Evolutionary and variable step size strategies for multichannel filtered-x affine projection algorithms

This study is focused on the necessity to improve the performance of the affine projection (AP) algorithm for active noise control (ANC) applications. The proposed algorithms are evaluated regarding their steady-state behaviour, their convergence speed and their computational complexity. To this end, different strategies recently applied to the AP for channel identification are proposed for multichannel ANC. These strategies are based either on a variable step size, an evolving projection order, or the combination of both strategies. The developed efficient versions of the AP algorithm use the modified filtered-x structure, which exhibits faster convergence than other filtering schemes. Simulation results show that the proposed approaches exhibit better performance than the conventional AP algorithm and represent a meaningful choice for practical multichannel ANC applications.This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0097, Spanish Ministerio de Ciencia e Innovacion TEC2009-13741 and Generalitat Valenciana PROMETEO 2009/2013.Gonzalez, A.; Albu, F.; Ferrer Contreras, M.; Diego Antón, MD. (2013). Evolutionary and variable step size strategies for multichannel filtered-x affine projection algorithms. IET Signal Processing. 7(6):471-476. https://doi.org/10.1049/iet-spr.2012.0213S47147676Shin, H.-C., Sayed, A. H., & Song, W.-J. (2004). Variable Step-Size NLMS and Affine Projection Algorithms. IEEE Signal Processing Letters, 11(2), 132-135. doi:10.1109/lsp.2003.821722Paleologu, C., Benesty, J., & Ciochina, S. (2008). A Variable Step-Size Affine Projection Algorithm Designed for Acoustic Echo Cancellation. IEEE Transactions on Audio, Speech, and Language Processing, 16(8), 1466-1478. doi:10.1109/tasl.2008.2002980Shin, H.-C., & Sayed, A. H. (2004). Mean-Square Performance of a Family of Affine Projection Algorithms. IEEE Transactions on Signal Processing, 52(1), 90-102. doi:10.1109/tsp.2003.820077Kong, S.-J., Hwang, K.-Y., & Song, W.-J. (2007). An Affine Projection Algorithm With Dynamic Selection of Input Vectors. IEEE Signal Processing Letters, 14(8), 529-532. doi:10.1109/lsp.2007.891325Seong-Eun Kim, Se-Jin Kong, & Woo-Jin Song. (2009). An Affine Projection Algorithm With Evolving Order. IEEE Signal Processing Letters, 16(11), 937-940. doi:10.1109/lsp.2009.2027638Kim, K.-H., Choi, Y.-S., Kim, S.-E., & Song, W.-J. (2011). An Affine Projection Algorithm With Periodically Evolved Update Interval. IEEE Transactions on Circuits and Systems II: Express Briefs, 58(11), 763-767. doi:10.1109/tcsii.2011.2168023Bouchard, M. (2003). Multichannel affine and fast affine projection algorithms for active noise control and acoustic equalization systems. IEEE Transactions on Speech and Audio Processing, 11(1), 54-60. doi:10.1109/tsa.2002.805642Kong, N., Shin, J., & Park, P. (2011). A two-stage affine projection algorithm with mean-square-error-matching step-sizes. Signal Processing, 91(11), 2639-2646. doi:10.1016/j.sigpro.2011.06.003MoonSoo Chang, NamWoong Kong, & PooGyeon Park. (2010). An Affine Projection Algorithm Based on Reuse Time of Input Vectors. IEEE Signal Processing Letters, 17(8), 750-753. doi:10.1109/lsp.2010.2053355Arablouei, R., & Doğançay, K. (2012). Affine projection algorithm with selective projections. Signal Processing, 92(9), 2253-2263. doi:10.1016/j.sigpro.2012.02.018Gonzalez, A., Ferrer, M., de Diego, M., & Piñero, G. (2012). An affine projection algorithm with variable step size and projection order. Digital Signal Processing, 22(4), 586-592. doi:10.1016/j.dsp.2012.03.00

An affine projection algorithm with variable step-size and projection order

It is known that the performance of adaptive algorithms is constrained by their computational cost. Thus, affine projection adaptive algorithms achieve higher convergence speed when the projection order increases, which is at the expense of a higher computational cost. However, regardless of computational cost, a high projection order also leads to higher final error at steady state. For this reason it seems advisable to reduce the computational cost of the algorithm when high convergence speed is not needed (steady state) and to maintain or increase this cost only when the algorithm is in transient state to encourage rapid transit to the permanent regime. The adaptive order affine projection algorithm presented here addresses this subject. This algorithm adapts its projection order and step size depending on its convergence state by simple and meaningful rules. Thus it achieves good convergence behavior at every convergence state and very low computational cost at steady state.This work was partially funded by Spanish MICINN TEC2009-13741, GV-PROMETEO/2009/0013, GV/2010/027 and UPV/2009-1034.Gonzalez, A.; Ferrer Contreras, M.; Diego Antón, MD.; Piñero Sipán, MG. (2012). An affine projection algorithm with variable step-size and projection order. Digital Signal Processing. 22(4):586-592. doi:10.1016/j.dsp.2012.03.004S58659222

