Linear systems theory and its relationship to ocean acoustics

The article of record as published may be found at https://doi: 10.1121/1.2028979The purpose of this talk is to demonstrate the consistency and relationships between linear systems theory and the physics of propagation of small‐amplitude acoustic signals in fluid media. Using the principles of linear, time‐variant, space‐variant filter theory and time‐domain and spatial‐domain Fourier transforms, derivations of the solutions of the linear, three‐dimensional, inhomogeneous wave equation for (1) an unbounded isospeed fluid medium, (2) and unbounded fluid medium with speed of sound an arbitrary function of depth, and (3) a full‐wave, pulse‐propagation model for three‐dimensional wave propagation in a Pekeris waveguide are presented. Characterizing a fluid medium as a linear filter is valid since this involves trying to solve the linear wave equation. Computer simulation results are presented.Work supported by ONR, Code1 1250 and the Naval Postgraduate Schoo

Comments on the relationships between linear systems theory and the free‐space solution of the inhomogeneous linear wave equation

The article of record as published may be found at https://doi.org/10.1121/1.400180Using the principles of linear, time‐variant, space‐variant, filter theory, and time‐domain and spatial‐domain Fourier transforms, derivations of the solutions of the linear, three‐dimensional, inhomogeneous wave equation for (1) an unbounded isospeed fluid medium and (2) an unbounded fluid medium with speed of sound an arbitrary function of a single variable are presented.This work was sponsored by ONR and the Naval Postgraduate Schoo

Recursive ray acoustics for three‐dimensional sound‐speed profiles

The article of record as published may be found at https://doi.org/10.1121/1.403301A recursive ray acoustics (RRA) algorithm for three‐dimensional sound‐speed profiles is presented. The RRA algorithm is simple, fast, and accurate and uses arc length as the independent variable. It can be used to compute the position, angles of propagation, travel time, and path length along a ray path. The accuracy of the RRA algorithm was tested by comparing its results with those obtained from a ray acoustics algorithm that requires the solution of a system of four, first‐order, ordinary differential equations (ODEs). The ODE algorithm, which uses horizontal range as the independent variable, can only handle one‐dimensional, depth‐dependent, sound‐speed profiles (SSPs). Therefore, five standard, depth‐dependent, SSPs were used to test the two algorithms. The results from the ODE algorithm were treated as the benchmark, with respect to accuracy, for comparison purposes. The RRA algorithm proved to be very accurate for the test cases tried. After the accuracy of the RRA algorithm was validated by using one‐dimensional SSPs, its three‐dimensional sound‐speed capability was also tested.Work supported by DARPA and the Naval Postgraduate School Direct Funded Research Program, sponsored by ONR, Code 1125 OAWork supported by DARPA and the Naval Postgraduate School Direct Funded Research Program, sponsored by ONR, Code 1125 O

Linear time-invariant space-variant filters and the parabolic equation approximation

The article of record as published may be found at https://doi.org/10.1016/0165-1684(85)90005-2Wave propagation in a random, inhomogeneous ocean is treated as transmission through a linear, time-invariant, space-variant, random communication channel. Using the parabolic equation approximation of the Helmholtz wave equation, a random transfer function of the ocean volume is derived. The ocean volume is characterized by a three-dimensional random index of refraction which is decomposed into deterministic and random components. Two additional calculations are performed using the transfer function. The first involves the derivation of the equations for the random, output electrical signals at each element in a receive planar array of complex weighted point sources in terms of the frequency spectrum of the transmitted electrical signal, the transmit and receive arrays, and the transfer function of the ocean medium. The second involves the derivation of the coherence function.Support for this research was obtained from the Naval Postgraduate School Foundation Research Program

LSVOCN: A pulse‐propagation model for a linear, space‐variant ocean

The article of record as published may be found at https://doi.org/10.1121/1.2027941A full‐wave, pulse‐propagation model for three‐dimensional wave propagation in a Pekeris waveguide based on linear systems theory is presented. The randomly rough ocean surface and bottom are accounted for via coherent (average) reflection coefficients. Attenuation due to absorption in all three fluid media is included. Transmit and receive planar arrays with beam steering can be simulated and, as a result, vertical arrays and a single, omnidirectional point source are automatically included. A built‐in signal generator can simulate arbitrary amplitude and angle‐modulated carriers. Outputs from this model include plots of the magnitude and phase of the ocean surface and bottom reflection coefficients, the complex acoustic field across the receive array. Because of its highly modular structure, the model can also be used to generate pulse‐propagation solutions using any time‐harmonic solution such as normal mode theory or the parabolic equation method. Preliminary computer simulation results are presented.Work supported by ONR, Code 11250A, and the Naval Postgraduate school.Work supported by ONR, Code 11250A, and the Naval Postgraduate school

