Replica Correlation Research Articles

SOUND TRANSMISSION AND SPATIAL COHERENCE IN SELECTED SHALLOW-WATER AREAS: MEASUREMENTS AND THEORY

Experiments from several shallow-water areas are summarized. Coherent sound transmission results, particularly wavenumber spectra, are compared to range-dependent calculations that use oceanographic and geophysical characteristics from measurements and archives as bounded inputs to the propagation codes. In general excellent agreement was obtained between the measured and calculated results for both narrowband and broadband transmissions between 50 Hz and 1 kHz to ranges of 40 km. A relative signal gain (RSG) method for the estimation of horizontal coherence length was applied to measured RSG results and yielded coherence lengths on the order of 30λ at 400 Hz at distances of 40 km. Perturbation theory was applied to the shallow-water waveguide under the condition of adiabatic normal modes and expressions were derived for the phase structure function that was simplified by the use of Gaussian correlation functions. These analytical results, along with estimates of the variances of the environmental variables permitted the estimation of the coherence function and the RSG. The calculated coherence function and RSG were found to be consistent with measured RSG and replica correlation results. The fluctuations in the oceanic water volume were found to be the dominant factor in the loss of coherence.

Broadband shallow-water reverberation statistics

Utilizing a wideband transmitting/receiving system, at-sea experimental results for broadband reverberation have been acquired for both harsh as well as moderate ocean acoustic environments. The principal contributor to the backscattered reverberation is bottom features at the shallow-water ocean sites. Replica correlation (so-called match filter) processing of linear FM (LFM) pulse codes is employed to study the degree of reverberation coherence, as measured by the replica correlation coefficient. The database covers several octaves of bandwidth. The corresponding reverberation coherence is computed in terms of the variance of the cross correlation coefficient (cc) as a function of time–bandwidth product. Of particular interest is the possible departure from Rayleigh statistics as signal bandwidth increases. Several bandwidths are compared, using the Kolmogoroff–Smirnof test as a measure of departure from Rayleigh. [This work was sponsored by A. Nucci, ONR 333.]

Forward scattering of pulses from a rough sea surface by Fourier synthesis of parabolic equation solutions.

A variable depth step implementation of the range-dependent acoustic model (RAM) is applied to the modeling of forward scattering from a rough sea surface. The sea surface is treated within RAM simply as an internal interface between a water layer and an air upper halfspace. A comparison with a numerically exact integral equation is undertaken for the scattering of single frequencies from Pierson-Moskowitz sea surfaces. The method is extended to model the variability of linear frequency modulated pulses from a series of frozen sea surfaces in a shallow water waveguide. The subsequent effect of rough boundary scattering on the replica correlation process is investigated.

Broadband ocean reverberation coherence

Utilizing a wide-band transmitting/receiving system, at-sea experimental results for broadband reverberation have been acquired for both a harsh as well as a moderate ocean acoustic environment. Both stationary and moving sonar systems were employed in the study. Replica correlation (so-called match filter) processing of linear FM (LFM) pulse codes is employed to study the degree of reverberation coherence achievable over 4 octaves of bandwidth. The LFM signals are systematically processed over a bandwidth interval from 0.5 kHz to over 12 kHz.The corresponding reverberation coherence is displayed in terms of the variance of the cross correlation coefficient as a function of time-bandwidth product. Comparison with the theoretically expected variance for band-limited white Gaussian noise is also included. [Work sponsored by L. Jacobi, ONR 333.]

Time reversal, back propagation, matched field processors, correlation receivers, and the principles of radar/sonar signal design

Time reversal and backpropagation have been demonstrated in several experiments. Similarly, while matched field processing (MFP) differs in terms of implentation—experimental vs computed replicas—both have two common properties: (i) they are based on a single, spatially coherent signal; and (ii) the conjugate transpose of the Green’s function and replica correlation are identical for self-adjoint systems. Hence, the principles for focusing and ambiguity plane properties of these processors are virtually identical to those for correlation receivers. The principles of optimal signal design for correlation receivers were the subject of much research for radar/sonar systems four decades ago and many of them seem to have been neglected in the analysis of time reversal, back propagation, and matched field processors. For example, time reversal from a point, a line array, or a random array of scatterers are duals of an impulse, a frequency modulated, and a pseudo-random noise signal, respectively. The equivalence and consequences of the time-bandwidth products for signals and array length wave number spread are demonstrated. The impact of sidelobes and multipath spread can be predicted. The generalizations of the important radar/sonar uncertainty principle, however, have yet been not demonstrated. This presentation reviews these optimal signal design principles and applies them to time reversal and MFP.

