Solving Weighted Least Squares (WLS) problems on ARM-based architectures

TheWeighted Least Squares algorithm (WLS) is applied to numerous optimization problems, but requires the use of high computational resources, especially when complex arithmetic is involved. This work aims to accelerate the resolution of a WLS problem by reducing the computational cost (relaying on BLAS/LAPACK routines) and the computational precision from double to single. As a test case, we design an IIR filter for a Graphic Equalizer, where the numerical errors due to single precision are easily visualized. In addition, given the importance of low power architectures for this kind of implementations, we evaluate the performance, scalability, and energy efficiency of each method on two different processors implementing the ARMv7 architecture, widely used in current mobile devices with power constraints. Results show that the method that exhibits a high theoretical computational cost overcomes in efficiency other methods with lower theoretical cost in architectures of this type.This work started in spring 2016 when Jose A. Belloch was a visiting postdoctoral researcher at Budapest University of Technology and Economics thanks to the European Network COST Action IC1305 inside the program Short Term Scientific Mission with the following reference: COST-SPASM-ECOST-STSM-IC1305-020416-072431. Dr. Jose A. Belloch is supported by GVA contract APOSTD/2016/069. The researchers from Universitat Jaume I are supported by the CICYT projects TIN2014-53495-R of MINECO and FEDER. The authors from the Universitat Politecnica de Valencia are supported by MINECO Projects TEC2015-67387-C4-1-R, PROMETEOII/2014/003 and CAPAP-H5 network TIN2014-53522-REDT. The researcher from UCM is supported by the EU (FEDER) and the Spanish MINECO, under Grants TIN 2015-65277-R and TIN2012-32180. The work of Balazs Bank was supported by the UNKP-16-4-III New National Excellence Program of the Ministry of Human Capacities, Hungary.Belloch Rodríguez, JA.; Bank, B.; Igual Peña, FD.; Quintana Ortí, ES.; Vidal Maciá, AM. (2017). Solving Weighted Least Squares (WLS) problems on ARM-based architectures. Journal of Supercomputing. 73(1):530-542. https://doi.org/10.1007/s11227-016-1910-9S530542731Smith TM, van de Geijn RA, Smelyanskiy M, Hammond JR, Van Zee FG (2014) Anatomy of high-performance many-threaded matrix multiplication. In: 28th IEEE International Parallel and Distributed Processing Symposium (IPDPS 2014)Burrus CS (2012) Iterative reweighted least squares. OpenStax-CNC document, May 2012, module m45285. http://cnx.org/content/m45285/1.12 . Accessed 2 Nov 2016Khang SW (1972) Best $$L_p$$ L p approximation. Math Comput 26(118):505–508Jackson LB (2008) Frequency-domain Steiglitz-McBride method for least-squares filter design, ARMA modeling, and periodogram smoothing. 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J Audio Eng Soc 51(5):324–343Holters M, Zölzer U (2006) Graphic equalizer design using higher-order recursive filters. In: Proceedings of International Conference Digital Audio Effects, Montreal, QC, pp 37–40Tassart S (2013) Graphical equalization using interpolated filter banks. J Audio Eng Soc 61(5):263–279Chen Z, Geng GS, Yin FL, Hao J (2014) A pre-distortion based design method for digital audio graphic equalizer. Digital Signal Process 25:296–302Välimäki V, Reiss J (2016) All about audio equalization: solutions and frontiers. Appl Sci 6(5):129–145Belloch JA, Välimäki V (2016) Efficient target-response interpolation for a graphic equalizer. In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), March 2016, pp 564–568Belloch JA, Alventosa FJ, Alonso P, Quintana-Ortí ES, Vidal AM (2016) Accelerating multi-channel filtering of audio signal on arm processors. 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Partial equalization of non-minimum-phase impulse responses

<p/> <p>We propose a modified version of the standard homomorphic method to design a minimum-phase inverse filter for non-minimum-phase impulse responses equalization. In the proposed approach some of the dominant poles of the filter transfer function are replaced by new ones before carrying out the inverse DFT. This method is useful when partial magnitude equalization is intended. Results for an impulse response measured in the car interior show that by using the modified version we can control the sound quality more precisely than when using the standard method.</p

Equalization of Loudspeaker and Room Responses Using Kautz Filters: Direct Least Squares Design

DSP-based correction of loudspeaker and room responses is becoming an important part of improving sound reproduction. Such response equalization (EQ) is based on using a digital filter in cascade with the reproduction channel to counteract the response errors introduced by loudspeakers and room acoustics. Several FIR and IIR filter design techniques have been proposed for equalization purposes. In this paper we investigate Kautz filters, an interesting class of IIR filters, from the point of view of direct least squares EQ design. Kautz filters can be seen as generalizations of FIR filters and their frequency-warped counterparts. They provide a flexible means to obtain desired frequency resolution behavior, which allows low filter orders even for complex corrections. Kautz filters have also the desirable property to avoid inverting dips in transfer function to sharp and long-ringing resonances in the equalizer. Furthermore, the direct least squares design is applicable to nonminimum-phase EQ design and allows using a desired target response. The proposed method is demonstrated by case examples with measured and synthetic loudspeaker and room responses