Frequency-difference Beamforming Research Articles

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.

Compressive frequency-difference direction-of-arrival estimation

Direction-of-arrival estimation is difficult for signals spatially undersampled by more than half the wavelength. Frequency-difference beamforming [Abadi, Song, and Dowling (2012). J. Acoust. Soc. Am. 132, 3018–3029] offers an alternative approach to avoid such spatial aliasing by using multifrequency signals and processing them at a lower frequency, the difference-frequency. As with the conventional beamforming method, lowering the processing frequency sacrifices spatial resolution due to a beam broadening. Thus, unconventional beamforming is detrimental to the ability to distinguish between closely spaced targets. To overcome spatial resolution deterioration, we propose a simple yet effective method by formulating the frequency-difference beamforming as a sparse signal reconstruction problem. Similar to compressive beamforming, the improvement (compressive frequency-difference beamforming) promotes sparse nonzero elements to obtain a sharp estimate of the spatial direction-of-arrival spectrum. Analysis of the resolution limit demonstrates that the proposed method outperforms the conventional frequency-difference beamforming in terms of separation if the signal-to-noise ratio exceeds 4 dB. Ocean data from the FAF06 experiment support the validity.

Frequency-difference backprojection of earthquakes

SUMMARY Backprojection has proven useful in imaging large earthquake rupture processes. The method is generally robust and requires relatively simple assumptions about the fault geometry or the Earth velocity model. It can be applied in both the time and frequency domain. Backprojection images are often obtained from records filtered in a narrow frequency band, limiting its ability to uncover the whole rupture process. Here, we develop and apply a novel frequency-difference backprojection (FDBP) technique to image large earthquakes, which imitates frequencies below the bandwidth of the signal. The new approach originates from frequency-difference beamforming, which was initially designed to locate acoustic sources. Our method stacks the phase-difference of frequency pairs, given by the autoproduct, and is less affected by scattering and -time errors from 3-D Earth structures. It can potentially locate sources more accurately, albeit with lower resolution. In this study, we first develop the FDBP algorithm and then validate it by performing synthetic tests. We further compare two stacking techniques of the FDBP method, Band Width Averaged Autoproduct and its counterpart (BWAP and non-BWAP), and their effects in the backprojection images. We then apply both the FDBP and conventional backprojection methods to the 2015 M7.8 Gorkha earthquake as a case study. The backprojection results from the two methods agree well with each other, and we find that the peak radiation loci of the FDBP non-BWAP snapshots have standard error of less than 0.33° during the rupture process. The FDBP method shows promise in resolving complex earthquake rupture processes in tectonically complex regions.

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.

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.

Use of frequency-difference beamforming for scatterer localization

Acoustic fields interacting with discontinuities or environmental variations lead to secondary scattered fields superimposed onto incident fields. As a result, the signal measured at a remote receiving array will be modified by the scatterer’s presence. Prior work has shown that the effects of strong random scattering are reduced using frequency-difference beamforming to downshift the analysis to a below-band frequency known to be less impacted by scattering. In some applications, information about a scatterer is desired, such as location, shape, or composition. However, a relatively weak scattered field from a single scatterer is difficult to detect when the incident field dominates the measurements, as both conventional and frequency-difference beamforming ambiguity surfaces primarily provide the incident field’s source information. Here, a subtraction-based algorithm is implemented with frequency-difference beamforming to locate the position of a single scatterer near a source. Simulations and water tank experiments with a 110 kHz center frequency signal are considered for incident and scattered fields with a 4 cm diameter spherical scatterer, where the scattered field’s received energy is roughly 2% that of the incident field. The ability to downshift the frequency demonstrates a more robust implementation of subtraction-based algorithms for locating weak scatterers without prior location knowledge. [Work supported by ONR.]

Out-of-band beamforming in shallow water with horizontal arrays

Out-of-band beamforming methods, frequency-difference and frequency-sum beamforming, are array signal processing methods that allow a user to process signals at either a lower, below-the-signal-band difference frequency, or higher, above-the-signal-band sum frequency. Frequency-difference beamforming has previously been investigated for shallow-ocean propagation with sparse vertical arrays, demonstrating its capability to overcome the negative effects of spatial aliasing [Douglass, Song, and Dowling (2017). J. Acoust. Soc. Am. 142, 1663–1673]. In the work presented here, data from the 2012 COAST (Cascadia Open-Access Seismic Transects) experiment is utilized to explore out-of-band methods with horizontal arrays, where variations in environmental characteristics (bathymetry, sound speeds, etc.) may be more pronounced across the array. The abundance of array elements (636 in total) and total array length (7.9 km) in this experiment provide a means for controlling the sparsity and signal coherence within a chosen subarray to determine the capabilities and limitations of out-of-band beamforming. In addition, the bandwidth of the signal of interest is sufficient to compare out-of-band methods to conventional methods where the difference or sum frequencies match the in-band frequency used with conventional beamforming. [Sponsored by ONR.]

