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

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.

Analysis of the Measurement Matrix in Directional Predictive Coding for Compressive Sensing of Medical Images

“…The matrix which compressively senses the signal should have several properties to keep its information content for recovery after being sensed. Donoho et al [21] proposed that a measurement matrix should satisfy the following conditions:…”

Direct Under-Sampling Compressive Sensing Method for Underwater Echo Signals and Physical Implementation

“…How to achieve the sampling of information using compressive sensing theory is a critical issue that needs a solution at present. Many studies have been carried out on this issue, such as: the observation matrix construction [7][8][9], analog broadband signal sampling [10][11][12], radar signal compression [13,14], image compression technology [15][16][17][18], and ultrasound transducer fields [19]. Some researchers [20][21][22] designed ultra-high-speed analog-to-digital converters, but their high level of complexity makes them difficult to implement in practice.…”