This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE JOURNAL OF OCEANIC ENGINEERING 1 Compressive Sensing for Detecting Ships with Second-Order Cyclostationary Signatures Umut Fırat and Tayfun Akg¨ ul, Senior Member, IEEE Abstract—Amplitude modulation of the broadband propeller noise as a result of the cavitation yields a second-order cyclosta- tionary ship-radiated noise. The spectrum of the modulating signal, consisting of the so-called propeller (or cavitation) tonals, enables the detection and the classification of the submarines or surface ships. However, data acquisition for this purpose causes vast data sizes due to high sampling rates and multiple sensor deployment. To mitigate the negative effects of this acquisition process such as on energy efficiency, hardware complexity, and storage capacity, we propose a scheme for compressive sensing of propeller tonals. We show that the spectral correlation function of cyclostation- ary propeller noise is sparse and derive a linear relationship be- tween the compressive and Nyquist-rate cyclic modulation spectra, i.e., the approximation of spectral correlation function that allows utilizing matrix representations required in compressive sensing. It also enables use of the cyclic modulation coherence, i.e., the normalized cyclic modulation spectrum, to demonstrate the effect of compressive sensing in terms of statistical detection. We com- pare the recovery and detection performance results of the sparse approximation algorithms based on the so-called iterative hard thresholding and compressive sampling matching pursuit. Results show that compression is achievable without affecting the detec- tion performance negatively. The main challenges are the weak modulation, low signal-to-noise ratio, and nonstationarity of the ambient noise, all of which reduce the sparsity level, hence causing degraded recovery and detection performance. Index Terms—Compressive sampling matching pursuit (CoSaMP), compressive sensing (CS), cyclostationarity, iterative hard thresholding (IHT), propeller tonals, ship detection. I. INTRODUCTION S HIP-RADIATED noise mainly consists of a broadband part induced by the propeller revolution and a narrowband part (machinery tonals) generated by the propulsion and auxiliary machinery represented in Fig. 1 [1], [2, Ch. 8, Ch. 10]. Ampli- tude of the broadband part is modulated as a consequence of the propeller revolution when it exceeds a certain rate [2, Ch. 8]. The physical phenomena causing this is known as cavitation, which leads to generation of bubbles around the propeller that collapse and create the broadband noise [2, Ch. 7]. Note that revolution of the propeller shaft and blades creates a periodic signal yield- ing the propeller (or cavitation) tonals. Given the propeller shaft Manuscript received February 2, 2017; revised June 5, 2017; accepted July 12, 2017. (Corresponding author: Umut Fırat.) Associate Editor: W. Xu. U. Fırat is with the T ¨ UB ˙ ITAK B ˙ ILGEM Information Technologies Institute, Kocaeli 41470, Turkey (e-mail: umut.firat@tubitak.gov.tr). T. Akg¨ ul is with the Electronics and Communication Engineering De- partment, Istanbul Technical University, Istanbul 34469, Turkey (e-mail: tayfunakgul@itu.edu.tr). Digital Object Identifier 10.1109/JOE.2017.2740698 Fig. 1. Representative ship-radiated noise spectrum with the additive ambient noise. rate (PSR) and the blade rate (BR), one can obtain the number of blades (NOB), i.e., (NOB) = (BR) ÷ (PSR). Detection of these components is arguably the most important stage of sonar target classification since the typical values of PSR, BR, and NOB are distinguishable for various ship categories [3]–[8]. These acoustic signatures emanating from nearby ships are acquired by hydrophones existing in multiple numbers for acoustical moni- toring or beamforming which causes vast data sizes. Moreover, broadband propeller noise may range up to 100 kHz [2, Ch. 8], which increases the required sampling rate and hence the data size further. This acquisition scheme negatively affects the en- ergy efficiency, hardware complexity, and storage capacity [9]. Compressive sensing (CS) has emerged to mitigate the inef- ficiency in sampling sparse signals. It has made the simultane- ous data acquisition and compression possible when the signal under consideration has a sparse representation in a known in- coherent basis [10]–[15]. Then, a suitable sparse approximation method can recover the Nyquist-rate signal with high probabil- ity from linear and nonadaptive sub-Nyquist-rate measurements [16]–[20]. The dense support in frequency, explicitly seen in Fig. 1, violates the sparsity requirement for CS to be applicable directly on the ship-radiated noise as well as on propeller tonals due to the ambient noise dominance in the low-frequency region. However, CS is possible if the problem can be formulated to acquire the propeller tonals in a suitable manner. For that, a function representing the inherent sparsity of propeller tonals and a basis that maps the Nyquist-rate samples of this function into the compressive ones must be found. In this paper, we propose such a scheme for CS and sparse recovery of propeller tonals to detect ships. We employ two well-known pursuit methods that are the so-called iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP). 0364-9059 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.