The triple autocorrelation of a sequence is shown to be a Lempel Costas array iff the sequence is maximal, assuming correlation is appropriately defined and the alphabet of the sequence is of prime size. A method that allows the construction of thumbtack autocorrelation sequences of arbitrary alphabet size is subsequently proposed.

In order to extrapolate a signal, Empirical Mode Decomposition is used to decompose it into simpler components. Each component is individually extrapolated linearly, and the final extrapolation value is produced as the sum of these individual values. This technique is applied on financial signals, with a view towards capturing the sign of the increment of the signal instead of the exact future value, and the results are compared to cubic spline extrapolation.

Families of Costas arrays with low pairwise cross-correlation are sought. The two families of all exponential Welch arrays and all Golomb arrays generated in a certain finite field are specifically studied, and the maximal cross-correlation is determined by exhaustive search. Mathematically rigorous explanations for some of the observed results are presented, a surprising link between Welch and Golomb arrays is revealed, and what remains to be proved is stated precisely. The results suggest that the families with uniformly low cross-correlation correspond to finite fields whose size is a safe prime power.

A Costas array is an arrangement of N dots on an N -by- N grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems, communication systems, cryptography, and other areas. We present a standardization of notation and language which can be used to discuss Costas array generation techniques and array manipulations. Using this standardization we can concisely and clearly state various theorems about Costas arrays, including several new theorems about the symmetries of Costas arrays. We also define labels for each array (generated, emergent, and sporadic), which describe whether the array is generated using a known technique, generated using a semiempirical variation of a known technique, or of unexplained origin. A new method for obtaining emergent Costas arrays, the DRT expansion, is also given here for the first time. A MATLAB Costas array toolbox has also been developed which implements the proposed standardization. The toolbox contains a comprehensive set of functions covering Costas array generation, manipulation and classification.

We present the results of the enumeration of Costas arrays of order 28: all arrays found are accounted for by the Golomb and Welch construction methods, making 28 the first order (larger than 5) for which no sporadic Costas arrays exist. The enumeration was performed on several computer clusters and required the equivalent of 70 years of single CPU time.

Given that frequency-hopping filters cannot easily implement big frequency hops instantaneously, those Costas permutations are determined in which the maximal frequency hop prescribed is as small as possible, as well as those that do contain the maximal hop possible, and are, consequently, less suitable for applications. It turns out that exponential Welch permutations not only lead in general to the smallest hops, but are also relatively easy to study, as a closed formula exists for the maximal hop. Through extensive collection of data for logarithmic Welch and Golomb permutations, on the other hand, it is found that: a) these two families behave (almost) identically; and that b) their maximal hops do not get as small as in exponential Welch permutations.

A structural constraint (symmetry property) of Golomb Costas arrays constructed in finite fields of odd size is presented, analogous to the anti-reflective symmetry of Welch Costas arrays.

In this paper, we show that in order to ascertain whether a permutation has the Costas property, only a restricted subset among the totality of pairs of entries in the same row of the difference triangle needs to be checked, and we explicitly describe such a subset. This represents a further refinement on the definition of a Costas permutation. This observation can be used to speed up algorithms that exhaustively search for Costas permutations. Asymptotically, the savings approaches 43% for large orders when compared with the previous standard efficient method.

An analytically solvable mathematical model is presented for an IEEE 802.11 network, where two terminals, both visible to an access point but not to each other, contend for the channel. This situation, known as the hidden node topology, has been identified as a key reason for the degradation of the protocol's performance. It is shown that the renewal theory, the analytical tool used traditionally, is not suitable for this topology and a discrete time Markov chain is introduced for the modelling of the channel contention. The model permits the accurate computation of several key performance metrics and is in agreement with simulation results over a variety of scenarios.

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.