Broadband sparse-array blind deconvolution using frequency-difference beamforming

Coherent reflection recovery in scattering from the ocean surface using the frequency-difference autoproducta)

The coherence of rough sea-surface-scattered acoustic fields decreases with increasing frequency. The frequency-difference autoproduct, a quadratic product of acoustic fields at nearby frequencies, mimics a genuine field at the difference frequency. In rough-surface scattering, the autoproduct's lower effective frequency decreases the apparent surface roughness, restoring coherent reflection. Herein, the recovery of coherent reflection in sea surface scattering via the frequency-difference autoproduct is examined for data collected off the coast of New Jersey during the Shallow Water '06 (SW06) experiment. An acoustic source at depth 40 m and receiver at depth 24.3 m and range 200 m interrogated 160 independent realizations of the ocean surface. The root mean square surface height h was 0.167 m, and broadcast frequencies were 14-20 kHz, so that 2.5 ≤kh cos θ≤ 3.7 for acoustic wavenumber k and incidence angle θ. Measured autoproducts, constructed from scattered constituent fields, show significant coherent reflection at sufficiently low difference frequencies. Theoretical results, using the Kirchhoff approximation and a non-analytic surface autocorrelation function, agree with experimental findings. The match is improved using a numerical strategy, exploiting the relationship between autoproduct-based coherence recovery, the ocean-surface autocorrelation function, and the ocean-surface height spectrum. Error bars computed from Monte Carlo scattering simulations support the validity of the measured coherence recovery.

High-resolution frequency-difference beamforming for a short linear array

Conventional beamforming (CBF) is a commonly employed approach for detecting and estimating the direction-of-arrival (DOA) of acoustic signals in underwater environments. However, CBF becomes ambiguous due to spatial aliasing when the received signal's half wavelength is smaller than the array spacing. Frequency-difference beamforming (FDB) allows for processing data in the lower frequency Δf without encountering spatial aliasing by utilizing the product of array data at frequency f with its complex conjugate at frequency f+Δf. However, lower frequency results in a wider mainlobe, which can lead to poorer DOA performance for short arrays. In this paper, a fourth-order cumulants FDB method and a conjugate augmented FDB method are proposed to extend an M-element uniform linear array to 2M-1 and 4M-3 elements. The proposed methods generate narrower beams and lower sidelobe levels than the original FDB for short arrays with large spacing. And by setting the signal subspace dimension reasonably, the proposed methods can improve the weak target detection ability under strong interference compared with FDB. Last, we verify the excellent performance of the proposed methods through simulations and experimental data.

Reformulation of frequency-difference matched-field processor for high-frequency known-source localization

Frequency-difference matched-field processing is a high-frequency source localization technique formulated by matching the frequency-difference autoproduct of the measured field and replicas at the difference-frequency. Although it successfully localizes sound sources by sparse vertical array in shallow or deep ocean with an environmental mismatch, there is still some ambiguity in replica modeling and signal processing. Here, the existing conventional processor is modified to match the bandwidth-averaged autoproduct of the measured field with replicas of the bandwidth-averaged autoproduct, or approximately its self-term for the expected source locations. The proposed processor is consistent with the perspective of matched-field processing and can naturally relieve some drawbacks of the existing one, such as low peak or low dynamic range on the ambiguity surface. Numerical tests are carried out in several shallow ocean environments and the source localization using experimental data are performed to confirm the properties of the proposed processor. It is found that the high-frequency diffracted field always leaves traces on its bandwidth-averaged autoproduct field. These high-frequency marks cause a bias in source localization in the presence of a sound speed mismatch even in low difference-frequencies.

