In this letter, a novel projection algorithm is proposed in which projection onto a triangle consisting of the three even-vertices closest to the vector to be projected replaces check polytope projection, achieving the same FER performance as exact projection algorithm in both high-iteration and low-iteration regime. Simulation results show that compared with the sparse affine projection algorithm (SAPA), it can improve the FER performance by 0.2 dB as well as save average number of iterations by 4.3%.

The Euclidean projection operation is the most complex and time-consuming of the alternating direction method of multipliers (ADMM) decoding algorithms, resulting in a large number of resources when deployed on hardware platforms. We propose a simplified line segment projection algorithm (SLSA) and present the hardware design and the quantization scheme of the SLSA. In simulation results, the proposed SLSA module has a better performance than the original algorithm with the same fixed bitwidths due to the centrosymmetric structure of SLSA. Furthermore, the proposed SLSA module with a simpler structure without hypercube projection can reduce time consuming by up to 72.2% and reduce hardware resource usage by more than 87% compared to other Euclidean projection modules in the experiments.

Approximate Simultaneous Diagonalization (ASD) is a problem to find a common similarity transformation which approximately diagonalizes a given square-matrix tuple. Many data science problems have been reduced into ASD through ingenious modelling. For ASD, the so-called Jacobi-like methods have been extensively used. However, the methods have no guarantee to suppress the magnitude of off-diagonal entries of the transformed tuple even if the given tuple has an exact common diagonalizer, i.e., the given tuple is simultaneously diagonalizable. In this paper, to establish an alternative powerful strategy for ASD, we present a novel two-step strategy, called Approximate-Then-Diagonalize-Simultaneously (ATDS) algorithm. The ATDS algorithm decomposes ASD into (Step 1) finding a simultaneously diagonalizable tuple near the given one; and (Step 2) finding a common similarity transformation which diagonalizes exactly the tuple obtained in Step 1. The proposed approach to Step 1 is realized by solving a Structured Low-Rank Approximation (SLRA) with Cadzow's algorithm. In Step 2, by exploiting the idea in the constructive proof regarding the conditions for the exact simultaneous diagonalizability, we obtain an exact common diagonalizer of the obtained tuple in Step 1 as a solution for the original ASD. Unlike the Jacobi-like methods, the ATDS algorithm has a guarantee to find an exact common diagonalizer if the given tuple happens to be simultaneously diagonalizable. Numerical experiments show that the ATDS algorithm achieves better performance than the Jacobi-like methods.

Dual-motor driving servo systems are widely used in many military and civil fields. Since backlash nonlinearity affects the dynamic performance and steady-state tracking accuracy of these systems, it is necessary to study a control strategy to reduce its adverse effects. We first establish the state-space model of a system. To facilitate the design of the controller, we simplify the model based on the state-space model. Then, we design an adaptive controller combining a projection algorithm with dynamic surface control applied to a dual-motor driving servo system, which we believe to be the first, and analyze its stability. Simulation results show that projection algorithm-based dynamic surface control has smaller tracking error, faster tracking speed, and better robustness and stability than mere dynamic surface control. Finally, the experimental analysis validates the effectiveness of the proposed control algorithm.

We propose a novel bias-free adaptive beamformer employing an affine projection algorithm with the optimal regularization parameter. The generalized sidelobe canceller affine projection algorithm suffers from a bias of a weight vectors under the condition of no reference signals for output of an array in the beamforming application. First, we analyze the bias in the algorithm and prove that the bias can be eliminated through a large regularization parameter. However, this causes slow convergence at the initial state, so the regularization parameter should be controlled. Through the optimization of the regularization parameter, the proposed method achieves fast convergence without the bias at the steady-state. Experimental results show that the proposed beamformer not only removes the bias but also achieves both fast convergence and high steady-state output signal-to-interference-plus-noise ratio.

Recently, a data-selective method has been proposed to achieve low misalignment in affine projection algorithm (APA) by keeping the condition number of an input data matrix small. We present an improved method, and a complexity reduction algorithm for the APA with the data-selective method. Experimental results show that the proposed algorithm has lower misalignment and a lower condition number for an input data matrix than both the conventional APA and the APA with the previous data-selective method.

This letter presents a new mathematical expression for the excess mean-square error (EMSE) of the affine projection (AP) algorithm. The proposed expression explicitly shows the proportional relationship between the EMSE and the condition number of the input signals.

We propose a gradient-limited affine projection algorithm (GL-APA), which can achieve fast and double-talk-robust convergence in acoustic echo cancellation. GL-APA is derived from the M-estimation-based nonlinear cost function extended for evaluating multiple error signals dealt with in the affine projection algorithm (APA). By considering the nonlinearity of the gradient, we carefully formulate an update equation consistent with multiple input-output relationships, which the conventional APA inherently satisfies to achieve fast convergence. We also newly introduce a scaling rule for the nonlinearity, so we can easily implement GL-APA by using a predetermined primary function as a basis of scaling with any projection order. This guarantees a linkage between GL-APA and the gradient-limited normalized least-mean-squares algorithm (GL-NLMS), which is a conventional algorithm that corresponds to the GL-APA of the first order. The performance of GL-APA is demonstrated with simulation results.

This paper introduces an efficient affine projection algorithm (APA) using iterative hyperplane projection. The inherent effectiveness against the rank deficient problem has led APA to be the preferred algorithm to be employed for various applications over other variety of fast converging adaptation algorithms. However, the amount of complexity of the conventional APA could not be negligible because of the accomplishment of sample matrix inversion (SMI). Another issue is that the "shifting invariance property," which is typically exploited for single channel case, does not hold ground for space-time decision-directed equalizer (STDE) application deployed in single-input-multi-output (SIMO) systems. Therefore, fast adaptation schemes, such as fast traversal filter based APA (FTF-APA), becomes impossible to utilize. The motivation of this paper deliberates on finding an effective algorithm on the basis of APA, which yields low complexity while sustaining fast convergence as well as excellent tracking ability. The performance of the proposed method is evaluated under wireless SIMO channel in respect to bit error rate (BER) behavior and computational complexity, and upon completion, the validity is confirmed. The performance of the proposed method is evaluated under wireless SIMO channel in respect to bit error rate (BER) behavior and computational complexity, and upon completion, the validity is confirmed.

This paper proposes a new algorithm called the fast Projection algorithm, which reduces the computational complexity of the Projection algorithm from (p+1)L+O(p3) to 2L+20p (where L is the length of the estimation filter and p is the projection order.) This algorithm has properties that lie between those of NLMS and RLS, i.e. less computational complexity than RLS but much faster convergence than NLMS for input signals like speech. The reduction of computation consists of two parts. One concerns calculating the pre-filtering vector which originally took O(p3) operations. Our new algorithm computes the pre-filtering vector recursively with about 15p operations. The other reduction is accomplished by introducing an approximation vector of the estimation filter. Experimental results for speech input show that the convergence speed of the Projection algorithm approaches that of RLS as the projection order increases with only a slight extra calculation complexity beyond that of NLMS, which indicates the efficiency of the proposed fast Projection algorithm.