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The BCGM-OR algorithm with a guaranteed value, which optimizes the number of iterations in the block-orthogonal projection algorithm to which the conjugate gradient technique is applied and which guarantees the stability of operation, provides an upper bound on estimation error through the setting of a guaranteed value. As such, it has significantly improved the estimation accuracy without sacrificing the excellent tracking capacity of the block-orthogonal projection algorithm. On the other hand, regarding the learning identification method in which the block size used in the block-orthogonal projection algorithm is set to 1, a step gain (the stochastically fastest step gain) that maximizes the rate of convergence in addition to the setting of a guaranteed value has been demonstrated. Based on the stochastically fastest step gain on the learning identification method, this paper proposes a technique wherein a similar method is introduced into the BCGM-OR algorithm with a guaranteed value and validates its effectiveness. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(3): 87–98, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.1143 [ABSTRACT FROM AUTHOR]

Adaptive algorithms for modifying filter coefficients play an important role in adaptive signal processing. Various methods have been proposed to date. Of these, the algorithm based on the orthogonal projection into a subspace spanned by the input signal vector has the property of estimating the optimum filter coefficients rapidly even if the input signals are correlated. Among the methods based on the orthogonal projection operation, the block orthogonal projection algorithm is a method that has a good balance between computational complexity and convergence properties. This algorithm is expressed in terms of a form containing the Moore–Penrose-type generalized inverse matrix. Several methods have been proposed for specific execution of this algorithm. In this paper, a lemma on the UD decomposition and the block inverse matrix for the autocorrelation matrix is used for a method executing the block orthogonal projection algorithm containing a Moore–Penrose generalized inverse matrix with less computational complexity than that in the conventional method. The present method is characterized by computation of the orthogonal projection matrix corresponding to the i-th step in an arbitrary block by means of the orthogonal projection matrix of the preceding step. By computer simulation, the present method and that proposed earlier are compared and the superiority of the present algorithm is discussed. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(5): 1–13, 2000 [ABSTRACT FROM AUTHOR]

The BCGM-OR algorithm was derived on the basis of the results of error analysis of the block orthogonal projection algorithm using the conjugate gradient method. The authors have improved the convergence characteristics of this algorithm significantly under noisy conditions while at the same time reducing the computational load by optimizing the number of iterations for the conjugate gradient method. However, countermeasures are required for unstable operation resulting from variations in the size of the input signal, which is a fundamental limit of the orthogonal projection-type adaptive algorithm in the authors' method. With respect to the learning identification method when the block length of the block orthogonal projection algorithm is 1, the authors have already shown that a guarantee value (the upper limit of the estimated error after convergence) can be obtained by using a method in which the coefficient updating is stopped when the state vector norm becomes smaller than the set threshold value. Based on this method, the authors first stabilize the guarantee value in the BCGM-OR algorithm when performing coefficient update cutoff. They then identify the threshold value used to satisfy the guarantee value and next describe the BCGM-OR algorithm with a guarantee value. Finally, using computer simulations, the authors show good convergence characteristics under noisy conditions for the proposed method, and then confirm that the operational stability is superior particularly with respect to nonstable input signals. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(5): 42–52, 2000 [ABSTRACT FROM AUTHOR]

Learning identification is one of the most widely utilized adaptive algorithms due to its small computational complexity. A problem with this algorithm is that a division by the square norm of the state vector is includ- ed in the coefficient update process which easily produces an instability when the nonstationary such as speech is given as the input. A method known to cope with such a problem and to realize the stable convergence involves not updating the coefficient when small input signals continue. This paper discusses this approach. As a first step, the situation is assumed where the square norm of the state vector is less than the specified threeshold and the effect of interrupting the coefficient update is analyzed. The convergence and the time-constant are formulated and the guarantee of the estimation accuracy for the convergence value is discussed. Then, the step gain is derived for the specified guarantee value so that the convergence speed is the fastest in the stochastic sense while ensuring stability. Lastly, the effectiveness of the proposed method is verified by computer simulation. It is shown that, even if the input signal is annotation, the fastest convergence speed is realized, together with the satisfactory estimation accuracy, which is always obtained according to the specified guarantee value after the convergence. [ABSTRACT FROM AUTHOR]