Localization of a remote source in a noisy deep ocean sound channel using phase-only matched autoproduct processing

Long-range passive source localization is possible in the deep ocean using phase-only matched autoproduct processing (POMAP) [Geroski and Dowling (2021). J. Acoust. Soc. Am. 150, 171-182], an algorithm based on matched field processing that is more robust to environmental mismatch. This paper extends these prior POMAP results by analyzing the localization performance of this algorithm in the presence of environmental noise. The noise rejection performance of POMAP is assessed using both simulated and measured signal data, with noise data based on environmental noise measurements. Herein, signal and noise measurements are from the nominally one-year-long PhilSea10 ocean acoustic propagation experiment. All signals were recorded from a single moored source, placed near the ocean sound channel 129.4 km away from a nearly water-column-spanning distributed vertical line array. The source transmitted linear frequency modulated chirps with nominal bandwidth from 200 to 300 Hz. The noise measurements used in this study were collected in the months after this source stopped transmitting, and synthetic samples of noise are calculated based on the characteristics of this measured noise. The effect that noise rejection algorithms have on the source localization performance of POMAP is also evaluated, but only 1 dB of performance improvement is achieved using these.

Difference frequency coherent matched autoproduct processing for source localization in deep ocean

Matched autoproduct processing (MAP) refers to a matched field processing (MFP) style array signal processing technique for passive source localization, which interrogates frequency-difference autoproduct instead of genuine acoustic pressure. Due to frequency downshifting, MAP is less sensitive to environmental mismatch, but it suffers from low spatial resolution and a low peak-to-sidelobe ratio of ambiguity surface. These source localization metrics are herein improved with coherent approaches. Specifically, the coherent normalized MFP is extended to coherent matched autoproduct processing (CMAP), a difference frequency coherent algorithm that exploits correlations among the autoproducts at various difference frequencies and eliminates the phase factor of the source spectrum for passive source localization. Phase-only coherent matched autoproduct processing is a CMAP derivation technique that only uses phase information. Through simulations in a Munk sound-speed profile environment, sensitivity analysis in the South China Sea environment, and high signal-to-noise ratio experimental measurements, these two algorithms are validated as compared to the conventional MFP and incoherent MAP. Simulation investigations demonstrate that difference frequency coherent algorithms can suppress sidelobes while simultaneously enhancing the localization resolution and robustness. The experimental results generally support the findings of the simulations.

Direction-of-Arrival Estimation Based on Frequency Difference-Wavenumber Analysis for Sparse Vertical Array Configuration

Frequency-wavenumber (f-k) analysis can estimate the direction of arrival (DOA) of broadband signals received on a vertical array. When the vertical array configuration is sparse, it results in an aliasing error due to spatial sampling; thus, several striation patterns can emerge in the f-k domain. This paper extends the f-k analysis to a sparse receiver-array, wherein a multitude of sidelobes prevent resolving the DOA estimates due to spatial aliasing. The frequency difference-wavenumber (Δf-k) analysis is developed by adopting the concept of frequency difference, and demonstrated its performance of DOA estimation to a sparse receiver array. Experimental results verify the robustness of the proposed Δf-k analysis in the estimation of the DOA of cracking sounds generated by the snapping shrimps, which were recorded by a sparse vertical array configuration during the shallow water experiment.

Unambiguous broadband direction of arrival estimation based on improved extended frequency-difference method

In this paper, we consider the problem of bearing ambiguity in the direction of arrival (DOA) estimation due to spatial aliasing when the minimum wavelength of the processing broadband signal is less than the element spacing of a uniform linear array (ULA). First, an extended frequency-difference (FD) method is presented. Unlike the existing FD methods, the extended FD signal is constructed by conjugate multiplying a diagonal matrix consisting of steering vectors at high frequency and pre-processing direction with the array sampled signal matrix at low frequency. Then, this paper establishes a decision criterion for distinguishing the aliasing component that varies linearly with frequency in the extended FD space. Finally, an unambiguous broadband DOA estimation method is achieved by suppressing spatial aliasing in the extended FD space. The simulation results show the effectiveness of the proposed method in low signal-to-noise ratio, low signal-to-interference ratio, and multi-interference conditions. The unambiguous processing ability of the proposed method is further verified in the South China Sea using ship signals in the frequency band of 200 to 700 Hz and a 10-element ULA with a 6.25 m spacing deployed on the seabed.