Frequency-difference beamforming with multiple realizations in an inhomogeneous environment

Frequency-difference beamforming is a sparse-array signal processing method that shifts the signal analysis to lower, out-of-band (difference) frequencies, provided that the difference frequency does not exceed the signal bandwidth. In an inhomogeneous environment with discrete scatterers, low-frequency fields may not be significantly affected, provided the scatterers are small relative to the signal wavelength. However, when the scatterer sizes are comparable to the signal wavelength, scattering effects are likely to be more significant. Previously, it was shown that frequency-difference beamforming can overcome some of the performance degradation from the high frequency scattering by shifting the processing to lower, out-of-band frequencies. In this case, the frequency-difference output was comparable to the output using a lower-frequency signal in the same environment. In this talk, theoretical fields that include scattering at the in-band frequencies are considered to predict the performance of the frequency-difference method. Furthermore, simulations and experiments in a 1.07-m-diameter water tank with a sparse array are used to determine the performance and robustness of the method in the presence of discrete high-contrast scatterers. Multiple realizations of randomly placed scatterers are combined to quantify performance statistics. [Sponsored by NAVSEA through the NEEC, and by ONR.]Frequency-difference beamforming is a sparse-array signal processing method that shifts the signal analysis to lower, out-of-band (difference) frequencies, provided that the difference frequency does not exceed the signal bandwidth. In an inhomogeneous environment with discrete scatterers, low-frequency fields may not be significantly affected, provided the scatterers are small relative to the signal wavelength. However, when the scatterer sizes are comparable to the signal wavelength, scattering effects are likely to be more significant. Previously, it was shown that frequency-difference beamforming can overcome some of the performance degradation from the high frequency scattering by shifting the processing to lower, out-of-band frequencies. In this case, the frequency-difference output was comparable to the output using a lower-frequency signal in the same environment. In this talk, theoretical fields that include scattering at the in-band frequencies are considered to predict the performance of the fr...

Cross-term analysis in frequency-difference-based source localization methods

In previous work, it has been shown that a quadratic product of frequency-domain acoustic fields at different frequencies but the same spatial location leads to an auxiliary field which may contain field information at frequencies below the original signal’s bandwidth (Worthmann, Song and Dowling, 2015, JASA 138, 3549-3562). This quadratic product, termed the frequency-difference autoproduct, has been shown to be valuable for beamforming and source localization in the presence of environmental mismatch and/or array sparseness. However, in a multipath environment, this quadratic product leads to undesired cross-terms. Bandwidth averaging procedures have been found to suppress some of their detrimental influences in some cases, but not all. Additionally, the poor dynamic range observed in frequency-difference beamforming and frequency-difference matched field processing are associated with the imperfect mitigation of these cross-terms. In this presentation, the nature of these cross-terms is analyzed, and signal processing tools are developed which attempt to robustly mitigate the detrimental effects of these cross terms. These signal processing tools can be used to potentially improve localization performance when using frequency-difference autoproduct-based source localization schemes. [Sponsored by ONR and NSF]

Frequency-difference wavenumber analysis with a sparse array

Frequency (f)- wavenumber (k) analysis can be used to estimate the direction of arrival (DOA) [J. Acoust. Soc. Am. 69, 732–737 (1980)]. When the receiver is a sparse array that is not suitable for conventional plane-wave beamforming, it adversely causes aliasing error due to spatial sampling, thus many striation patterns can emerge in f-k domain. In this study, we propose frequency-difference wavenumber analysis that is motivated frequency-difference beamforming [J. Acoust. Soc. Am. 132, 3018–3029 (2012)]. It is found that this approach can mitigate (or eliminate) such aliasing effect, which extends its applicability to the robust DOA estimation. Numerical simulation and experimental results are presented, and a major drawback is discussed.

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.

Frequency-difference beamforming in inhomogeneous media

Frequency-difference beamforming (Abadi et al., 2012, JASA, 132, 3018-3029) is an array signal processing technique that overcomes the limitations of the spatial Nyquist criterion by lowering the processing to out-of-band frequencies. This is accomplished using a quadratic product of complex signal amplitudes at different frequencies, resulting in wave propagation information at the out-of-band difference-frequency. Acoustic waves are susceptible to strong scattering in inhomogeneous media when the sizes of the inhomogeneities are comparable to or larger than the signal wavelength. Thus, conventional beamforming in random media at high frequencies with sparse arrays may be impossible, even in the presence of small inhomogeneities. However, at lower frequencies, acoustic propagation and beamforming in the same environment might not be significantly impacted. Thus, frequency-difference beamforming in the presence of high-frequency scattering is expected to maintain robustness similar to that of a low-frequency field. In this presentation, we present a theoretical framework that supports this hypothesis using the Born approximation. The theory is tested using experiments in a 1.07-m-diameter, 0.8-m-deep cylindrical water tank with an array of up to 16 receivers and 100 kHz to 200 kHz signal pulses that propagate through an inhomogeneous medium. [Sponsored by NAVSEA through the NEEC, and by ONR.]

