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.
Highlights
Compressive sensing (CS) is a rapidly emerging technique [1,2,3]
CS is widely applied in medical image processing [5], image compression [6,7], pattern recognition [8,9], and radar detection [10]; it can be combined with the Bayesian algorithm [11,12], e.g., to reconstruct underwater echoes
The second step, construction of a measurement matrix, is important as it directly affects the performance of signal reconstruction and is an indispensable part of sampling systems based on CS, thereby determining their possible applications in actual engineering situations
Summary
Compressive sensing (CS) is a rapidly emerging technique [1,2,3]. Its main attraction is that it can overcome the shortcomings of the traditional Nyquist sampling theorem [4] in signal processing, in terms of the large amount of data to be sampled and large data storage space required, because signal sampling and compression are performed simultaneously. The performance in terms by reducing the cross-correlation coefficient between the measurement matrix and the sparse of signal reconstruction and the number of measurements needed depend closely on the crossmatrix [25] and improving the nonlinear correlation of the measurement matrix. A minor number of measurements are required for signal reconstruction correlation is a very important property of and the measurement matrix, by which the performance is if the cross-correlation coefficient is small, the signal is adaptive over a larger sparsity range. Corresponding experimental matrix to ensure high-accuracy reconstruction with low computational complexity This is tested with measurements of echoes from a scaled submarine target in a freshwater lakemeasurements (Section 5). Sci. 2018, 8, 2510 from a scaled submarine target in a freshwater lake (Section 5)
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