The real-time processing of the recursive least-square algorithm is difficult to execute by the conventional Neumann-type processor because of its large computational complexity. On the other hand, it is known that the recursive least-square algorithm based on the UD decomposition, which is equivalent to the recursive least-square algorithm, can be realized by the systolic array proposed by Rung. The systolic array to execute this algorithm is difficult to realize since it requires a tremendous number of elements and corrections. In the construction of the dedicated hardware, in general, the word length of the processor affects the processing speed and the hardware area. From such a viewpoint, it is important in the construction of the systolic array to execute the recursive least-square algorithm based on the UD decomposition that the word length of the processor should be minimized. To evaluate the word length, an error analysis is required for the finite word length operation of the recursive least-square algorithm based on the UD decomposition. This paper presents the finite word length floating-point error analysis for the recursive least-square algorithm, based on the UD decomposition, and evaluates the operation error. Then the convergence of the algorithm and the number of updates of the algorithm are evaluated analytically. Finally, by a computer simulation, the validity of the theoretical analysis for the convergence and the number of updates is verified. [ABSTRACT FROM AUTHOR]

With the progress of hardware technology, the development of real-time adaptive algorithms is becoming more and more important. The essential questions concerning adaptive algorithms are: 1) increasing the speed of the algorithm; and 2) speeding up the convergence. It seems that the block signal processing and the orthogonal projection on the subspace of state-vectors are effective methods to solve the for mentioned questions 1) and 2), respectively. This paper proposes a block orthogonal projection algorithm (BOPA) which combines the block processing with the orthogonal projection operation. Next, an analysis of this algorithm is presented and the fundamental properties of the relation between the block length and the convergence characteristics arc shown. Also, the efficiency of the proposed algorithm is verified by computer simulation. This algorithm is based on a block- by-block adaptation where the operation of orthogonal projection on the state-vector space is performed repetitively for arbitrary blocks. [ABSTRACT FROM AUTHOR]

The solution of a system of linear equations is very important in various fields of engineering, and various methods have been presented for the numerical solution. Especially, in the adaptive signal processing and adaptive control, the system of equations is often considered where the coefficient matrix is (pseudo-) positive-definite and symmetrical. The conjugate graduate method is known as a means to solve such a system of equations. Letting the rank of the coefficient matrix be R, it is a method where the solution for the system of equations is determined by iterating a certain procedure R times. In this method, even if the iterative process is terminated on the way due to the constraint for the computation time, for example, an approximate solution which is somewhat accurate is obtained. This method, however, still has room for improvement with respect to the accuracy of the approximate solution. From such a viewpoint, this paper examines geo- metrically the iterative computation process in the con- jugate gradient method and, based on that result, propos- es a new method of iterative solution for the system of linear equations with symmetrical coefficient matrix. Then the original method is slightly modified and an adaptive algorithm is derived by applying the result to the parameter estimation of the adaptive filter. Those methods are based on the orthogonal projection operation from the solution to be determined to the linear space. Consequently, it is expected that, even if the iterative computation is terminated on the way, a more accurate approximate solution is obtained, compared to the case of the conjugate gradient method or the adaptive algorithm based on the conjugate gradient method. By the property of the method proposed in this paper, the method is useful when the coefficient matrix is ill-conditioned. It is expected that the method will effectively be utilized in the signal processing and control. [ABSTRACT FROM AUTHOR]

The block orthogonal projection algorithm exhibits an excellent convergence property in a noise-free environment. However, a compromise must be sought between the speed and the precision of the convergence when there exists a noise. This paper considers the block orthogonal projection algorithm using the conjugate-gradient method, which is one of the realizations of the block orthogonal projection algorithm. The convergent value is evaluated, and it is shown that one of the major factors that determine the convergent value is the number of iterations in the algorithm. Based on the result of the error analysis, the optimal number of iterations for the block orthogonal projection algorithm is derived so that the effect of the noise can be reduced without sacrificing the high-speed property. For this purpose, the block adaptive algorithm is proposed, based on the block orthogonal projection algorithm using the conjugate-gradient method. computational complexity can be reduced drastically compared to the block orthogonal projection algorithm. [ABSTRACT FROM AUTHOR]

Two methods have been proposed for calculating the block orthogonal projection algorithm using the Moore-Penrose generalized inverse matrix. One is the conjugate gradient method, and the other is the block adaptive algorithm composed of the Gram-Schmidt orthogonalization and the learning identification. Both computation methods are suited to the high-speed estimation of the optimal coefficients with a small computational complexity. However, the operation may become very unstable due to the effects of observation error, quantization error, and other factors. This paper proposes a method of realizing the block orthogonal projection algorithm with the learning identification, where the orthogonal basis of the space spun by the state matrix in the block adaptive filter is used as the direction of modification. A feature of this method is that the orthogonal basis is calculated using the orthogonal projection matrix. Computer simulation shows that the proposed adaptive algorithm realizes a very stable operation without being affected by either the quantization error, the operational error, or the observation noise. [ABSTRACT FROM AUTHOR]