Ray-based blind deconvolution with maximum kurtosis phase correction

Ray-based blind deconvolution (RBD) is a method that estimates the source waveform and channel impulse response (CIR) using the ray arrival in an underwater environment. The RBD estimates the phase of the source waveform by using beamforming. However, low sampling, array shape deformation, and other factors can cause phase errors in the beamforming results. In this paper, phase correction is applied to the beamforming estimated source phase to improve RBD performance. The impulsiveness of the CIR was used as additional information to correct the initially estimated source phase. Kurtosis was used to measure impulsiveness, and the phase correction that maximized the kurtosis of the CIRs was calculated through optimization. The proposed approach is called ray-based blind deconvolution with maximum kurtosis phase correction (RBD-MKPC) and is based on a single-input multiple-output system. The RBD-MKPC was tested with several CIRs and source waveform combinations in the shallow-water acoustic variability experiment 2015 using broadband high-frequency pulses (11-31 kHz) as the source and a sparse vertical 16-element line array as receivers. The results indicate that the RBD-MKPC improves the estimation performance. In addition, from an optimization point of view and compared with other initialization methods, the proposed method showed superior convergence speed and estimation performance.

Recovery of coherent reflection from rough-surface scattered acoustic fields via the frequency-difference autoproduct

The acoustic field reflected from a random rough surface loses coherence with the incident field in the Kirchhoff approximation as kh cos θ increases, where k is the incident field wavenumber, h is the root mean square roughness height, and θ is the incidence angle. Thus, for fixed rough-surface properties and incidence angle, a reflected field at lower wavenumber should retain more coherence. Recent results suggest that the frequency-difference autoproduct formed from complex acoustic field amplitudes at two nearby frequencies can recover acoustic information at the difference of those frequencies even when the difference frequency is below the recorded field's bandwidth. Herein analytical, computational, and experimental results are presented for the extent to which the frequency-difference autoproduct recovers coherence from randomly rough-surface-scattered constituent fields that have lost coherence. The analytical results, developed from the Kirchhoff approximation and formal ensemble averaging over randomly rough surfaces with Gaussian height distributions and Gaussian correlation functions, indicate that the coherence of the rough-surface-reflected frequency-difference autoproduct depends on the surface correlation length and Δkh cos θ, where Δk is the difference of the autoproduct's constituent field wavenumbers. These results compare favorably with Monte Carlo simulations of rough surface scattering, and with laboratory experiments involving long surface correlation lengths where 1  ≤kh cos θ≤ 3.

Aliasing-free broadband direction of arrival estimation using a frequency-difference technique

When the intersensor spacing of a uniform linear array (ULA) is larger than the half-wavelength of an incident narrowband signal, spatial aliasing is generated. For broadband signals, the broadband spatial spectrum is still affected as a result of the spatial aliasing in each frequency bin. In this paper, an aliasing-free broadband direction-of-arrival (DOA) estimation algorithm for ULAs is proposed. First, an array output is constructed with a given Gaussian random sequence from the direction ϑ. Then, a frequency-difference (FD) operation is conducted, which multiplies the array observation in the frequency bin f by the conjugate form of the constructed array output in the frequency bin f+Δf. Thus, an equivalent array output at a desired frequency Δf is obtained, whose wavelength is equal to twice the intersensor spacing. In this manner, an aliasing-free spatial spectrum in the FD domain is achieved. Scanning the direction ϑ, the DOA of signals is finally estimated based on the difference between the peaks in the aliasing-free spatial spectrum and direction ϑ. The proposed method can achieve a satisfactory estimation even in a strong interference environment. The simulations and experimental results are included to demonstrate the superiority of the proposed method.

