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Finite precision analysis of the conventional QR decomposition RLS algorithm

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Finite precision analysis of the conventional QR decomposition RLS algorithm
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  Finite Precision zyxwvu nalysis of the Conventional QR Decomposition RLS Algorithm z M. zyxwv . Siqueira and P. zyx   zy   iniz Programa de Engenharia ElCtrica COPPE/EE/Federal University of Rio de Janeiro Caixa Postal 68504 zyxwvu   Rio de Janeiro - RJ 21945 - Brazil zyxwvu Abstract- This paper presents relations for predicting the mean squared values in the devi- ations of the outputs in the Conventional QRD- zyxw LS algorithm [l] due to quantization effects. A set of formulas are presented, along with a pro- cedure to employ them to calculate the expect- ed 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. I. INTRODUCTION The Conventional QRD-RLS algorithm has been known for a long time as a numerically stable algorithm for solving the Recursive Least Squares Problem for adaptive filtering. Bounds for its internal variables have been calculated [2], but so far no detailed analysis of the errors propagation in this algorithm has been presented. This paper shows a set of relations, rather than closed formulas, for calculating the mean squared values of deviations in the 6lter outputs with the help of a simple procedure. This paper uses relations for mean squared values of the internal variables of the Conventional QRD-RLS algorithm, which can be found on 131. The QRD-RLS algorithm consists of the following steps [2], when implemented with inhite precision arithmetic: Matiwral Fonnulatzon: For zyxwvutsrqp   = 0 . N o zyxwvutsrq X k) = Q”(k). Q’ )Xp(k) = Q(1)X”k) Ewer. calculation: IJse ne of the equations c(k) = d(k) WH(k)X(k) zyxw (k) = Eq k)C05: Onr(k). Co.O,(k) 11. MEAN SQUARED VALUES OF DEVIATIONS N THE INTERNAL ARIABLES In this section, the ked-point quantization errors are defmed, and recursive difference equations that describe the recursive error propagation are derived and solved. We assume that the input signal has been scaled to avoid overflow [3], so that no overflow is considered in the analy- sis. Two’s complement arithmetic is used, and consequently no errors are considered when additions and subtractions are performed. We also assume that the instantaneous quan- tizations are performed by rounding, for any type of arith- metic. The quantization error has zero mean and variance ux = 2Pb 12, where b is the number of bits excluding the sign. We also assume that multiplication, division and square root operations introduce, respectively, quantization errors described by where a and b are scalars. The accumulated quantization error in each quantity is defined as he difference between its value in infinite precision (3) 0 7803 2428 5195 4.00 1995 IEEE 990  implementation and its value in finite precision, zyxwvut s shown below zyxwvutsrq   1 zyxwvu j O zyxwvutsrq A. zyxwvutsrqpo ean Squared Value of Aai,l(k) C. Mean Squared Value of Ac;(k) The quantity aij(k) is the intermediate value that the j-th element of the first row of XP(k) assumes, during the triangularization process, by the time the i-th Givens Rota, tion is applied. These quantities can be expressed as The quantity ci(k) is defined as he diagonal terms of the ma'trix zyxwvu (k) when calculations are performed as in a systolic arcliitecture [l]. This value is calculated by By using dehitions 13) and (10) and neglecting second higher order terms, it follows that and zyxwvut  14) By using the following approximation i 1 E{[Aaij(k)]'} U: zyxwvutsrqpon   ~E{[ACOSB,(~)]~} -0 i 1 Ai-m {E{ uLj(k)}E{[Asin Bm(k)]'} + m=O E{[Aumj(k)l2}(1 A)} + i- 1 it follows that {E{uLj(k)}(l A) D ean Square Value of Asin O, k) For a division operation, the following approximation is valimd for small Aa B. Mean Squared Value of Abi(lC) 1 1 a+Aa a The quantities bi(k) are the values of the first element of the intermediate vectors that result from the application of Givens rotations to the vector d2(lc). The general expression for bi(k) is given by Using the above relation on the definition of sin B,(k) a good approximation results. Using definitions 1)- 4) on this approximation and considering only the Gst-order terms, it fcdlows Substituting definitions 13) and 10) in the above equation and using only the first order terms as an approximation, it follows that E{ [Asin (k)]'} = E{[Aa,,,(k)lZ){(l )' E{[Ac,(k)] }A (l- A)' ; U; - 1 E{[AcosB,(k)]'} t nt=O E I j O E{[A~inBj(k)]~} E{[A j(k)J'}(l A)} + 99  E. zyxwvutsrqp ean Squared Value of 4 cos Oi(k) zyxwvut of COS zyxwvutsrqponml (k) it follows Using the approximation 19) in the following definition H. Mean Squared Value of Ae(k) The error signal in the infmite precision implementation is given by Substituting in the resulting equation the dehitions (1) zyxwv   (4) and considering only the -order terms, it follows Using definitions 1)- 4) in the above equation, and consid- ering only the resulting