Albert H. Nuttall

Estimation of the two‐dimensional spectrum of the space‐time noise field for a sparse line array

For a line array immersed in a stationary homogeneous noise field composed of plane waves of differing speeds and angles of arrival, standard beamforming is not useful for estimating the noise spectrum directionality. However, the two‐dimensional spectrum of the received noise field enables resolution in temporal‐frequency and in spatial‐frequency and can thereby indicate the temporal‐frequency content and the apparent speed along the line array of plane‐wave arrivals. The interrelationships and attributes of the space‐time correlation, the cross‐spectrum, and the two‐dimensional spectrum are presented and interpreted via simple examples. A technique for estimation of the two‐dimensional spectrum for a sparse line array and examples of its use and interpretation are given.

Phase Distortion of a Pulse Caused by Bottom Reflection

The phase distortion of a plane‐wave pulse owing to bottom reflection at incident angles more grazing than the critical angle is considered. Two general methods of investigation, using the time and frequency domains, are given; they employ, respectively, the Hilbert transform of the incident pulse and the complex envelope signal. Applications to two typical SONAR pulses—a carrier frequency amplitude‐modulated by a rectangular pulse and a Gaussian pulse—are given, along with graphs of the results. It is concluded that negligible distortion occurs, there being only a phase shift of the carrier, when the Q of the waveform is moderately larger than unity, the exact value depending on the smoothness of the transmitted pulse.

A brief summary of a generalized framework for power spectral estimation

Abstract This correspondence presents a brief summary of a generalized framework for power spectral estimation and shows how three previous estimation methods fit into this framework as special cases. Further, this correspondence clarifies some recent comments made on spectral estimation; in particular specific references are given that strongly support the use of overlapped weighted segment averaging for spectral estimation.

Detection performance characteristics for a system with quantizers, OR‐ing, and accumulator

The false alarm and detection probabilities of an N‐channel system subject to L+1 level quantization, OR‐ing, and accumulation of M time samples, are evaluated exactly, with no Gaussian assumptions, for arbitrary values of N, L, M, and for a quantizer with arbitrary breakpoint locations. The channel noises are independent but can have arbitrary statistics. The signal occupies one unknown (possibly changing) channel, if present. Two FFTs suffice to sweep out a complete detection probability versus threshold curve. The optimum placement of quantizer breakpoints, for a fixed total number of levels, L+1, is a subtle one and shown to depend on N, M, and the desired level of performance; a simple rule‐of‐thumb is presented which yields near‐optimum capability over the useful range of the operating characteristics.

Detection performance of normalizer for multiple signals subject to partially correlated fading with chi‐squared statistics

The false alarm and detection probabilities for a multipulse signal subject to partially correlated fading, in the presence of Gaussian noise of unknown level, are derived in closed form. The number K of signal pulses, as well as the number L of noise‐only pulses used to estimate the noise background power level, are arbitrary. The power fading is characterized by a chi‐squared distribution with 2m degrees of freedom and a normalized set of covariance coefficients {ρkt}, all of which can be selected arbitrarily, in order to match an experimental realization or an actual measured situation. The performance capability of this processor depends additionally on the received signal‐to‐noise ratio. This study covers the case of a nonconstant threshold; comparisons of this normalizer with earlier results (for L = ∞) enable a quantitative evaluation of the losses incurred by lack of knowledge of the noise level. The important capability of constant false alarm rate is achieved by this normalizer.

Spectral analysis via lag reshaping of the correlation estimate

Abstract : The fundamental performance of the generalized spectral analysis technique employing quadratic frequency-smoothing of Fourier-transformed, overlapped, weighted data segments was thoroughly investigated analytically and reported. Possibility pointed out there was that of doing lag-reshaping of the first-stage correlation estimate, and thereby realizing effective spectral window with low sidelobes and good decay rates, without the need for overlap or any temporal weighting at all. However, no programs or simulation results. This situation is rectified by presenting programs for achieving maximally stable low-sidelobe spectral estimates via the lag-reshaping procedure and then employ these routines to exhibit some simulation results for data with tones and noise. In this report, attention is to the case of abutting rectangular temporal weightings with no overlap; this procedure, with appropriate lag-reshaping. Keywords: Effective window; FFT processing; Lag-reshaping; Lag weighting; Minimum execution time; Minimum variance.