Robust long-range source localization in the deep ocean using phase-only matched autoproduct processing

Passive source localization in the deep ocean using array signal processing techniques is possible using an algorithm similar to matched field processing (MFP) that interrogates a measured frequency-difference autoproduct instead of a measured pressure field [Geroski and Dowling, J. Acoust. Soc. Am. 146, 4727-4739 (2019)]. These results are extended herein to a new MFP-style algorithm, phase-only matched autoproduct processing, that is more robust at source-array ranges as large as 225 km. This new algorithm is herein described and compared to three existing approaches. The performance of all four techniques is evaluated using measured ocean propagation data from the PhilSea10 experiment. These data nominally span a 12-month period; include six source-array ranges from 129 to 450 km; and involve signals with center frequencies between 172.5 and 275 Hz, and bandwidths of 60 to 100 Hz. In all cases, weight vectors are calculated assuming a range-independent environment using a single sound-speed profile measured near the receiving array. The frequency-differencing techniques considered here are capable of localizing all six sources, with varying levels of consistency, using single-digit-Hz difference frequencies. At source-array ranges up to and including 225 km, the new algorithm requires fewer signal samples for success and is more robust to the choice of difference frequencies.

Measurements of the correlation of the frequency-difference autoproduct with acoustic and predicted-autoproduct fields in the deep ocean

Frequency-domain spatial-correlation analysis of recorded acoustic fields is typically limited to the bandwidth of the recordings. A previous study [Lipa, Worthmann, and Dowling (2018) J. Acoust. Soc. Am. 143(4), 2419-2427] suggests that limiting such analysis to in-band frequencies is not strictly necessary in a Lloyd's mirror environment. In particular, below-band field information can be retrieved from the frequency-difference autoproduct, a quadratic product of measured complex pressure-field amplitudes from two nearby frequencies. The frequency-difference autoproduct is a surrogate field that mimics a genuine acoustic field at the difference frequency. Here, spatial-correlation analysis is extended to deep-ocean acoustic fields measured during the PhilSea10 experiment. The frequency-difference autoproduct, at difference frequencies from 0.0625 to 15 Hz, is determined from hundreds of Philippine Sea recordings of 60 or 100 Hz bandwidth signals with center frequencies from 172.5 to 275 Hz broadcast to a vertical receiving array 129-450 km away. The measured autoproducts are cross correlated along the array with predicted acoustic fields and with predicted autoproduct fields at corresponding below-band frequencies. Stable measured cross correlations as high as 80%-90% are found at the low end of the investigated difference-frequency range, with consistent correlation loss due to mismatch at the higher below-band frequencies.

Frequency-difference autoproduct cross-term analysis and cancellation for improved ambiguity surface robustness

Frequency-differencing, or autoproduct processing, techniques are one area of research that has been found to increase the robustness of acoustic array signal processing algorithms to environmental uncertainty. Previous studies have shown that frequency differencing techniques are able to mitigate problems associated with environmental mismatch in source localization techniques. While this method has demonstrated increased robustness compared to conventional methods, many of the metrics, such as ambiguity surface peak values and dynamic range, are lower than would typically be expected for the observed level of robustness. These previous studies have suggested that such metrics are reduced by the inherent nonlinearity of the frequency-differencing method. In this study, simulations of simple multi-path environments are used to analyze this nonlinearity and signal processing techniques are proposed to mitigate the effects of this problem. These methods are used to improve source localization metrics, particularly ambiguity surface peak value and dynamic range, in two experimental environments: a small laboratory water tank and in a deep ocean (Philippine Sea) environment. The performance of these techniques demonstrates that many source localization metrics can be improved for frequency-differencing methods, which suggests that frequency-differencing methods may be as robust as previous studies have shown.

Deconvolved frequency-difference beamforming for a linear array

Frequency-difference beamforming (FDB) provides a robust estimation of wave propagation direction by shifting signal processing to a lower frequency which, however, produces a decline in the spatial resolution. In this letter, the beam pattern of FDB for a distant point source is proved to be shift invariant and therefore can be regarded as the point spread function corresponding to FDB's beam output. Then, deconvolved frequency-difference beamforming (Dv-FDB) is proposed to improve array performance. Dv-FDB yields a narrower beam and lower sidelobe levels while maintaining robustness. The superior performance of Dv-FDB is verified by simulations and experimental data.

