Complex Signal Amplitudes Research Articles

Beamforming applications of the frequency-difference acoustic autoproduct

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 utilizing the acoustic autoproduct to shift the processing to below-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. The resulting field is capable of mitigating many of the in-band challenges often associated with high frequency acoustic signal processing, including sparse receiver arrays and features of the physical environment that are on a scale significant for the in-band wavelength. This presentation uses both laboratory and ocean-based experiments to demonstrate capabilities of the method. The mitigation of sparse array aliasing effects on beamforming (caused by elements that are spaced by many wavelengths) in both a laboratory water tank and an ocean environment utilizing data collected during the KAM11 experiment are considered. Additionally, the method is used to localize a high frequency source in the presence of strong, random scatterers in a water tank experiment. [Work sponsored by NAVSEA and ONR.]

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.

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.]

Using frequency-sum beamforming in passive cavitation imaging

Passive Cavitation Imaging (PCI) is a method for locating cavitation emissions to study biological effects of ultrasound on tissues. In this method, an image is formed by beamforming passively recorded acoustic emissions with an array. The image resolution depends on the ultrasound frequency and array geometry. Acoustic emissions can be scattered due to tissue inhomogeneity, which may degrade the image resolution. Emissions at higher frequencies are more susceptible to such degradation. Frequency-sum beamforming is a nonlinear technique that alters this sensitivity to scattering by manufacturing higher frequency information from lower frequency components via a quadratic product of complex signal amplitudes. This presentation evaluates the performance of frequency-sum beamforming in a scattering environment using simulations and experiments, conducted in the kHz and MHz frequency regimes. First, 50 and 100 kHz signals were broadcasted from a single source to an array of 16 hydrophones in a water tank with and without discrete scatterers. Second, a tissue-mimicking phantom perfused with microbubbles was insonified at 2 MHz, and the emissions were received by a 128-element linear array. The performance of frequency-sum beamforming was compared to conventional delay-and-sum and minimum variance beamforming in mild and strong scattering environments. [Work partially supported by NAVSEA through the NEEC.]

Distributed MIMO Multicast With Protected Receivers: A Scalable Algorithm for Joint Beamforming and Nullforming

We consider the problem of multicasting a common message signal from a distributed array of wireless transceivers by beamforming to a set of beam targets , while simultaneously protecting a set of null targets by nullforming to them. We describe a distributed algorithm in which each transmitter iteratively adapts its complex transmit weight using common aggregate feedback messages broadcast by the targets, and the local knowledge of only its own channel gains to the targets. This knowledge can be obtained using reciprocity without any explicit feedback. The algorithm minimizes the mean square error between the complex signal amplitudes at the targets and their desired values. We prove convergence of the algorithm, present geometric interpretations, characterize initializations that lead to minimum total transmit power, and prescribe designs for such initializations. We show that the convergence speed is nondecreasing in the number of transmitters $N$ if a step size parameter is kept constant. For Rayleigh fading channels, as $N$ goes to infinity: 1) convergence can be made arbitrarily fast and 2) beams and nulls can be achieved with vanishing total transmit power even with noise, both with probability one. These results add up to some remarkable scalability properties: the feedback overhead does not grow with the number of transmitters, and with high probability, the algorithm can be configured to converge arbitrarily fast and use vanishingly small total transmit power.

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...

Interpretations of the frequency difference autoproduct in multipath environments

When locating remote acoustic sources in a shallow ocean sound channel, the established array signal processing technique known as matched field processing (MFP) has shown much success. However, MFP is sensitive to mismatch between the modeled and actual environments, and may fail to localize acoustic sources in the presence of such mismatch, particularly at high frequencies. A recent nonlinear array signal processing technique, frequency difference MFP (Abadi et. al. 2012, Worthmann, et. al., under review), has shown some success in localizing high frequency sources by moving the replica calculations to a lower, out-of-band, difference frequency where the detrimental effects of environmental mismatch are less severe. To extract the requisite out-of-band difference frequency information from the measured signals, a quadratic product, termed the autoproduct, is formed from complex signal amplitudes separated by the difference frequency but still lying within the signal bandwidth. Through the use of simple multipath propagation environments, the nature of this autoproduct is explored, and the reasons that it provides out-of-band field information are presented. More complex propagation environments are simulated as well to demonstrate some of the expected and unexpected behaviors of the autoproduct. [Sponsored by the Office of Naval Research and the National Science Foundation.]

Frequency-sum beamforming in a random scattering environment

In a uniform environment, sound propagation direction(s) or the location of a sound source may be determined from array-recorded signals by beamforming. However, the beamforming results may be degraded when there is random scattering between the source and the receivers. Such sensitivity to mild scattering may be altered through use of an unconventional beamforming technique that manufactures higher frequency information from lower-frequency signal components via a quadratic product of complex signal amplitudes. This presentation will describe frequency-sum beamforming, and then illustrate it with simulation results and near-field acoustic experiments. The simulations suggest that frequency-sum beamforming may be beneficial when there is one loud source and the environment provides one primary propagation path. The experiments were conducted in either a 1.0-m-deep 1.07-m-diameter cylindrical water tank using 50 kHz and 100 kHz signals broadcast from a single source to an array of 16 hydrophones when discrete scatterers are present and absent from the tank or in a tissue-mimicking phantom with a dominant scatterer embedded and insonified at 2 MHz and the scatter received by a 128-element array. The results from frequency-sum beamforming are compared to the output of conventional delay-and-sum beamforming and minimum variance beamforming. [Work supported by NAVSEA through the NEEC.]

