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Appl. Sci. 2018, 8(12), 2510; https://doi.org/10.3390/app8122510

Construction of Measurement Matrix Based on Cyclic Direct Product and QR Decomposition for Sensing and Reconstruction of Underwater Echo

1
Department of Automation, Hangzhou Dianzi University, Xiasha Higher Education Zone, Hangzhou 310018, China
2
Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK
*
Author to whom correspondence should be addressed.
Received: 28 October 2018 / Revised: 27 November 2018 / Accepted: 3 December 2018 / Published: 6 December 2018
(This article belongs to the Section Acoustics and Vibrations)
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Abstract

Compressive sensing is a very attractive technique to detect weak signals in a noisy background, and to overcome limitations from traditional Nyquist sampling. A very important part of this approach is the measurement matrix and how it relates to hardware implementation. However, reconstruction accuracy, resistance to noise and construction time are still open challenges. To address these problems, we propose a measurement matrix based on a cyclic direct product and QR decomposition (the product of an orthogonal matrix Q and an upper triangular matrix R). Using the definition and properties of a direct product, a set of high-dimensional orthogonal column vectors is first established by a finite number of cyclic direct product operations on low-dimension orthogonal “seed” vectors, followed by QR decomposition to yield the orthogonal matrix, whose corresponding rows are selected to form the measurement matrix. We demonstrate this approach with simulations and field measurements of a scaled submarine in a freshwater lake, at frequencies of 40 kHz–80 kHz. The results clearly show the advantage of this method in terms of reconstruction accuracy, signal-to-noise ratio (SNR) enhancement, and construction time, by comparison with Gaussian matrix, Bernoulli matrix, partial Hadamard matrix and Toeplitz matrix. In particular, for weak signals with an SNR less than 0 dB, this method still achieves an SNR increase using less data. View Full-Text
Keywords: compressive sensing; measurement matrix; cyclic direct product; QR decomposition; underwater echo; sonar measurements compressive sensing; measurement matrix; cyclic direct product; QR decomposition; underwater echo; sonar measurements
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Sun, T.; Cao, H.; Blondel, P.; Guo, Y.; Shentu, H. Construction of Measurement Matrix Based on Cyclic Direct Product and QR Decomposition for Sensing and Reconstruction of Underwater Echo. Appl. Sci. 2018, 8, 2510.

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