The effects of refraction and caustics on autoproducts

Quadratic products of complex amplitudes from acoustic fields with nonzero bandwidth, denoted "autoproducts," can mimic acoustic fields at frequencies lower or higher than the bandwidth of the original field. While this mimicry has been found to be very promising for a variety of signal processing applications, its theoretical extent has, thus far, only been considered under the most elementary ray approximation. In this study, the combined effects of refraction and diffraction are considered in environments where refraction causes neighboring rays to cross and form caustics. Acoustic fields on and near caustics are not well-predicted by elementary ray-acoustic theory. Furthermore, caustics introduce frequency dependence to the nearby acoustic field and a phase shift on the acoustic waves that passes through them. The effects these caustics have on autoproducts is assessed here using two simple, range-independent waveguides with index of refraction (n) profiles that are n²-quadratic and n²-linear. It is found that in multipath regions where rays have passed through differing numbers of caustics, the ability of autoproducts to mimic out-of-band fields is substantially hindered.

Autoproducts in and near acoustic shadow zones created by barriers

The autoproducts are nonlinear mathematical constructs developed from acoustic fields with non-zero bandwidth. When averaged through the field's bandwidth, the autoproducts may mimic a genuine acoustic field at frequencies that are lower or higher than the original field's bandwidth. The resulting opportunity to extend signal processing to user-selectable below- or above-band frequencies is intriguing for many signal processing algorithms. Based on prior work, the limitations of the autoproducts' mimicry of out-of-band fields are understood when the in-band acoustic field is well-represented by ray acoustics. Thus, the focus in this study is on autoproducts in acoustic shadow zones behind barriers containing only diffracted acoustic fields where a sum of ray-path contributions is not an adequate field description. Diffraction is expected to be a detriment to autoproduct techniques due to its sensitivity to frequency. Two ideal shadow-zone environments with exact analytic Helmholtz-equation solutions are considered: Sommerfeld's half-plane problem, also known as knife-edge diffraction, and Mie scattering from a sphere with ka = 40, where k is the wavenumber and a is the sphere's radius. With the exception of the shadow regions, autoproducts experience only mild degradation in field-mimicry performance when compared to what the ray-based theory would predict.

Long-range frequency-difference source localization in the Philippine Sea

Matched field processing (MFP) refers to a variety of source localization schemes for known complicated environments and involves matching measured and calculated (replica) fields to identify source locations. MFP may fail for several reasons, most notably when the calculated fields are insufficiently accurate. This error commonly prevents MFP-based long-range (>100 km) source localization in the deep ocean (from 5 to 6 km depth) for signal frequencies of hundreds of Hz, even when extensive high-signal-to-noise ratio field measurements are available. Recently, below-band MFP utilizing the frequency-difference autoproduct [Worthmann, Song, and Dowling (2015). J. Acoust. Soc. Am, 138(6), 3549-3562] achieved some shallow-ocean localization success at a 3 km source-to-array range with signal frequencies in the tens of kHz. The performance of this technique, when extended to matching the measured frequency-difference autoproduct with a composite mode-ray replica, is described here for deep ocean source localization. The ocean propagation data come from the PhilSea10 experiment and involve source-to-array ranges from 129 to 379 km and nominal 100-Hz-bandwidth signals having center frequencies from 250 to 275 Hz. Based on an incoherent average of five signal samples, the frequency-difference technique was 90%-100% successful at four different source-to-array ranges using single-digit-Hz difference frequencies.