first order terms, it follows X(1 X)3E{ Aci(k)] } zyxwvutsrq   X(l-X)a'f, l+X zyxwvutsrqpon   X F. Mean Squared Value of 414 zyxw k) For the infinite precision implementation of the QR-RLS algorithm, the elements of the triangularized matrix U k) axe calculated by 7rtJ(k) = X'/ trw(k )cmB, k) + av k)sinB, k) zyxwvutsr 24) Substituting definitions (1)-(4) in the above equation, and considering only the resulting first order terms, it follows that I. Mean Squared Value of Aw, k) The taps vector w(k) is calculated through the buck sub- stitution algorithm [l], nd the mean squared value of its deviation was obtained on [6] as ; E{ [Asin Oi(k)] } [ 1 1+X G. Mean Squared Value of Adz,i(k) The elements of the vector d2(k) re resultant of the ap plication of N + 1 Givens rotations to X1I2d2 k l , that is The expeted value of norm of the deviation vector Aw(lc) can be obtained by using ,,(k) = X1r2%,,(k 1)cm (k) + b,(k) sin O,(k) 26) Using the same procedure for calculating nu, (k), t follows 111. SIMULATION ESULTS ND CONCLUSION A program in C was written in order to simulate the QR- RLS algorithm implemented with fixed-point arithmetic to 27) zyxwvut 99  verify the accuracy of the formulas presented. In all exam- ples, the experimental set up consisted of the system identi- fication application where both the input signal and mea, surement noise are pseudo-random sequences with normal distribution and zero-mean. zyxwvutsrq   moving-averaging processes was utilized zyxwvutsr s the unknown system in the simulations. The order of the unknown system was 4 in all experiments. In the examples presented, the QR-RLS algorithm ran for 1500 iterations and the simulation results are obtained by aver- aging the results of zyxwvutsrqp 00 independent runs. InGnite precision simulations are actually executed with 64 bits floating-point arithmetic. In the ked-point implementation, the quantities are represented by numbers with magnitude less than uni- ty. Frequent overflow is avoided by choosing the input signal variance appropriately. In the simulations to be presented, we measure the mean squared value of the deviation in the prediction error (excess of MSE) and the mean squared norm of Aw(k), and compare with the results obtained by using Table I. The validity of the proposed formulas was tested by changing some of the adaptive filtering environment param- eters. Except for where it is explicitly indicated, the simula, tions use the same values for input signal variance (-30 dB), precision (15 bits), forgetting factor zyxwvutsr A = 0.95) and reference signals (an MA process derived from the input signal, added with a measurement noise of variance equal to zyxwvutsrq 70dB). Analysing tables 11-IV we can verify that the procedure shown on table I and the developed formulas are a powerful tool for predicting the numerical errors due to the recursive nature of the QRD-RLS algorithm. -30 -32 -35 eferences zyxwvu l] S. Haykin, Adaptive FiEter Theory, Prentice-Hall, Englewood-Cliffs, New Jersey, 1991. [2] S. Leung and S. Haykin, “Stability of recursive QRD- LS algorithms using finite-precision systolic array im- plementation”, IEEE Trans. on Acoust., Speech, Signal Processing, vol. 37, pp. 760-763, May 1989. [3] M. G. Siqueira and P. S. R. Diniz, “Infinite precision analysis of the QRD-RLS algorithm”, IEEE Interna- tional Symposium on Circuit and Systems, Chicago - USA,1993. [4] P. S. R. Diniz and M. G. Siqueira, “Fixed-point error analysis of the QR- recursive least squares algorithm”, submitted to IEEE Transactions on Circuits and Systems. [5] C. Samson and V. Reddy, “Fixed point error andy- sis of the normalized ladder algorithm”, IEEE Trans. on Acoust., Speech, Signal Processing, vol. 31, p. 1177-1191, October 1983. [6] M. G. Siqueira and P. S. R. Diniz, “Stabilty Analysis of the QR-Recursive Least Squares Algorithm”, IEEE Midwest Conference on Circuit and Systems, De- troit - USA ,1993. I -64.7 -64.3 -92.6 -92.2 I -62.6 -62.3 -92.5 -92.2 -59.7 -89.4 -92.5 -92.2 TABLE C PSEUDO-CODE OR COMPUTATION F zyxw { [AW~(~)]~) for i=O; i 5 N; ++) I Calculate E{[Au~~(~)]’} hrough 14) Calculate E{[AC~(~)]~} hrough (18) Calculate E([A in zyxw , k)]’} and E{ [ACOS Z k)lz} hrough (21) and (23) for +i+l; j _ N; ++ i Calculate E{ [Aa, k)]’} through 14) Calculate E{[AU,~(~)]~} hrough 25) X 0.90 Calculate E([Abi(k)]*} hrough 15) Calculate E{ [A ,i(k)]’} hrough 27) Calculate E{[Awi(k)Iz} hrough (30) E{llak)ll;} dB) EiIWk)l2} dB) Simulated I[ Calculated Simulated Calculated -68.2 11 -67.7 -93.7 II -93.2 TABLE I1 SIMULATIONS OR VARIOUS zy : Et IP k)ll3 dB) i ” dB) ’ Simulated Calculated Simulated Calculated -2.5 -69.7 -69.S -92.5 -92.2 -27 -67.7 -67.3 -92.6 -92.2 I \ ,,,L, \ I LL \ I‘ I \ I i ” dB) ’ Simulated Calculated Simulated Calculated -2.5 -69.7 -69.S I -92.5 -92.2 -27 -67.7 -67.3 -92.6 -92.2 TABLE 11 SIMULATIONS FOR VARIOUS TABLE IV SIMULATIONS OR DIFFERENT RECISIONS -68.2 -64.3 Ij -91.7 -92.2 -124.9 -124.6 /I -152.8 -152.4 50 - 154.0 -1.54.7 I[ -182.8 -182.5 -94.8 -94.4 [ -122.3 -122.:3 993
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