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In this paper, we consider requirement-driven selection of Costas arrays for the detection of closely spaced targets. We review existing order selection strategies in terms of the time–bandwidth product and treat selection of specific Costas arrays separably. Our computational analyses prioritize a clear central region in single-pulse discrete ambiguity functions (DAFs) and also address maximum frequency hops. Using finite field arithmetic, we significantly generalize results about 1s in the central square DAF regions for Welch and Lempel–Golomb arrays, and prove two general results about the nonexistence of such clear regions. We find that order 14 is very beneficial for target detection in which Costas arrays having an almost entirely clear $5 \times 5$ square central region in the single-pulse DAF exist and feature the corresponding time–bandwidth product of 196 within an example. We feature a first-principles system simulation with such an order-14 Costas array to obtain a 3-dB target detection improvement. Since radars commonly use burst waveforms, we also consider three different kinds of bursts and discuss their merits using ambiguity function analyses. We provide supporting computational database surveys on central rectangular $3 \times m$ regions.

In this paper we consider requirements-driven selection of Costas arrays for detection of closely-spaced targets. We review order selection strategies and treat selection of specific Costas arrays. Our analyses prioritize a clear central region in discrete ambiguity functions and address maximal frequency hops. We announce and use a new database of all known Costas arrays out to order 1030.

This correspondence presents the results of the enumeration of Costas arrays of order 27: all arrays found, except for one, are accounted for by the Golomb and Welch construction methods.

Costas arrays are permutation matrices that provide sequencing schemes for frequency hop in FSK waveforms. Such frequency-shift keying (FSK) waveforms can be designed to have nearly ideal ambiguity function properties in both the time and frequency directions: the Costas property permits at most one coincident tone in autocorrelations in both time and frequency. Costas arrays are found by number-theoretic generators and their extensions, and by exhaustive search methods. Two new extensions of number-theoretic methods are introduced here that find two new Costas arrays. All Costas arrays for orders 24, 25, and 26 are disclosed here, including previously unknown examples.

Two innovations in search methodology for Costas arrays are presented here: extensive exploitation of symmetries, and look-ahead row index exclusion tables. Together, they achieve a reduction of more than a factor of four in computational requirements over conventional search methods. We examined the benefits of these innovations on Costas arrays of higher orders, and particularly on a search over order 28.

A technique is presented which allows an FFT to be performed in place without bit reverse index reordering either before or after the computation. This is done by combining the reordering operation with the first iteration of a decimation in time FFT. Typical savings are 10 percent.

A general problem is addressed in which a real, continuous signal is to be sampled and its spectrum analyzed. It is shown that real sampling followed by an innovative complex first decimation filter stage minimizes the sample rate and produces minimum digital arithmetic or CCD hardware requirements. Only small differences distinguish broadband and narrow-band processing algorithms.

Costas arrays are permutation matrices that also provide a frequency indexing sequence that permits at most one coincident tone in cross-correlations of FSK waveforms. As such, they have obvious application as frequency indexing sequences in radar and communications when long codes with bounded autocorrelation are required or when Doppler is a significant portion of the transmitted bandwidth. All Costas arrays for orders less than 25 are known, with those for N=24 disclosed in the paper. Higher orders are found through number-theoretic generators and partial searches.

The reordering problem and its solutions were known early in the development of the FFT. Results by the author appeared in 1969 and 197017 in the IEEE Information theory in 197720 and ASSP symposia in 197821, and by C. S. Burrus in 1987 and 198822,23. This chapter makes the in-place Cooley-Tukey FFT available to the practitioner, and provides a reference that is easily cited.

This chapter is both an introduction to the book as a whole, and to the concepts underlying the finite Fourier transform. Our topics begin with the classical Fourier transform, which underlies most of our application concepts and is the conceptual basis for most applications of the FFT. We proceed by showing the relationship between Fourier series and the Fourier transform and briefly exploring that technology. We conclude by showing the relationship between the discrete Fourier transform (DFT) and the Fourier series, and exploring the nature of this transform. Thus we show the relationship between the Fourier transform and Fourier series with the DFT, the algorithm implemented by the FFT, and show some of the similarities, differences, and properties of each.

The FFT first emerged as an integrated, well-defined algorithm in Cooley and Tukey’s classical 1965 paper13. Early implementations were principally power of two algorithms, but, in 1969, Singleton published a paper exploiting Cooley and Tukey’s algorithm using a variable radix14. Bergland15,16 has shown advantages for radix 4 and radix 8 FFTs, which are commonly exploited in modern implementations.

A spectral widow weight is a function applied to the time domain data that changes the frequency domain representation in a controllable way. Nearly always, a spectral window function is a symmetrical bell-shaped curve that de-emphasizes the ends of the intervals of the time data. Its effect is observed by the DFT response to a sine wave of variable frequency and phase. Without a window weight function, the output of an FFT follows the familiar sine of x over x shape except when the frequency of the tone is exactly positioned at the center of a frequency bin. In a plot of amplitude in decibels versus frequency, this is the familiar Christmas-tree shape. With a spectral window, the main lobe is broadened slightly, while the sidelobes are lowered, particularly the near-in sidelobes.

The concept of bistatic ground moving target indicator BGMTI is presented here. Monostatic GMTI radars use separate modes for ground movers whose Doppler does and does not allow separation of the target return from ground clutter using frequency processing. Bistatic flight scenarios that minimize Doppler spreading in the vicinity of a designated target area are presented as candidate flight test plans. Flight scenarios supporting a sustained broadside spot mode and an oblique spot mode are presented.

The hydroacoustical bearings only target motion analysis (TMA) problem is that of determining target location, heading and velocity from target bearings. It must be solved using noisy data. Also, the hydroacoustical bearing trackers used for the measurements produce poor quality data because of the nature of the ocean medium. Since bearings only TMA is used on submarines for fire control, the performance of the TMA subsystems must be accurate, reliable, and computationally efficient. Historically, this problem has been most effectively solved using plotting boards and extended Kalman filters. Recent work has established the performance superiority of batch estimation techniques. The work presented here is a batch maximum likelihood estimation technique which solves the computational problems associated with batch estimation of the past. In addition, the reliability of the approach given here is appropriate for "hands-off" application in a real time subsystem. The problem is solved using state of the art computational and statistical estimation techniques and a coordinate system differing slightly from those used in the past. The formulation of the technique is a response to the hydroacoustical, computational, and mathematical aspects of the TMA problem, within the constraints of performance requirements and real time implementation considerations. Simulation results are presented which show the performance of the maximum likelihood TMA estimator to be superior to a Kalman TMA estimator.

In this paper, closed form solutions are derived for several configurations of the parametric receiving array with a phase demodulator receiver. The expression for parametric nonlinear acoustic phase modulation of a high‐frequency spherical wave from a point source by a low‐frequency plane wave is calculated. The result is in turn used to generate the solution for parametric receiving arrays with various pump transducers, including a truncated line source, a rectangular piston, and a circular piston. The solution of the parametric receiving array and a truncated line receiver is also found. A series of experiments was conducted with a 48‐ft parametric receiving array at a pump frequency of 90 kHz. These experiments duplicated several of the geometries used to obtain the theoretical results. Good agreement between theory and experiment was achieved. [Work supported by ONR]