Broadband shallow-water sea-test results with a towed vertical aperture array

We review matched-field processing (MFP) sea test results obtained using a towed vertical aperture array. The array consisted of five horizontal lines towed in a vertical plane, nominally one over the other, with an approximately 5-m separation between the lines. We review results from two tests, one conducted in the Gulf of Mexico in about 120 m of water, and the other conducted southwest of Key West, FL in about 600 m of water. The vertical aperture array was towed on straight-line courses, at various ranges, parallel to a towed sound source. The radiated signal was a Gaussian pseudorandom waveform in the 100–600-Hz band. Results using replica correlation techniques are first used to study the channel transfer function; next, MFP results obtained by treating the signal as a random process are shown. These results demonstrate accurate range and depth localizations from 5 to 20 km as well as the ability to resolve a loud surface interferer from a quieter source at depth which is on the same bearing. These test results demonstrate the inherent robustness of broadband MFP. [Work supported by ONR 321.]

Analyzing single molecule trajectories on complex energy landscapes using replica correlation functions

Single molecule experiments bring to mind a new set of questions having to do with the statistical mechanics of uniqueness and individuality. One such question is the quantitative description of the diversity of mechanisms for reactions on complex landscapes. An example of this is the question of the existence of a unique “pathway” for protein folding. These questions also arise in the theory of spin glasses. We show how single molecule experiments can be interpreted using replica correlation functions analogous to those appearing in spin glass theory. A caricature of a many conformational state reaction scheme appropriate for folding is analyzed in some detail.

Fluctuations in acoustic propagation seen in the SWARM 95 experiment

Fluctuations in acoustic propagation as seen in data collected during the SWARM 95 experiment are analyzed and related to the oceanographic environment. Acoustic data were collected on a 32 element vertical receiver array which spanned the 88 m water column from depths of 23 to 85 m. The receiver array was located 42 km seaward from two fixed acoustic projectors which transmitted a 224 Hz PRN signal (16 Hz bandwidth) and a 400 Hz PRN signal (100 Hz bandwidth). Intensity statistics, such as scintillation index (SI) and the probability distribution function (PDF), are discussed as a function of depth and time for narrow-band and broadband (replica correlation) processing. Broadband SI is shown to be less than the narrow-band SI which has a typical value of 2. The PDF is shown to be neither a Rayleigh nor a log-normal distribution and evidence is presented that unsaturated scattering processes are involved. [Work supported by the Office of Naval Research.]

Copolymer Melts in Disordered Media

The symmetric AB block copolymer melt in a gel matrix with preferential adsorption of A monomers on the gel gives an example of a random-field system, which is described near the point of the microphase separation transition by the random field Landau-Brazovskii Hamiltonian. By using the technique of the 2-nd Legendre transform, the phase diagram of the system is calculated. We found that the preferential adsorption of the copolymer on the gel results in two effects: a) It decreases the temperature of the first order phase transition between disordered and ordered phase. b) There exists a region on the phase diagram at some small but finite value of the adsorption energy in which the replica symmetric solution for two replica correlation function is unstable with respect to replica symmetry breaking. We interpret this state as a glassy state and calculate a spinodal line of this transition. We also consider the stability of the lamellar phase in the weak segregation limit by mapping the copolymer Hamiltonian onto the Hamiltonian of the random field XY model. We suggest that the long range order is always destroyed by weak randomness.

SECTOR FOCUSING FOR ROBUST HIGH RESOLUTION ANALYSIS OF THE MATCHED FIELD PROCESSING WORKSHOP DATA SET

Replica correlator, minimum variance distortionless response, and sector-focused processing are applied to the test data cases devised for the Matched Field Processing Workshop. The methods are compared on this data set and their performance is clearly delineated. In all the cases examined from the workshop data set which were appropriate for the application of sector-focused procedures (i.e. the presence of mismatch together with modal noise), sector-focused processing performed successfully. It located the source with robustness similar to the replica correlator processor, but with the higher resolution characteristic of minimum variance distortionless response processing. In all cases where replica correlator localizes correctly, sector focusing also localizes correctly and achieves higher resolution. Sector focusing always reduced signal gain degradation compared to minimum variance distortionless response processing. It also reduced signal gain degradation compared to replica correlator processing for the lowest signal-to-noise ratio case. However, significant range bias error was observed in sector-focused as well as replica correlator processing in the case with general mismatch and colored noise. Sector focusing was not applied to the case with sound speed mismatch but no colored noise.