Numerical study on the effect of various parameters on beamforming performance and the estimation of particle velocity using a circular array

Numerous studies on beamforming techniques have been done for the estimation of source direction with an array. The conventional beamforming (delay and sum), MVDR, and MUSIC are some examples. Recently, the frequency difference beamforming was also introduced for applications to high frequency ranges or the elimination of the aliasing effect. This talk compares performance of the beamforming techniques including frequency difference beamforming using a circular array. The effects of the various parameters, such as number of sensors, array radius, frequency range, and SNR, on the beamforming performance are studied. The error analysis for the estimation of particle velocity is also discussed. [This work was financially supported by the research project PES9020 funded by KRISO.]

Comparing frequency-difference beamforming and delay-and-sum beamforming as an estimator of direction-of-arrival

Direction-of-arrival estimation (DOA) is an important and long-studied problem in array signal processing. Existing algorithms such as conventional delay-and-sum beamforming are limited in performance and become inadequate for a high-frequency broadband source and when the receiving linear array is sparse. Abadi, Song, and Dowling (2012), however, showed that DOAS estimation is possible in this case with an algorithm they introduced called frequency-difference beamforming. This was demonstrated empirically in their previous work using a real dataset. In addition, the resemblance between frequency-difference beamforming and delay-and-sum beamforming was briefly discussed. The purpose of this study is to analyze this relationship further and determine how well frequency-difference beamforming compares to delay-and-sum beamforming as an estimator of DOA. The approach taken in this study is to explicitly formulate the estimator equation for frequency-difference beamforming and compare it mathematically to the estimator equation of delay-and-sum beamforming. The equations are compared for the case of one DOA and two DOAs. For both cases, we show the effect of both source frequency and source bandwidth on frequency-difference beamforming. Using these results, we probe the following question: what are the conditions on the source frequency or source bandwidth for which frequency-difference beamforming will outperform delay-and-sum beamforming?

Comparisons of conventional and frequency-difference beamforming

Conventional beamforming can isolate ray arrival directions from transducer array recordings, provided the array element spacing does not significantly exceed half of a signal wavelength. However, when the signal frequency is high and the array elements are many wavelengths apart, conventional beamforming may break down due to spatial aliasing. Frequency difference beamforming avoids this limitation by utilizing a quadratic product of complex signal amplitudes at different frequencies to extract wave propagation information at a difference frequency chosen low enough to avoid aliasing. For this presentation, the performance of frequency difference beamforming was evaluated two ways. First, simulations of acoustic propagation in a 106-m-deep shallow-ocean sound channel using the BELLHOP ray-tracing code were completed with frequency-sweep signals from 11 kHz to 33 kHz and 1100 Hz to 3300 Hz broadcast from a single source to a 56-m-long 16-element vertical array. Second, lab experiments in a 1.07-m-diameter...

Wave physics of the frequency difference autoproduct: Bilinear time-frequency analysis

Frequency difference beamforming (Abadi et al., 2012, JASA, 132, 3018-3029) and matched field processing (Worthmann et al., 2015, JASA, 138, 3549-3562) are array signal processing techniques that determine acoustic ray-path directions and source locations, respectively, from sparse array recordings by shifting the signal processing to a lower, out-of-band, difference-frequency bandwidth. This frequency down-shift is accomplished through the use of the frequency difference autoproduct (Δf-AP), a quadratic product of complex frequency-domain acoustic field amplitudes at two different frequencies. In this presentation, the physical and mathematical underpinnings of this quadratic product are discussed. Specifically, in the branch of applied mathematics termed bilinear time-frequency analysis, this Δf-AP is actually one of four quadratic time-frequency functions, each inter-related through forward and inverse Fourier transforms. The other three functions are the time-domain autoproduct, the Wigner distribution, and the ambiguity function. These four functions and several of their interesting properties are reviewed, including their relationship to the Wiener-Khintchine Theorem, and the spatial and spectral coherence and correlation functions. Additionally, the behavior of these functions when evaluated from simulated acoustic recordings from ideal single-path and multi-path acoustic environments is shown and compared. [Sponsored by ONR and NSF.]