Frequency-difference beamforming in the presence of strong random scattering

Frequency-difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018-3029] is a nonlinear, out-of-band signal processing technique used to beamform non-zero bandwidth signals at below-band frequencies. This is accomplished with the frequency-difference autoproduct AP(Δω)=P(ω₂)P^*(ω₁), a quadratic product of complex field amplitudes that mimics a genuine field at the difference frequency, Δω=ω₂-ω₁. For frequency-difference beamforming, AP(Δω) replaces the in-band complex field in the conventional beamforming algorithm. Here, the near-field performance of frequency-difference beamforming is evaluated in the presence of 1 to 30 high-contrast spherical scatterers with radius a placed between, and in the plane defined by the source and a 12-element linear receiving array with element spacing d. Based on the center frequency wave number, k, of the 150-200 kHz frequency sweep source signal, the scatterers are large, ka ≈ 15; the array is sparse, kd = 37; and the average source-to-receiver distance is up to 4.3 mean-free-path lengths. Beamforming results from simulations and experiments show that in-band beamforming loses peak-to-sidelobe ratio and fails to reliably locate the source as the scatterer count increases. Using the same signals, frequency-difference beamforming with difference frequencies from 5 to 25 kHz localizes sources reliably with higher peak-to-side-lobe ratios, though with reduced resolution.

In-Depth Exploration of Signal Self-Cancellation Phenomenon to Achieve DOA Estimation of Underwater Acoustic Sources

In the ocean environment, the minimum variance distortionless response beamformer usually has the problem of signal self-cancellation, that is, the acoustic signal of interest is erroneously suppressed as interference. By exploring the useful information behind the signal self-cancellation phenomenon, a high-precision direction estimation method for underwater acoustic sources is proposed. First, a pseudo spatial power spectrum is obtained by performing unit circle mapping on the beam response in the direction interval. Second, the online calculation process is given for reducing the computational complexity. The computer simulation results show that the proposed algorithm can obtain satisfactory direction estimation accuracy under the conditions of low intensity of acoustic source, strong interference and noise, and less array snapshot data.

DOA Estimation for Underwater Target by Active Detection on Virtual Time Reversal Using a Uniform Linear Array

Aiming at addressing the problem caused by multipath effects in direction of arrival (DOA) estimation for underwater targets, a method based on the active detection on virtual time reversal (ADVTR) Capon algorithm is proposed. Unlike the conventional passive target estimation method ignoring the multipath effects but only considering the direct wave, the proposed method is closer to the actual situation in that the multipath signal propagation model is fully taken into account; in addition, active detection (AD) and virtual time reversal (VTR) processes are added, which use active detection to estimate channels, and virtual time reversal to realize focusing in a computer after the source-receive array (SRA) receives the reflected signal of the target. The combination of the two methods can greatly improve the energy of SRA and the precision of target direction estimation. With the popular acoustic field simulation tool Bellhop, the model proposed in this paper is verified. Compared with the conventional Capon method without time reversal, the simulation results show that the ADVTR Capon estimation method is far better, in terms of resolution and suppressing the sidelobes. It is suitable for the target DOA estimation under low signal-to-noise ratio (SNR) conditions. Further, we also show the ADVTR Capon estimation method works well in a real tank experiment.

Frequency-sum beamforming for passive cavitation imaging

Beamforming includes a variety of spatial filtering techniques that may be used for determining sound source locations from near-field sensor array recordings. For this scenario, beamforming resolution depends on the acoustic frequency, array geometry, and target location. Random scattering in the medium between the source and the array may degrade beamforming resolution with higher frequencies being more susceptible to degradation. The performance of frequency-sum (FS) beamforming for reducing such sensitivity to mild scattering while increasing resolution is reported here. FS beamforming was used with a data-dependent [minimum variance (MV)] or data-independent (delay-and-sum, DAS) weight vector to produce higher frequency information from lower frequency signal components via a quadratic product of complex signal amplitudes. The current findings and comparisons are based on simulations and passive cavitation imaging experiments using 3 MHz and 6 MHz emissions recorded by a 128-element linear array. FS beamforming results are compared to conventional DAS and MV beamforming using four metrics: point spread function (PSF) size, axial and lateral contrast, and computation time. FS beamforming produces a smaller PSF than conventional DAS beamforming with less computation time than MV beamforming in free space and mild scattering environments. However, it may fail when multiple unknown sound sources are present.