Frequency-sum beamforming in an inhomogeneous environment

The arrival directions of ray paths between a sound source and a receiving array can be determined by beamforming the array-recorded signals. And, when the array and the signal are well matched, directional resolution increases with increasing signal frequency. However, when the environment between the source and the receivers is inhomogeneous, the recorded signal may be distorted and beamforming results may be increasingly degraded with increasing signal frequency. However, this sensitivity to inhomogeneities may be altered through use of an unconventional beamforming technique that manufactures higher frequency information by summing frequencies from lower-frequency signal components via a quadratic (or higher) product of complex signal amplitudes. This presentation will describe frequency-sum beamforming, and then illustrate it with simulation results and near-field acoustic experiments made with and without a thin plastic barrier between the source and the receiving array. The experiments were conducted in a 1.0-meter-deep and 1.07-m-diameter cylindrical water tank using a single sound projector, a receiving array of 16 hydrophones, and 100 micro-second signal pulses having nominal center frequencies from 30 to 120 kHz. The results from frequency-sum beamforming will be compared to the output of conventional delay-and-sum beamforming for different center frequencies. [Work sponsored by ONR and NAVSEA.]

Complex signal amplitude analysis for complete fusion nuclear reaction products

A complex analysis has been performed on the energy amplitude signals corresponding to events of Z = 117 element measured in the 249Bk + 48Ca complete fusion nuclear reaction. These signals were detected with PIPS position sensitive detector. The significant values of pulse height defect both for recoils (ER) and for fission fragments2 were measured. Comparison with the computer simulations and empirical formulae has been performed both for ER and FF signals.

A novel method for estimating vertical eddy diffusivities using diurnal signals with application to western Long Island Sound

We present an approach that allows the estimation of vertical eddy diffusivity coefficients from buoy measurements made at two or more depths. By measuring the attenuation and phase lag of a scalar signal generated periodically at the surface as it propagates downwards, the vertical eddy diffusivity coefficients can be calculated as K v = ωΔz 2/2ln 2(α 2/α 1), where α 2/α 1 is the ratio of the real amplitudes at frequency ω at the two depths separated by Δ z = z 2 − z 1; as K V = ωΔ z 2/ 2φ 2, where φ is the phase lag at the frequency ω; or as K v = iωΔ z 2/ln 2( U 2/ U 1), where U 2/ U 1 is the ratio of the complex signal amplitudes at the two depths. The method requires that horizontal fluxes be small at the ω frequency and that the signal-to-noise ratios at the two depths allow the determination of the amplitude and phase of ω. Application of this method to summertime 2004 western Long Island Sound oxygen and temperature buoy measurements at two depths provides a time-series of two-day average vertical eddy diffusivity estimates. Using these eddy diffusivities in conjunction with measured vertical concentration gradients, we obtain a time-series of vertical transport rates for oxygen and heat and estimate mean downward fluxes for June and July as 150–260 mMol m − 2 day − 1 and 100–400 W m − 2 respectively. These estimates are of a similar magnitude to sub-pycnocline O 2 and heat demands of 240 ± 200 mMol m − 2 day − 1 and 180 ± 60 W m − 2 that we infer from simple budgets, implying that vertical transport is significant to both budgets. The eddy coefficients obtained from the independent O 2 and temperature measurements have a 68% correlation, and the O 2 flux estimates show a correlation of 41% to measured rates of change in bottom dissolved oxygen levels. Our results indicate that extended time-series of eddy diffusivity coefficients can be obtained from in situ buoy measurements and the method shows promise as a way to constrain the vertical transport variability in budgets of dissolved materials in estuaries.

Bayesian Complex Amplitude Estimation and Adaptive Matched Filter Detection in Low-Rank Interference

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We propose a Bayesian method for complex amplitude estimation in low-rank interference. We assume that the received signal follows the generalized multivariate analysis of variance (GMANOVA) patterned-mean structure and is corrupted by low-rank spatially correlated interference and white noise. An iterated conditional modes (ICM) algorithm is developed for estimating the unknown complex signal amplitudes and interference and noise parameters. We also discuss initialization of the ICM algorithm and propose a (non-Bayesian) adaptive-matched-filter (AMF) signal detector that utilizes the ICM estimation results. Numerical simulations demonstrate the performance of the proposed methods. </para>

Joint signal detection and parameter estimation in multiuser communications

The problem of simultaneously detecting the information bits and estimating signal amplitudes and phases in a K-user asynchronous direct-sequence spread-spectrum multiple-access communication system is addressed. The joint maximum-likelihood (ML) estimator has a computational complexity that is exponential in the total number of bits transmitted and thus does not represent a practical solution to the problem. An estimator that combines a suboptimum tree-search algorithm with a recursive least-squares estimator of complex signal amplitude is considered. The complexity of this estimator is O(K/sup 2/) computations per decoded bit, and its performance is very close to that of the joint ML receiver. This receiver has the advantage that the transmitted signal powers and phases are extracted from the received signal in an adaptive fashion without using a test sequence.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>