An analysis of mismatch induced range localization error in conventional matched-field processing

In this work the environmental mismatch problem in matched-field processing is addressed by attempting to obtain analytic expressions for mismatch-induced range localization error and signal gain degradation for a linear (replica correlator) matched-field processor using simple models such as the rigid and Pekeris waveguides. This analysis is based on the concept that these types of errors are the result of mismatch-induced dephasing of the modally interfering components of the acoustic field in the waveguide. As these modal components dephase, the signal peak degrades, and it forms at a location slightly different from the correct location. In particular, which types of mismatch produce a substantial range localization error have been determined and the frequency dependence of such range shifts has been studied. The analytic results have been substantiated with computer simulations. [This work was supported by the Office of Naval Research, Program Element 61153N, with technical management provided by the Naval Research Laboratory, Stennis Space Center, MS.]

On detecting linear frequency-modulated waveforms in frequency- and time-dispersive channels: alternatives to segmented replica correlation

In modern active sonars, so-called high-gain waveforms are used to obtain processing gain for improved detection performance in reverberation. In time and frequency spread channels, the full processing gain of these waveforms is not achievable and robust detectors are needed. It is common practice to use a segmented replica correlator (SRC) detector for robust detection in such environments. In this paper, we show that an alternative processor formed by integrating the full replica correlator output magnitude squared, denoted RCI for replica correlator integrator, has advantages over the SRC when the waveform is linear frequency modulated (LFM). The RCI is better suited to the case of unknown time spreading through the use of a bank of integrators tuned for a wide range of spreading widths. The SRC, on the other hand, may be designed only for a fixed amount of distortion. Computer simulations are used to show how a multihypothesis RCI processor improves upon the fixed SRC by up to 4 dB over a realistic range of time spreading widths. >

Demonstration of the large and small sector limits for large vertical array sector focused matched-field processing in a deep water environment

Sector focused matched-field processing is a technique that has been developed to stabilize high-resolution estimators against random phase errors and environmental mismatch. This technique produces satisfactory high-resolution matched-field localization over a wider latitude of environments than does conventional high-resolution techniques. Its implementation requires the selection of several parameters such as sector size, shape and dimensionality. In this work, general guidance is given for the use of this technique by establishing the properties of the small and large sector limits. Specifically, it is demonstrated that in the small sector limit, sector focusing behaves like replica correlator processing (robust and low resolution). Whereas, in the large sector limit, sector focusing behaves like reduced maximum-likelihood processing (high resolution, but not as robust). The demonstration is accomplished by plotting signal gain degradation versus three sediment sound-speed parameters. These plots show a small hot spot at the correct environment immersed in very cool surroundings for the conventional high-resolution estimator (mvdr). Whereas, for sector focusing, the plots show a rather wide swath of acceptable environments. In most cases, the technique peaks on the correct environment. [This work was supported by the Office of Naval Research, with technical management provided by the Naval Research Laboratory, Stennis Space Center, MS.]

Real time time-frequency active sonar processing: a SIMD approach

A paradigm for massively parallel processing of matched filters, replica correlators, ambiguity functions, and time-frequency distributions is presented, using a SIMD (single instruction stream, multiple data stream) programming methodology. It is shown that active sonar detection algorithms, as implemented by frequency domain processing, can be a natural match to a SIMD methodology, meeting the extensive computational needs of enhanced active sonar systems. The decomposition process is presented, and examples are given of the output of the computer program CMASP (Connection Machine Ambiguity Surface Processor). CMASP can provide real-time simultaneous multiple-beam, Doppler, and waveform replica correlations. Synthetic data are processed, and the corresponding CMASP outputs are displayed as three-dimensional ambiguity surfaces on networked graphic workstations. Because of efficient problem decomposition, other time-frequency processing can be exploited. Specifically, real-time instantaneous-like time-frequency distributions have been realized in which the data set is presented and processed as time-varying spectral representations. >

Use of rectangular vandermonde matrix decomposition to resolve damped sinusoidal oscillation components

Signals that are sums of decaying sinusoids arise frequently in passive and active sonar applications, both as spontaneous acoustic transients and as echoes from elastic objects. When the oscillations are highly damped, identification of component frequencies and damping factors is more amenable to least‐squares Prony analysis than to Fourier analysis, and is particularly suited to a related technique that we call “rectangular Vandermonde matrix decomposition.” Our results indicate that the Vandermonde method is robust in the presence of noise, but not in the presence of distortions created by ideal bandpass filters. On the other hand, analysis of simulated replica‐correlated echoes of FM pulses bounced off an object having a pair of heavily damped resonances give different results. Surprisingly, the Vandermonde method identified the resonant frequencies even without replica correlation, despite the fact that the analyzed waveform looked not at all like a sum of decaying sinusoids. [Work supported by the ONR.]

