Cross-correlation properties of algebraically constructed Costasarrays

The problem of determining the cross-correlation properties of signals based on algebraically constructed Costas arrays is addressed by examining the discrete cross-correlation of the algebraically constructed Costas arrays for a given construction and dimension. Finding two arrays that minimally correlate implies that the signals based on these arrays also minimally correlate. The properties of finite fields are reviewed, and the major algebraic constructions for Costas arrays are presented, i.e. the Welch construction and the Golomb construction. The discrete cross-correlation properties of the Costas arrays are derived for arrays of the same dimension derived from the same construction. The use of Costas arrays in the signal design problem is discussed, and examples are given to show the cross-correlation of the signals based on the algebraically constructed arrays     

Volume Properties for the Wideband Ambiguity Function

It is shown that the volume under the wideband ambiguity function is close to the square of the signal energy. The ambiguity volume is asymptotically conserved as the signal approaches the narrowband case. The narrowband ambiguity volume is a lower bound for the volume of the wideband ambiguity function.     

Time-frequency hop codes based upon extended quadratic congruences

Time-frequency hop codes are developed that can be used for coherent multiuser echolocation and asynchronous spread spectrum communication systems. They represent a compromise between Costas codes, which have nearly ideal autoambiguity but not so good cross-ambiguity properties, and linear congruential codes, which have nearly ideal cross-ambiguity but unattractive autoambiguity properties. Extended quadratic congruential code words are shown to have reasonably good autoambiguity and cross-ambiguity properties across the whole class of code sets considered. A uniform upper bound is placed on the entire cross-ambiguity function surface, and bounds are placed on the position and amplitude of spurious peaks in the autoambiguity function. These bounds depend on the time/bandwidth product and code length exclusively and lead naturally to a discussion of the design tradeoffs for these two parameters. Examples of typical autoambiguity and cross-ambiguity functions are given to illustrate the performance of the new codes     

The hit array: an analysis formalism for multiple access frequencyhop coding

A formalism is presented for the analysis of general frequency hop waveforms, such as those suitable for use in coherent active radar and sonar echolocation systems as well as multiple-access spread-spectrum communications. This formalism is based on the concept of coincidence, or `hit', between two frequency hopping patterns. The collection of all possible hits, together with their locations, is recorded in time-frequency space, which produces the high array associated with the two patterns considered. If the code length is sufficiently small with respect to the time-bandwidth product chosen, the hit array can be viewed as a digital representation of the corresponding ambiguity function. Salient properties of the hit array formalism are derived, including simple relationships between hit arrays resulting from basic symmetry-preserving transformations. These properties make it possible to predict the performance of a given set of frequency hop waveforms directly from the associated set of frequency hopping patterns     

Time-Frequency HOP Signals Part I: Coding Based upon the Theory of Linear Congruences

Time-frequency hop codes are developed based upon the theory of linear congruences. These codes can be used for multiuser radar and asynchronous spread spectrum communications systems. A uniform upper bound is placed on the cross-correlation function between any two elements of the code set. The upper bound is minimized by choice of time-bandwidth product and is shown to diminish as 2/N, where N is the number of elements in the code set. The size and position of spurious peaks in the autocorrelation functions are discussed. The results are extended to narrowband ambiguity functions.     

Synchronization improvements using traceability in spread spectrumsignal design

Synchronization improvements using traceability information found in the narrowband auto-ambiguity function (NB AAF) are demonstrated. This new auto-ambiguity function property, traceability, is the key to defining a signal design approach for solving the synchronization problems in spread spectrum system such as communication systems, radar networks, and sonar systems. This work defines the property and introduces a new tracing synchronization scheme that significantly reduces initial synchronization time and the number of false synchronization events     

Time-Frequency HOP Signals Part II: Coding Based upon Quadratic Congruences

High-efficiency multicomponent signals for maximization of signalto-noise ratio are investigated. Maximization of signal-to-noise ratio in colored noise requires control of volume distribution of the signal ambiguity function and transmission of unity efficiency signals. Signal efficiency is defined as the ratio of average power to the peak power. It is concluded that the signals must be frequency hop pulse trains. Quadratic congruences are chosen to place the components in time-frequency space. The number-theoretic properties of these signals provide bounds on the position and amplitude of the various peaks of the signal ambiguity function. The tradeoffs are shown between volume removal, number of component signals, and the time-bandwidth product.     

Frequency hop multiple access codes based upon the theory of cubiccongruences

The need for families of frequency-hop codes which have mutually small auto-ambiguity and cross-ambiguity functions is discussed. Current coding methods are reviewed. A new family of frequency-hop codes based upon the number-theoretic concept of cubic congruences is introduced. It is shown that for about 50% of the prime numbers, families of full codes exist which have at most two coincidences for any time-frequency shift in their auto-ambiguity functions and at most three coincidences in the set of mutual cross-ambiguity functions     

On cross-ambiguity properties of Welch-Costas arrays

We discuss cross-ambiguity properties of a specific family of Costas arrays called Welch-Costas (W-C) arrays. These properties are of interest in multiuser radar and sonar system, especially since Costas arrays are known to possess ideal auto-ambiguity functions. The theory of W-C arrays is reviewed. It is then proved that only pairs of W-C arrays can have at most two hits in their cross-ambiguity function (best possible case). The maximum number of hits in the cross-ambiguity functions of a family of W-C arrays is shown to be a function of the number of W-C arrays in the family. The upper bound on the number of hits in the cross ambiguity functions for a family of W-C arrays is also derived. Specific examples of how reducing the number of W-C arrays improves the cross-ambiguity properties are given for various types of prime numbers     

Graphical Derivations of Radar, Sonar, and Communication Signals

The designer of a communication system often has knowledge concerning the changes in distance between transmitter and receiver as a function of time. This information can be exploited to reduce multipath interference via proper signal design. A radar or sonar may also have good a priori information about possible target trajectories. Such knowledge can again be used to reduce the receiver's response to clutter (MTI), to enhance signal-to-noise ratio, or to simplify receiver design. There are also situations in which prior knowledge about trajectories is lacking. The system should then utilize a single-filter pair which is insensitive to the effects induced by relative motion between transmitter, receiver, and reflectors. For waveforms with large time-bandwidth products, such as long pulse trains, it is possible to graphically derive signal formats for both situations (trajectory known and unknown). Although the exact form of the signal is sometimes not specified by the graphical procedure, the problem in such cases is reduced to one which has already been solved, i. e., the generation of an impulse equivalent code.