Measurement of autoproduct fields in a Lloyd's mirror environment

Conventional frequency-domain acoustic-field analysis techniques are typically limited to the bandwidth of the field under study. However, this limitation may be too restrictive, as prior work suggests that field analyses may be shifted to lower or higher frequencies that are outside the field's original bandwidth [Worthmann and Dowling (2017). J. Acoust. Soc. Am. 141(6), 4579-4590]. This possibility exists because below- and above-band acoustic fields can be mimicked by the frequency-difference and frequency-sum autoproducts, which are quadratic products of frequency-domain complex field amplitudes at a pair of in-band frequencies. For a point source in a homogeneous acoustic half-space with a flat, pressure-release surface (a Lloyd's mirror environment), the prior work predicted high correlations between the autoproducts and genuine out-of-band fields at locations away from the source and the surface. Here, measurements collected in a laboratory water tank validate predictions from the prior theory using 40- to 110-kHz acoustic pulses measured at ranges between 175 and 475 mm, and depths to 400 mm. Autoproduct fields are computed, and cross-correlations between measured autoproduct fields and genuine out-of-band acoustic fields are above 90% for difference frequencies between 0 and 60 kHz, and for sum frequencies between 110 and 190 kHz.

Performance comparisons of frequency-difference and conventional beamforming

Frequency-difference beamforming [Abadi, Song, and Dowling (2012b). J. Acoust. Soc. Am. 132, 3018-3029] is an unconventional beamforming method for use with sparse receiver arrays. It involves beamforming a quadratic product of complex field amplitudes, P(ω₂)P^*(ω₁), at the difference frequency, ω₂-ω₁, instead of beamforming the complex field amplitude P(ω) at frequencies ω within the signal bandwidth. Frequency-difference beamforming is readily implemented with ordinary transducer array recordings of non-zero bandwidth signals. Results for, and comparisons of, frequency-difference beamforming from simulations and experiments are reported herein. In particular, spherical-wave beamforming is investigated using 15 and 165 kHz pulse signals in a 1.07-m-diameter water tank with a linear array having 14 elements spaced 5.08 cm apart. Here, frequency-difference beamforming using the high-frequency pulses provides comparable results to conventional beamforming at 15 kHz. Plane-wave beamforming is investigated using 11.2-32.8 kHz frequency-sweep signals broadcast 3 km through a 106-m-deep ocean sound channel to a vertical array having 16 elements spaced 3.75 m apart. Here, frequency difference beamforming in the 1.7-2.3 kHz difference frequency band provides results comparable to conventional beamforming in this band.

The frequency-difference and frequency-sum acoustic-field autoproducts

The frequency-difference and frequency-sum autoproducts are quadratic products of solutions of the Helmholtz equation at two different frequencies (ω₊ and ω_-), and may be constructed from the Fourier transform of any time-domain acoustic field. Interestingly, the autoproducts may carry wave-field information at the difference (ω₊ - ω_-) and sum (ω₊ + ω_-) frequencies even though these frequencies may not be present in the original acoustic field. This paper provides analytical and simulation results that justify and illustrate this possibility, and indicate its limitations. The analysis is based on the inhomogeneous Helmholtz equation and its solutions while the simulations are for a point source in a homogeneous half-space bounded by a perfectly reflecting surface. The analysis suggests that the autoproducts have a spatial phase structure similar to that of a true acoustic field at the difference and sum frequencies if the in-band acoustic field is a plane or spherical wave. For multi-ray-path environments, this phase structure similarity persists in portions of the autoproduct fields that are not suppressed by bandwidth averaging. Discrepancies between the bandwidth-averaged autoproducts and true out-of-band acoustic fields (with potentially modified boundary conditions) scale inversely with the product of the bandwidth and ray-path arrival time differences.

Multichannel myopic deconvolution in underwater acoustic channels via low-rank recovery

This paper presents a technique for solving the multichannel blind deconvolution problem. The authors observe the convolution of a single (unknown) source with K different (unknown) channel responses; from these channel outputs, the authors want to estimate both the source and the channel responses. The authors show how this classical signal processing problem can be viewed as solving a system of bilinear equations, and in turn can be recast as recovering a rank-1 matrix from a set of linear observations. Results of prior studies in the area of low-rank matrix recovery have identified effective convex relaxations for problems of this type and efficient, scalable heuristic solvers that enable these techniques to work with thousands of unknown variables. The authors show how a priori information about the channels can be used to build a linear model for the channels, which in turn makes solving these systems of equations well-posed. This study demonstrates the robustness of this methodology to measurement noises and parametrization errors of the channel impulse responses with several stylized and shallow water acoustic channel simulations. The performance of this methodology is also verified experimentally using shipping noise recorded on short bottom-mounted vertical line arrays.

