Multiplicative Processors Research Articles

Cross Term Decay in Multiplicative Processors

Multiplicative processors combine the beamformed outputs of undersampled subarrays to estimate the spatial power spectrum. While the multiplicative processor requires fewer sensors to achieve the same resolution as a conventional linear processor, it also produces cross terms that can degrade the spectral estimate. Cross terms are false peaks formed when a signal passing through the grating lobe of one subarray interacts with a different signal passing through the other subarray. Snapshot averaging reduces cross terms when the signals are uncorrelated. This letter quantifies the number of snapshots required for effective cross term mitigation, accounting for the use of subarray tapers to control sidelobe levels. To facilitate the analysis, the letter derives closed form expressions for the probability density function and cumulative distribution function of the sum of products of independent complex Gaussian random variables. The analysis demonstrates that cross terms decay at a rate of 5 dB per decade of snapshots averaged. While subarray tapering reduces sidelobe leakage, it also increases the number of snapshots required for cross term mitigation.

Performance weighted blending of nested array processors

Nested arrays consist of two uniform subarrays: a short aperture array with half-wavelength spacing and a long aperture array with greater than half-wavelength spacing. Two approaches for estimating the scanned response require computing either the product or the minimum of the subarray responses in each look direction. In multi-source environments, the multiplicative and min scanned responses may be corrupted by cross terms [Wage, Acoustics Today (2018)]. When the sources are uncorrelated, snapshot averaging is often used to mitigate cross term interference. The min processor's response can contain cross terms that do not decay with snapshot averaging. Alternatively, the multiplicative processor’s response contains cross terms that decay, though large numbers of snapshots may be required for the cross terms to fall below the level of those in the min processor. This talk proposes a performance weighted combination of the two processors based on Buck and Singer's (IEEE, 2018) blended dominant mode rejection beamformer. The same performance weighting function can be used to combine the nested multiplicative and min processors. The resulting universal processor adjusts the weighting in each look direction as the number of snapshots increases to favor the multiplicative processor as its cross terms average out. [Work supported by ONR]

Statistical characterization of cross terms in snapshot-averaged multiplicative processors

Multiplicative processors generate spatial spectrum estimates for sparse arrays, such as nested and coprime arrays, by multiplying the beamformed outputs of two interleaved subarrays. Nested and coprime arrays achieve significant sensor savings compared to dense Uniform Linear Arrays (ULAs) since one or both subarrays are undersampled. Chavali et al. show that careful design of the subarrays and proper selection of beamformer weights can guarantee power pattern performance (response to single planewave source) comparable to a conventional ULA processor [JASA (2018)]. The multiplicative processors’ response to multiple sources contains cross terms that appear as erroneous sources in the spectral estimate. The height of cross term peaks in the spectrum depends on the subarray design, beamformer weights, and signal powers. Ksienski and Pedinoff show that averaging the multiplicative processor output over snapshots reduces the power of uncorrelated cross terms [IEEE (1962)], though they do not explore how much averaging is required. This talk derives the statistics of the averaged cross terms assuming complex Gaussian planewave signals and noise. The statistical model is used to predict the number of snapshots required to eliminate uncorrelated cross term peaks from the multiplicative spectra. Analytical predictions show excellent agreement with planewave simulations. [Work supported by ONR.]

Multiplicative and min processing of experimental passive sonar data from thinned arrays.

Sparse arrays reduce the number of sensors required to achieve a specific angular resolution by using sensor spacing greater than the half-wavelength. These undersampled sparse arrays require processing algorithms to eliminate aliasing ambiguities. Thinned arrays are sparse arrays whose sensor positions lie on an underlying equally spaced grid. Using data from a shallow water passive sonar experiment, this paper investigates two thinned array geometries (coprime and nested) along with two processing algorithms (multiplicative and min). Coprime and nested arrays consist of two interleaved Uniform Line Arrays (ULAs) where one or both of the ULAs are undersampled. Multiplicative and min processors combine the outputs of the conventionally-beamformed subarrays to estimate the spatial spectrum. While these nonlinear processors can suppress aliasing, they are often plagued by high sidelobes and cross term interference. This paper presents sparse array designs for a shallow waveguide that require 33% fewer sensors than a fully-sampled ULA and provide significant sidelobe attenuation. Experimental data analysis reveals that cross term interference dominates the spectral estimates for the coprime and nested multiplicative processors and the coprime min processor. The nested min processor outperforms its sparse counterparts due to its ability to contend with coherent multipath in the environment.