Signal‐Waveform Distortion Caused by Reflection off Lossy‐Layered Bottoms

The acoustic theory for the reflection of a single‐frequency plane wave from a fluid layer lying between two semi‐infinite fluid media, where the layer and the bottom medium are subject to attenuation, is reviewed. Applying the experimental fact that the attenuation is proportional to the first power of frequency to this theoretical model of the ocean bottom, we show that the effect on impingent plane waves may be treated as a parallel linear‐filtering operation with a component transfer function of the basic form Hu2009(f)u2009=u2009exp[−2πb|f|+ieu2009sgnu2009(f)]. The parameters b and e depend on the media physical constants, the angle of incidence, and the particular reflected component under consideration. Here, f is frequency in cycles per second, 2πb is attenuation in nepers per cycles per second, e is phase shift in radians, and sgn (f) = 1 for f>0, −1 for f <0. For this basic type of transfer function, the component reflected pulse is first derived generally in terms of the incident pulse, the attenuation, and the ph...

Spectrum of a Signal Reflected from a Time-Varying Random Surface

Abstract : The power density spectrum of the received acoustic waveform after reflection off the time-varying random sea surface is evaluated at an arbitrary observation point in the farfield. For a monochromatic transmitted signal and a narrow-band Gaussian surface-height variation, the received acoustic spectrum is shown to consist of an impulse at the transmitted acoustic frequency plus sidelobes centered at frequencies separated from the transmitted frequency by multiples of the surface center frequency. The powers in the coherent component and scattered sideband components of the received pressure waveform are evaluated in terms of the surface roughness and spatial-temporal correlation function of the surface. For the special case of elliptical contours of iso- correlation at zero time delay, the sideband powers and scattering strengths are evaluated in terms of two fundamental parameters that include the geometry of the experiment, the incident acoustic frequency, the root mean square (rms) surface height, and the surface correlation distances. The rms bandwidths of the sideband scatter components are evaluated for small surface roughness and shown to be approximately proportional to the square root of the sideband number. Numerous examples of sideband scattering strengths for a variety of spatial correlation functions, including exponential and Gaussian decay as special cases, are given.

Spectrum of a Reflected Signal from a Time‐Varying Stochastic Surface

The spectrum of a single‐frequency wave reflected from a time‐varying stochastic homogeneous surface is determined as a function of the statistical parameters of the surface. It is assumed that the surface is stationary in the wide sense in both space and time. For the special case of surface wave heights possessing a narrow‐band spectrum at a point, joint Gaussian distribution of heights, and an elliptical spatial field, the integral expression obtained for the reflected acoustic spectrum at the observation point is derived. It is shown that the acoustic spectrum consists of a series of narrow‐band spectral lobes, shifted from the incident acoustic frequency by a multiple of the center frequency of the surface variation. The power in each component is evaluated in terms of two parameters: surface roughness and correlation distance of surface wave heights. The computations are made for a Gaussian‐modulated sinusoidal and an exponentially modulated sinusoidal autocorrelation function. Curves summarizing th...

Signal Distortion Caused by Bottom Reflection

The acoustic theory for the reflection of a single‐frequency plane wave from a fluid layer lying between two semiinfinite fluid media, where the layer and the bottom medium are subject to attenuation, is reviewed. Applying the experimental fact that the attenuation is proportional to the first power of frequency to this theoretical model, it is shown that the effect on impingent plane waves may be treated as a parallel linear filtering operation with a typical transfer function of the basic form H(f)=exp[−2πb|f|+iesgn(f)]. Here f is the frequency in cycles per second, 2πb is the attenuation in nepers/cycles per second, e is the phase shift in radians, and sgnu2009(f)u2009=u2009{1,u2009f>0−1u2009f<0}. For this basic type of transfer function, the reflected pulse is first derived generally in terms of incident pulse, attenuation, and phase shift. Reflected pulse shapes for some specific incident pulses are then derived and numerically evaluated. The correlation loss for a random function with flat finite bandwidth is also obta...