed scheme is inherently symmetric and is suited very well to initialization of fast sequential algorithms as well as algorithms searching for the optimum lag filter.

Error sources and error propagation in the Levinson-Durbin algorithm

It is proved that there are two types of numerical error, due to finite precision, in the Levinson-Durbin algorithms; an erratic and a systematic one. The erratic one depends on the value the input autocorrelation accidentally takes at an iteration, and, essentially, it affects only the results obtained at this particular recursion. On the contrary, the systematic numerical error increases with the information the system carries and propagates essentially throughout the algorithm. It is shown that, for both types of error, as well as the overall one, there are specific intermediate quantities, calculated in the evolution of the algorithm, which may serve as precise indicators of the exact number of erroneous digits with which the various quantities are computed including the PARCOR coefficients and the filter coefficients. Therefore, the generated numerical error can be accurately traced. >

Comparing LS FIR filtering and l-step ahead linear prediction

This comparative study of the l-step-ahead linear prediction and least-squares finite impulse response (LS FIR) filtering problems emphasizes the numerical behavior of the resulting Toeplitz systems. It is shown that, although these systems are similar, the restraints on the autocorrelation coefficients fundamentally differentiate them. In the process of doing so, a new algorithmic scheme for the computation of the lagged lattice coefficients is developed, which exhibits fundamentally improved numerical behavior. Moreover, explicit formulas for the supremums of the absolute values of both the lagged lattice and filter coefficients are found theoretically and are experimentally confirmed by using the proposed algorithm. Finally, the bounds of the LS FIR filter coefficients are treated in comparison with the supremums of the lagged quantities. >

Error propagation and numerical recovery for a class of parametric DSP algorithms

The error propagation in various useful order- and time-recursive DSP algorithms is studied. It is demonstrated that in all these algorithms there are two kinds of error due to finite precision: an erratic and a systematic one. Examples of both kinds of error are provided, and special emphasis is given to the study of the systematic truncation or round-off error. It is shown that in the Levinson-type and Schur-type algorithms for the solution of the FLP (forward linear prediction) and the FIR (finite impulse response) problem, there is one dominant source of systematic error, while in the l-step ahead case there are two sources of such error. Moreover, it is pointed out that in the fast Kalman algorithms there are two kinds of systematic error. Precise indicators of the exact magnitude of the finite precision error are given, and possible recovery techniques are proposed.<<ETX>>

Error propagation and methods of error correction in LS FIR filtering and l-step ahead linear prediction

In the paper, a general method is presented, which leads to the exact prediction and tracing of the finite precision error generated in the solution of the l-step ahead linear prediction and the optimal FIR filtering problems. It is shown that two sources of numerical error exist, in the algorithms used for the solution of these problems. The first source lies in the formulas used for the forward linear prediction, while the second source is intimately connected with the formulae used specifically for the solution of the l-step ahead and LS FIR problems. The propagation of this numerical error is determined precisely, and it is shown that there exist specific intermediate quantities, calculated in the evolution of the algorithms, which are indicators of the exact magnitude of the overall finite precision error. Clearly, the way for the error correction is open and, in fact, in the paper, some methods are presented for improving the numerical accuracy of the solution of these problems. >

Numerical behaviour of Toeplitz solutions

The numerical behavior of the order-recursive algorithms for the solution of a number of typical Toeplitz systems is studied. Use is made of a novel methodology that gives a direct interpretation of the positive definiteness that leads to the explicit computation of the minimum absolute bounds of the 1-step and l-step-ahead linear predictors. Ill-conditioning is discussed, and an algorithmic scheme with improved numerical behavior is given for the l-step-ahead LP case. It is proved that the Levinson algorithm, when it is used for the solution of the least-squares finite-impulse-response filtering problem, manifests a fundamentally different numerical behavior than when it is used for the l-step-ahead problem.<<ETX>>

A New Multichannel Recursive Least Squares Algorithm for Very Robust and Efficient Adaptive Filtering

In this paper, a new multichannel recursive least squares (MRLS) adaptive algorithm is presented which has a number of very interesting properties. The proposed computational scheme performs adaptive filtering via the use of a finite window, where the burdening past information is dropped directly by means of a generalized inversion lemma; consequently, the proposed algorithm has excellent tracking abilities and very low misjudgment. Moreover, the scheme presented here, due to its particular structure and to the proper choice of mathematical definitions behind it, is very robust; i.e., it is less sensitive in the finite precision numerical error generation and propagation. Also, the new algorithm can be parallelized via a simple technique and its parallel form and, when executed with four processors, is faster than all the already existing schemes that perform both infinite and finite window multichannel adaptive filtering. Finally, due to the particular structure of this scheme and to the intrinsic flexibility in the choice of the window length, the proposed algorithm can act as a full substitute of the infinite window MRLS ones.

Exact analysis of the finite precision error generation and propagation in the FAEST and the fast transversal algorithms: a general methodology for developing robust RLS schemes

In this paper, an analysis for the actual and deeper cause of the finite precision error generation and accumulation in the FAEST-5p and the fast transversal filtering (FTF) algorithm is undertaken, on the basis of a new methodology and practice. In particular, it is proved that, in case where the input data in these algorithms is a white noise or a periodic sequence, then, out of all the formulas that constitute these two schemes, only four specific formulas generate an amount of finite precision error that consistently makes the algorithms fail after a certain number of iterations. If these formulas are calculated free of finite precision error, then all the results of the two algorithms are also computed error-free. In addition, it is shown that there is a very limited number of specific formulas that transmit the finite precision error generated by these four formulas. Moreover, a number of very general propositions is presented that allow for the calculation of the exact number of erroneous digits with which all the quantities of the FAEST and FTF schemes are computed, including the filter coefficients. Finally, a general methodology is introduced, based on the previous results, that allows for the development of new RLS algorithms that, intrinsically, suffer less of finite precision numerical problems and that therefore are, in practice, suitable for high quality fast Kalman filtering implementations.

A very robust, fast, parallelizable adaptive least squares algorithm with excellent tracking abilities

In this paper a new computational scheme is introduced for performing recursive least squares adaptive filtering. The proposed algorithm is far more robust than all the already existing RLS schemes, in the sense that it is drastically less sensitive in the numerical error due to the finite precision with which all operations are executed. Hence, it has a lifetime tens of times greater than all the previous RLS schemes. Moreover, the algorithm introduced here has excellent tracking abilities and, due to its particular structure, it is parallelizable. When it is executed in parallel by four processors, it is faster than all the existing RLS algorithms, and in particular, it is by m steps faster, where m is the system order, than the FAEST and the FTF computational schemes.<<ETX>>

Superlattice/superladder computational organization for linear prediction and optimal FIR filtering

A family of computational organizations for the solution of the Toeplitz systems appearing in the digital signal processing (DSP) techniques of linear prediction and optimal FIR filtering is presented. All these organizations are based on a structure called superlattice which governs the Toeplitz solving procedure and provides many possible implementations. Algorithmic schemes for the implementation of these organizations, suitable for single-processor and multiprocessor environments, are developed. Among them there are order recursive algorithms, parallel-algorithms of O(p) complexity which use O(p) processing elements, and partitioned-parallel algorithms. The last can make full use of any number of available, parallel-working processors, independently of the system order. Superlattice-type algorithms are described for many Toeplitz-based problems. >