Comparison of multiplicative and min processors for coprime and nested geometries using the Elba Island data set

Vaidyanathan and Pal describe nested and coprime array geometries that provide significant sensor savings when compared to densely populated Uniform Linear Arrays (ULAs) [IEEE Trans. Sig. Proc., 2010, 2011]. They show that a multiplicative processor can reconstruct the spatial power spectrum from the sparse measurements made by a Nested Array (NA) or Coprime Sensor Array (CSA). Multiplication of the beamformed outputs of two undersampled subarrays eliminates the ambiguity due to aliasing, but requires temporal averaging to mitigate cross terms. Prior work analyzes a multiplicative CSA processor using data from an underwater vertical array deployed near Elba Island [Chavali et al., Asilomar Conf. SS&C, 2014]. Compared to the conventional spectrum obtained with the ULA, the multiplicative CSA spectrum for the Elba data contains endfire interference due to cross terms associated with coherent mode arrivals. Liu and Buck propose an alternative processor, which uses the minimum of the two subarray outputs as the spectral estimate [IEEE SAM, 2016]. This talk compares performance of the multiplicative and min processors using the NA and CSA subarrays designed for the Elba experiment. The NA-min processor produces the best spectral estimate for this scenario. [Work supported by ONR Basic Research Challenge Program.]

Algorithm and architecture for a Galois field multiplicative arithmetic processor

We present a new algorithm for generic multiplicative computations of the form ab/c in GF(p/sup m/), including multiplication, inversion, squaring, and division. The algorithm is based on solving a sequence of congruences that are derived from the theory of Grobner bases in modules over the polynomial ring GF(p)[x]. Its corresponding hardware and software architectures can be successfully used in applications such as error control coding and cryptography. We describe a versatile circuit associated with the algorithm for the most important case p=2. The same hardware can be used for a range of field sizes thus permitting applications in which different levels of error control or of security are required by different classes of user. The operations listed are all performed by the hardware in the same number of clock cycles, which prevents certain side-channel attacks. The loss in performance by having 2m iterations for multiplication is compensated by the full parameterization of the Galois field and the ability to perform division and multiplication in parallel.

Receiver operating characteristics for a class of three channel receivers

The probability density function for signal absent is obtained for a class of three channel receivers commonly used for direction finding. The receiver considered here is closely related to the multiplicative array processors first considered by Miller (1973). The false alarm probability and noise mean are derived. The statistics for signal present appear to be intractable, but the false alarm statistics presented here may be used to properly set the threshold for a Monte-Carlo study of detection performance. Detection probability is presented for an example case.

Mathematical comparison of the effects of multipath interference on estimates yielded by. additive and multiplicative processors

This analysis considers the use of additive and multiplicative processing techniques to estimate a signal level in an ocean environment that is conducive to an interfering arrival, such as a surface reflected arrival. Representing the auto- and cross-correlation functions as decaying exponentials having different effective correlation times facilitates the evaluation of the integral expressions for the variance. By varying these effective correlation times and the measurement geometry (to change the delay time of the interfering arrival) the sensitivity of the estimate on the input characteristics can be observed. The parameters that are found to govern the system performance are input signal-to-noise ratio, the degree of input noise correlation, and the amount of signal decorrelation. The results show that the estimates yielded by the system depend heavily on input signal-to-noise ratio with the multiplicative processor generally outperforming its additive counterpart for most cases at low input signal-to-noise ratios. [Work sponsored by Office of Naval Research.]

Line Array Performance When the Signal Coherence is Spatially Dependent

The effects of a signal coherence which decreases exponentially with sensor separation are examined for a number of line array processors. For the conventional beamformer the limiting gain as the array is made infinitely long is simply (1 + ρ)/(l − ρ), where ρ is the signal coherence between adjacent sensors. The gain of the optimum processor is presented. A simple suboptimum processor which incoherently combines the outputs of a number of subarrays is found to provide near optimum performance. The performance of the split-beam multiplicative processor is shown to degrade rapidly as signal coherence decreases. [This work was done at the University of California, San Diego, Marine Physical Laboratory of the Scripps Institution of Oceanography, San Diego, California 92132.]

Line array performance when the signal coherence is spatially dependent

The effects of a signal coherence which decreases exponentially with sensor separation are examined for a number of line array processors. For the conventional beamformer the limiting gain against uncorrelated noise as the array is made infinitely long is simply (1 + ρ)/(1 − ρ), where ρ is the signal coherence between adjacent sensors. The gain of the optimum processor is also presented. A simple suboptimum processor which incoherently combines the outputs of a number of subarrays is found to provide near optimum performance. The performance of the split-beam multiplicative processor is shown to degrade rapidly as signal coherence decreases.