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 locating remote acoustic sources in known environments. Unfortunately, environment-to-propagation model mismatch prevents successful application of MFP in many circumstances, especially those involving high frequency signals. For beamforming applications, this problem was found to be mitigated through the use of a nonlinear array-signal-processing technique called frequency difference beamforming (Abadi et. al. 2012). Building on that work, this nonlinear technique was extended to Bartlett MFP, where ambiguity surfaces were calculated at frequencies two orders of magnitude lower than the propagated signal, where the detrimental effects of environmental mismatch are much reduced. Previous work determined that this technique has the ability to localize high-frequency broadband sources in a shallow ocean environment with a sparse vertical array, using both simulated and experimental propagation data. Using simulations, the performance of this technique with horizontal arrays and adaptive signal processing techniques was investigated. Results for signals with frequencies from 10 kHz to 30 kHz that propagated in a 100-m-deep shallow ocean sound channel with a downward refracting sound speed profile will be shown for source array ranges of one to several kilometers. [Sponsored by the Office of Naval Research.]

Localization of a high frequency source in a shallow ocean sound channel using frequency-difference matched field processing

Matched field processing (MFP) is an established technique for locating remote acoustic sources in known environments. Unfortunately, environment-to-propagation model mismatch prevents successful application of MFP in many circumstances, especially those involving high frequency signals. For beamforming applications, this problem was found to be mitigated through the use of a nonlinear array-signal-processing technique called frequency difference beamforming (Abadi et al. 2012). Building on that work, this nonlinear technique was extended to MFP, where Bartlett ambiguity surfaces were calculated at frequencies two orders of magnitude lower than the propagated signal, where the detrimental effects of environmental mismatch are much reduced. In the Kauai Acomms MURI 2011 (KAM11) experiment, underwater signals of frequency 11.2 kHz to 32.8 kHz were broadcast 3 km through a 106-m-deep shallow-ocean sound channel and were recorded by a sparse 16-element vertical array. Using the ray-tracing code Bellhop as the propagation model, frequency difference MFP was performed, and some degree of success was found in localizing the high frequency source. In this presentation, the frequency difference MFP technique is explained, and comparisons of this nonlinear MFP technique with conventional Bartlett MFP using both simulations and KAM11 experimental data are provided. [Sponsored by the Office of Naval Research.]

Frequency-difference matched field processing in the presence of random scatterers

Matched field processing (MFP) is an established technique for locating remote acoustic sources in known environments. Unfortunately, unknown random scattering and environment-to-propagation model mismatch prevents successful application of MFP in many circumstances, especially those involving high frequency signals. Recently a novel nonlinear array-signal-processing technique, frequency difference beamforming, was found to be successful in combating the detrimental effects of random scattering for 10 kHz to 20 kHz underwater signals that propagated 2.2 km in a shallow ocean sound channel and were recorded by a 16-element vertical array. This presentation covers the extension of the frequency-difference concept to MFP using sound propagation simulations in a nominally range-independent shallow ocean sound channel that includes point scatterers. Here again, 10 kHz to 20 kHz signals are broadcast to a vertical 16-element array, but the frequency difference approach allows Bartlett and adaptive MFP ambiguity surfaces to be calculated at frequencies that are an order of magnitude (or more) below the signal bandwidth where the detrimental effects of environmental mismatch and random scattering are much reduced. Comparison of these results with equivalent simulations of conventional Bartlett and adaptive MFP for different of source-array ranges are provided. [Sponsored by the Office of Naval Research.]

Broadband sparse-array blind deconvolution using frequency-difference beamforming

Synthetic time reversal (STR) is a technique for blind deconvolution of receiving-array recordings of sound from an unknown source in an unknown multipath environment. It relies on generic features of multipath sound propagation. In prior studies, the pivotal ingredient for STR, an estimate of the source-signal's phase (as a function of frequency ω), was generated from conventional beamforming of the received-signal Fourier transforms, P(j)(ω), 1 ≤ j ≤ N, where N is the number of array elements. This paper describes how STR is implemented even when the receiving-array elements are many wavelengths apart and conventional beamforming is inadequate. Here, the source-signal's phase is estimated by beamforming P(j)(*)(ω(1))P(j)(ω(2)) at the difference frequency ω(2) - ω(1). This extension of STR is tested with broadband signal pulses (11-19 kHz) and a vertical 16-element receiving array having a 3.75-m-spacing between elements using simple propagation simulations and measured results from the FAF06 experiment involving 2.2 km of down slope propagation from 46 to 92 m water depth. The cross-correlation coefficient between the source-broadcast and STR-reconstructed-signal waveforms for the simulations and experiments are 98% and 91%-92%, respectively. In addition, frequency-difference beamforming can be used to determine signal-path-arrival angles that conventional beamforming cannot.