The use of replica correlation to benchmark acoustical propagation models

The three usual techniques employed for benchmarking acoustical propagation models involve measured acoustical data, analytical solutions, and other established acoustical propagation models. Each of these has inherent limitations. The alternative approach described here involves replica correlation to test the accuracy of acoustical propagation models. A received acoustic pressure spectrum, which is generated by the acoustical propagation model to be tested, is multiplied with the source pressure spectrum. The inverse Fourier transform of this product is a replica correlogram, which contains information of the travel time, phase and amplitude of all arrivals. Then a ray‐tracing model is used to determine the travel time of the arrival. If the model to be tested is accurate, the travel times computed by replica correlation will agree with the travel time computed by ray tracing. Results will be presented for an existing acoustical propagation model under realistic ocean conditions. [Work supported by NRL.]

Mismatched Filtering of Sonar Signals

A replica correlator (matched filter) is an optimum processor for a receiver employing a pulse of continuous wave (CW) signal in a white Gaussian noise background. In an active sonar, however, when the target of interest has low Doppler shift and is embedded in a high reverberation background, this is not so. High sidelobes of the correlator frequency response pass a significant portion of the signal contained in the mainlobe of the reverberation spectrum. In order to reduce the sidelobes of the correlator output spectrum and at the same time keep the increase in its 3 dB bandwidth to a small amount, we propose lengthening of the replica of the transmitted signal and weighting it by a Kaiser window. It is demonstrated that by extending the weighted replica by 50 percent compared with the transmitted signal, it is possible to reduce the sidelobe levels to at least 40 dB below the mainlobe peak, with the concomitant increase of the 3 dB band-width by less than 5 percent. The degradation in signal-to-noise ratio (SNR) performance for such a ?mismatched? filter receiver with respect to the matched filter is less than 1.5 dB.

On the relationship between long-time correlations and replica correlations in disordered systems

A proof is given of the equivalence between the replica correlations, (( sigma ialpha sigma ibeta )=q, which have been used by Edwards and Anderson (1975) to construct a theory of spin-glass equilibrium properties) and the correlations at long-time predicted by the dynamical equations.

Beam pattern control by the use of real (vice complex) shading coefficients

The usual method of beam control using complex shading coefficients or complex shading functions involves the algorithm of “phasing or time delaying to a plane,” and then performing a weighted sum. This paper considers the performance of beams formed using real shading coefficients (i.e., no time delay or phase shift being implemented). The algorithm used is a combination of “replica correlations” and the normal “aperture function” for sidelobe control. The array considered in the majority of the work consists of equally spaced finite‐sized acoustic elements or staves arranged on the surface of a right circular cylinder. It has been found, however, that the method works for several geometries including volumetric, line, and plane arrays. The functional dependence of beamwidth, bandwidth, sidelobe level, and sensitivity upon array size in wavelengths is presented. Advantages and disadvantages of the use of the method will be enumerated. Experimental proof of the mathematical model used in this study is contained in a companion paper.

The digital computation of discrete spectra using the fast Fourier transform

This paper is devoted to a discussion of discrete spectrum analysis which is important in applicational areas such as sonar and replica correlation. The discrete Fourier transform is shown to arise naturally as a consequence of finite impulsive sampling and the fast Fourier transform is introduced as the most efficient means of computing the discrete Fourier transform. These are described in terms of parameters pertinent to digital sonar signal processing, including resolution, dynamic range, and processing gain. Computational accuracy is investigated as a function of word lengths associated with the data, kernels, and intermediate transforms for both conditional and automatic array scaling. In real-time equipment, it is frequently necessary to employ some sort of automatic gain control and such a device is investigated here. Results are presented which enable specification of word length and automatic gain control requirements as a function of desired dynamic range, input signal-to-noise ratio, and mean-square error at the quantizer output.