Three necessary conditions for the validity of the Fresnel phase approximation for the near-field beam pattern of an aperture

The article of record as published may be found at https://doi.org/10.1109/48.211494A simple, straightforward derivation of three necessary conditions that define the region of validity for the near-field directivity function (beam pattern) of an aperture (array) is presented. The derivation of all three criteria is based on determining what conditions must be satisfied in order to obtain a valid Fresnel approximation of the time-independent free-space Green’s function. Comparisons with other near-field conditions in the literature are made

Localization of multiple broadband targets via frequency domain adaptive beamforming for planar arrays

The article of record as published may be found at https://doi.org/10.1121/1.397454Computer simulation studies of a frequency domain adaptive beamforming algorithm for planar arrays are presented. The algorithm, which can localize multiple broadband targets, is a modified complex least‐mean‐square (LMS) adaptive algorithm, and can process an arbitrary number of harmonics. The algorithm provides estimates of both the depression and bearing angles of incident plane‐wave fields. Computer simulation results are presented comparing the average depression and bearing angle estimation errors as a function of the input signal‐to‐noise ratio (SNR) at a single element in the array, sampling rate, and harmonic number. The ‘‘full angular coverage’’ capability of the algorithm was also tested

Maximation of the signal-to-interference ratio for a doubly spread target: Problems in nonlinear programming

The article of record as published may be found at https://doi.org/10.1016/0165-1684(83)90094-4Three optimization problems concerning the maximization of the signal-to-interference ratio for a doubly spread target via signal design are expressed in terms of equivalent nonlinear programming problems defined on a real space by restricting the transmit and processing waveforms to be complex weighted, uniformly spaced pulse trains. Each subpulse can be different in shape and occupy the entire interpulse spacing interval. The approach taken is analogous to the Rayleigh-Ritz technique. The first two optimization problems involve maximization with respect to the complex weights, The third problem involves maximization with respect to the subpulse parameters (e.g., frequency deviation, swept bandwidth, etc.) and allows one to find optimum frequency hop codes. One need not develop algorithms to solve these problems, but rather, one can simply use standard computer programs or methods which are available for solving nonlinear programming problems.The work described in this paper was supported by NAVSEA Undersea Weapons Guidance and Control Block, Code NSEA 63R-14

Three-dimensional ray acoustics: new expressions for the amplitude, eikonal, and phase functions

The article of record as published may be found at https://doi.org/10.1109/48.35990New three-dimensional ray-acoustics expressions for the amplitude, eikonal, and phase along a ray path are derived. These expressions clearly indicate the numerical calculations that must be performed in order to evaluate these functions. The ocean medium is characterized by a three-dimensional random index of refraction which is decomposed into deterministic random components

Broadband and narrow‐band signal‐to‐interference ratio expressions for a doubly spread target

The article of record as published may be found at https://doi.org/10.1121/1.388260This paper is based on Chaps 3. and 4 of L.J. Ziomek's Ph.D. dissertation "A Scattering Function Approach to Underwater Acoustic Detection and Signal Design, "The Pennsylvania State University (1981).Signal‐to‐interference ratio (SIR) expressions for a doubly spread target are derived for both broadband and narrow‐band transmit signals. For broadband signals, the SIR is dependent upon target and reverberation two‐frequency correlation functions and upon the transmit and processing waveforms. For wide‐sense stationary uncorrelated spreading (WSSUS) communication channels (which implies narrow‐band transmissions), the SIR is dependent upon target and reverberation scattering functions and the cross‐ambiguity function of the transmit and processing waveforms. Volume reverberation and target two‐frequency correlation functions and scattering functions are derived. Volume reverberation is modeled as the spatially uncorrelated scattered field from randomly distributed point scatterers in deterministic plus random translational motion. A single scattering approximation is used and frequency‐dependent directivity functions and attenuation due to absorption are included. A probability density function of random Doppler shift due to the random motion of the scatterers is also derived. Computer plots of the density function are presented as a function of the standard deviation of the random motion. The target is modeled as a linear array of discrete highlights in deterministic translational motion. Example scattering function calculations are presented. The volume reverberation scattering function predicts Doppler spreading as a function of both beam steering angle and random motion of the scatterers. The target scattering function also predicts a spread in Doppler values. Both scattering functions predict time spread and/or contraction as a function of Doppler spread.The work described in this paper was supportedby NAVSEA Undersea Weapons Guidance and Control Block, Code NSEA 63R- 14