Abstract: We présent an implementation of the transversal form of fast RLS on a high performance floating-poini Digital Signal Proccssor. It's well known that ...
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AMSE Conférence 'Signais & Systems", Jme lT-19, 1992, AMSE Press, VoL 1, pp. 17-28
Proceedings / n l c m .
Ceneva
(SvitzerUmd),
A stabilized fast RLS algorithm implementation for the Motorola DSP96002 with applications. O. CASPARY. M. TOMCZAK, P. NUS, C.WISS. C. VOMSCHEID Centre de Recherche en Automatique de Nancy, CNRS URA 821 Faculté des Sciences, B.P. 239,54506 Vandoeuvre Cedex (France). Ph. 83.91.20.69 - Fax 83.91.20.30
Abstract: We présent an implementation of the transversal form of fast RLS on a high performance floating-poini Digital Signal Proccssor. It's well known that roundoff crrors cause instabililics, so the implementation closely follows the so called stabilized version of Benallal's Fast Ka]man. This version was originally developed by Ljung. A forgetling factor permits to foUow slowly changing signal statistics. The algorithm behaviour was tested with a white noise. Based on our results, several applications can be considered. We have successfully applied this algorithm for the design of noise protection equipraent, real time EEG processing, NMR spectroscopy.
I. I N T R O D U C T I O N Adaptive filiering involves that filter parameters adjust with each next sample arrivai. The parameters are computed recursively, and real time processing requires the computation job to be completed before the acquisition of the next input sample. Recursive algoriihms minimizing a weighted least squares criterion, such as Recursive Least Squares (RLS) are heavily used because of their fast convergence as compared to LMS in case of abrupt changes ( 13]. On the other hand, thèse algorithms have a N^ computational complexity, denoted o(N^) where N is the number of parameters to be estimated. This is why an intensive research work has been going on to obtain fast RLS algorithms of oO^ opérations [6] [11] [12]. For our implementation we have chosen the Ljung's transversal Fast Kalman (FK) algorithm of o(lON) [12]. However, it is well known that divergence of this algorithm occurs rapidly, because of the accumulation of roundoff errors. To remedy this unstable behaviour without significantrisein computational requirements, we took the approach suggested by Benallal, i.e. the Stabilized Fast Kalman (SFK) [4]. We first show that this relatlvely simple stabilization procédure is suffïcient to ensure robust performance on high speed floating point DSP, where we are not limited by the range of representable values. Next we show how to design applications in the given framework. Last but not least, the capacities and performances of the algorithm are assessed with typical case studies, i. e. impulse response identification of an anti-noise helmet, real time spectrum estimation of
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Electroenccphalogram (EEG) signais, as well as spectral analysis in Nuclear Magnetic Résonance (NMR) spectroscopy. This section is followed by a brief description of the algorithm itself and its stabilizing procédure. A white noise was used at the input, as to permit the excitation of the whole frequency range. No sign of going out of convergence was observed. A brief description of the DSP used is presented next with some panicularities pertaining to the programming of transversal filters. In the application part, we first eraploy high order filters for the impulse response détermination of an antinoise helmet. Real time EEG spectral analysis is considered next Important information about différent brain sûtes is found in EEG spectral characteristics. Unfortunately, EEG signais are generally non-stationary. If it is assuroed that there are only slow changes in spectral characteristics, a forgetting factor allows to track thèse changes in order to observe how the spectrum evolves on a time scale. The last application deals with NMR specuoscopy. It is shown that the results of specu-al analysis of NMR signais may be improved by using a parametric approach in lieu of more conventional Fast Fourier Transform (FFT) algorithms. In the last part, we présent a more gênerai discussion about the possibilities offered by our processing System.
(2)
E(n) = ï wn-p.( y(p) - hj!j(n).XN(p) )^ p=l w is the forgetting factor (chosen between 0.9S and 1) and y(n), x(n) are causal. Prédiction error is defined as :
(3)
£(n) = y(n)-hJ(n-l).Xf^(n)
The vector hj^(n) has to be computed recursively as : (4)
hj^(n+l) = hj^(n) + R;^'(n+1). x^(n+l). e(n+l) where
is the autocorrélation matrix defined by RM(n)=
2;w""P.
(5)
Xjj(p).xJ(p)
p=l
,-1, With g^(n) = R^'(n). Xj^(n), we obtain : hj^(n-H) = hj^(n) + g^(n+l).E(n+l)
(6)
which is in close ressemblance to the Kalman updating équation, and this is where the Kalman gain namefor gj^ cornes from. II. S F K BASICS II. 1. The Fast Kalman Adaptive filter opération involves, choosing an adaptive criterion and updating filter parameters wiih the acquisition of each new sample, in coirespondence to a predefined criterion (seefig.1). rcfercnce )(n) , input rignal x(n)
Filter
Adaptive altorithm
itstimated y ( n ) /
error E(n)
Matrix inversion in (4), using the inversion lemma, leads to the classical RLS of o(N^). This complexity is due to the propagation of Rj^ through the computations. One method for reducing the computations is to exploit the symmetries of with a clever partitioning so as to eliminate redundant calculations. Applying such a method, vectors of length N or N+1 are propagated instead of matrices which leads to algorithms with complexity proportional to o(N). Thèse "fast algorithms" can be structured in two distinct parts, prédiction for gain updating, and adaptive filtering. Conuary to the classical RLS, the fast RLS computes and updates the backward linear prédiction coefficients (14) that minimize the sum of squared backward linear prédiction errors %. along with ihe forward coefficients aj^. Note that the forward and backward criteria are minimized separately. The other advantage of FK is the ability to make an exact initialization thanks to a simultaneous order and time updating routine, so that the bias induced by the RLS arbitrary initialization can be avoided [14]. The complète theory underlying the FK is found in [2] [12]. Ail necded variables are referenced in table 1, and the équations are in table 2.
Adaptive filter Fig. 1: Adaptive filtering.
»1) X(r>.î)
X(~2)
«(11.3)
«