Blind deconvolution of shipping sources in an ocean waveguide

This paper investigates the applicability of a ray-based blind deconvolution (RBD) method for underwater acoustic sources of opportunity such as ships recorded on a receiver array. The RBD relies on first estimating the unknown phase of the random source by beamforming along a well-resolved ray path, and then matched-filtering each received signal using the knowledge of this random phase to estimate the full channel impulse responses (CIRs) between the unknown source and the array elements (up to an arbitrary time-shift) as well as recovering the radiated signal by the random source. The performance of this RBD is investigated using both numerical simulation and experimental recordings of shipping noise in the frequency band [300-800 Hz] for ranges up to several kilometers. The ray amplitudes of the estimated CIRs are shown to be consistent with known bottom properties in the area. Furthermore, CIRs obtained for an arbitrarily selected shipping track are used as data-derived replicas to perform broadband matched-field processing to locate another shipping source recorded at a later time in the vicinity of the selected track.

Adaptive frequency-difference matched field processing for high frequency source localization in a noisy shallow ocean

Remote source localization in the shallow ocean at frequencies significantly above 1 kHz is virtually impossible for conventional array signal processing techniques due to environmental mismatch. A recently proposed technique called frequency-difference matched field processing (Δf-MFP) [Worthmann, Song, and Dowling (2015). J. Acoust. Soc. Am. 138(6), 3549-3562] overcomes imperfect environmental knowledge by shifting the signal processing to frequencies below the signal's band through the use of a quadratic product of frequency-domain signal amplitudes called the autoproduct. This paper extends these prior Δf-MFP results to various adaptive MFP processors found in the literature, with particular emphasis on minimum variance distortionless response, multiple constraint method, multiple signal classification, and matched mode processing at signal-to-noise ratios (SNRs) from -20 to +20 dB. Using measurements from the 2011 Kauai Acoustic Communications Multiple University Research Initiative experiment, the localization performance of these techniques is analyzed and compared to Bartlett Δf-MFP. The results show that a source broadcasting a frequency sweep from 11.2 to 26.2 kHz through a 106 -m-deep sound channel over a distance of 3 km and recorded on a 16 element sparse vertical array can be localized using Δf-MFP techniques within average range and depth errors of 200 and 10 m, respectively, at SNRs down to 0 dB.

High frequency source localization in a shallow ocean sound channel using frequency difference matched field processing

Matched field processing (MFP) is an established technique for source localization in known multipath acoustic environments. Unfortunately, in many situations, particularly those involving high frequency signals, imperfect knowledge of the actual propagation environment prevents accurate propagation modeling and source localization via MFP fails. For beamforming applications, this actual-to-model mismatch problem was mitigated through a frequency downshift, made possible by a nonlinear array-signal-processing technique called frequency difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018-3029]. Here, this technique is extended to conventional (Bartlett) MFP using simulations and measurements from the 2011 Kauai Acoustic Communications MURI experiment (KAM11) to produce ambiguity surfaces at frequencies well below the signal bandwidth where the detrimental effects of mismatch are reduced. Both the simulation and experimental results suggest that frequency difference MFP can be more robust against environmental mismatch than conventional MFP. In particular, signals of frequency 11.2 kHz-32.8 kHz were broadcast 3 km through a 106-m-deep shallow ocean sound channel to a sparse 16-element vertical receiving array. Frequency difference MFP unambiguously localized the source in several experimental data sets with average peak-to-side-lobe ratio of 0.9 dB, average absolute-value range error of 170 m, and average absolute-value depth error of 10 m.