Robust adaptive filtering algorithms for system identification and array signal processing in non-Gaussian environment

This dissertation proposes four new algorithms based on fractionally lower order statistics for adaptive filtering in a non-Gaussian interference environment. One is the affine projection sign algorithm (APSA) based on L₁ norm minimization, which combines the ability of decorrelating colored input and suppressing divergence when an outlier occurs. The second one is the variable-step-size normalized sign algorithm (VSS-NSA), which adjusts its step size automatically by matching the L₁ norm of the a posteriori error to that of noise. The third one adopts the same variable-step-size scheme but extends L₁ minimization to Lp minimization and the variable step-size normalized fractionally lower-order moment (VSS-NFLOM) algorithms are generalized. Instead of variable step size, the variable order is another trial to facilitate adaptive algorithms where no a priori statistics are available, which leads to the variable-order least mean pth norm (VO-LMP) algorithm, as the fourth one. These algorithms are applied to system identification for impulsive interference suppression, echo cancelation, and noise reduction. They are also applied to a phased array radar system with space-time adaptive processing (beamforming) to combat heavy-tailed non-Gaussian clutters. The proposed algorithms are tested by extensive computer simulations. The results demonstrate significant performance improvements in terms of convergence rate, steady-state error, computational simplicity, and robustness against impulsive noise and interference --Abstract, page iv

Design of a Variable Step-size Filtering Algorithm for Acoustic Feedback Cancellation in Audio Systems

[[abstract]]Acoustic feedback often limits the maximum usable gain of acoustic systems and degrades the overall system response. It is well known to be detrimental that the system stability and performance must be taken into account in system design. Most of the conventional methods for acoustic feedback cancellation in an acoustic system are based primarily on an adaptive filter with the least-mean-square (LMS) error algorithm. Unfortunately, convergence speed is often limited when a sound source or a filtering plant is varied, because the learning process of the adaptive algorithm fails to respond fast enough to changing operational conditions. This report proposes a variable step-size affine-projection algorithm (VSS APA) for acoustic feedback cancellation in audio systems. The proposed adaptive filter is based on the filtering affine-projection algorithm with variable step-size for improving convergence speed in acoustic feedback cancellation. A performance evaluation and simulation comparison has been conducted to compare the proposed algorithm and various traditional adaptive filtering algorithms

Adaptive filters for sparse system identification

Sparse system identification has attracted much attention in the field of adaptive algorithms, and the adaptive filters for sparse system identification are studied. Firstly, a new family of proportionate normalized least mean square (PNLMS) adaptive algorithms that improve the performance of identifying block-sparse systems is proposed. The main proposed algorithm, called block-sparse PNLMS (BS-PNLMS), is based on the optimization of a mixed ℓ2,1 norm of the adaptive filter\u27s coefficients. A block-sparse improved PNLMS (BS-IPNLMS) is also derived for both sparse and dispersive impulse responses. Meanwhile, the proposed block-sparse proportionate idea has been extended to both the proportionate affine projection algorithm (PAPA) and the proportionate affine projection sign algorithm (PAPSA). Secondly, a generalized scheme for a family of proportionate algorithms is also presented based on convex optimization. Then a novel low-complexity reweighted PAPA is derived from this generalized scheme which could achieve both better performance and lower complexity than previous ones. The sparseness of the channel is taken into account to improve the performance for dispersive system identification. Meanwhile, the memory of the filter\u27s coefficients is combined with row action projections (RAP) to significantly reduce the computational complexity. Finally, two variable step-size zero-point attracting projection (VSS-ZAP) algorithms for sparse system identification are proposed. The proposed VSS-ZAPs are based on the approximations of the difference between the sparseness measure of current filter coefficients and the real channel, which could gain lower steady-state misalignment and also track the change in the sparse system --Abstract, page iv

Numerical Verification of Affine Systems with up to a Billion Dimensions

Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to scalable analysis for more complex systems. In this paper, we improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system. The reachable states of affine systems can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We overcome these challenges by combining several methods that leverage common problem structure. Memory is reduced by exploiting initial states that are not full-dimensional and safety properties (outputs) over a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use Krylov-subspace simulation approaches based on the Arnoldi or Lanczos iterations. Our method produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, can analyze a system with one billion real-valued state variables

Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

Approximating the set of reachable states of a dynamical system is an algorithmic yet mathematically rigorous way to reason about its safety. Although progress has been made in the development of efficient algorithms for affine dynamical systems, available algorithms still lack scalability to ensure their wide adoption in the industrial setting. While modern linear algebra packages are efficient for matrices with tens of thousands of dimensions, set-based image computations are limited to a few hundred. We propose to decompose reach set computations such that set operations are performed in low dimensions, while matrix operations like exponentiation are carried out in the full dimension. Our method is applicable both in dense- and discrete-time settings. For a set of standard benchmarks, it shows a speed-up of up to two orders of magnitude compared to the respective state-of-the art tools, with only modest losses in accuracy. For the dense-time case, we show an experiment with more than 10.000 variables, roughly two orders of magnitude higher than possible with previous approaches