Extended Sonar Sequences - CiteSeerX

Extended Sonar Sequences Oscar Moreno*’, Solomon

W. Golomb** and Carlos J. Corrada*

*Department of Mathematics and Computer Science, University of Puerto Rico, Rio Piedras, Puerto Rico 00931 **Communication Science Institute, Department of Electrical Engineering-Systems, University of Southern California, Los Angeles, CA 90089 wave) time slots, then we would call it an (n,m, I C ) extended sonar. The case when k = 0 is that of a sonar, and the case of n = 1 reduces to what has been studied previously under the name of rulers, which have other applications besides radar and sonar to synchronization, crystallography, etc (see [l]). In other words extended sonar sequences are a natural generalization of sonars and also of rulers. The main point of the present talk is to e v e several constructions of extended sonars.

Abstmct - Sonar as well as other related sequences were introduced by Golomb and Taylor in [ 2 ] . Following a similar approachiwe introduce the concept of an extended sonar sequence. It is similar to that of a sonar sequence but blank columns are permitted. Several constructions of extended sonars are given. Our constructions are very close to ordinary sonar sequences. However they provide good improvements to the list of the best known constructions for sonar sequences up to 100 symbols given in [3].

111. THECONSTRUCTIONS

I. INTRODUCTION Sonar sequences were introduced in [Z] to deal with the following problem: “You have an object which is moving towards (or away) from you, and you want to know effectively your distance and velocity of the object.” The solution to the problem comes from using the Doppler effect: when a wave hits a moving object its frequency changes in direct proportion to the velocity of the object. In other words you send a wave, wait until it returns and from the time it takes you know the distance, from the new frequency you know the velocity. On the other hand since the world is noisy you might send out a wave that does not return. Consequently you send out m waves with frequencies rangng from 1 to n. Waves are sent out at times ranging from 1 to m. Once the whole pattern of waves returns, from the change in frequency you determine the velocity of your object and from the time change the distance. On the other hand if not all the frequencies return there might be some ambigwties as to what is the whole pattern. Sonars are those patterns for which you send out exactly one wave at every time and also for which even if only two waves return you can reconstruct the whole pattern. This last point means that there is no ambiguity. The problem for sonars is, g v e n n frequencies, construct an n by m sonar sequence for m as large as possible.

11. EXTENDED S O N A R SEQUENCES The point of this talk is that for the sonar application an alternative to sending exactly one wave a t every time (the sonar case) is that of sending at most one wave, or in other words, choose not to send any wave in some time slots. This is done to achieve a larger number of waves sent for a given number of frequencies, while increasing the probability of receiving at least the two frequencies needed to reconstruct the whole sequence. Because of the similarity with the common sonar sequences, our sequences will be using the same equipment, in other words, it will be more cost effective than common sonar sequences. Again we would send m waves with frequencies ranging from 1 to n, and let us say that there are k blank (no

We will show how some of the constructions used in [3] to generate Costas and Sonar sequences, have a circular periodicity property that is the basis of our constructions of extended sonar sequences, namely the Extended Logarithmic Welch, the Extended Shift Sequence and the Extended Lempel-Golomb. This three constructions with k = 1, are very similar to ordinary sonars but for which the table of our constructions for n up to 100,outperforms the corresponding table of the best known construction for sonars given in [3]. For example for n = 46 and n = 75 it fills 7 more slots that common sonar sequences. Also we have tested the performance of this constructions comparing them with the best possible extended sonar sequences obtained doing an extensive search. The problem of generating extended sonar sequences exhaustively with the computer resides in the fact that the time of computation increases exponentially. The only practical way to obtain a sonar or extended sonar sequence for large lengths is therefore by generating it with some particular construction. At the moment we have done the extensive search for up to m = 10. The constructions obtained the best possible value 60% of the time. We will define the Circular extended sonar sequences and then we will prove that the Logarithmic Welch, the Shift Sequences and the Lempel-Golomb constructions Qve us a circular extended sonar sequences. We will show then that from any circular extended sonar sequences, we can obtain n extended sonar sequences. Then we will apply a series of transformations to the resulting extended sonar sequence to obtain a sequence with a reduced number of symbols obtaining the best known extended sonar sequences.

REFERENCES

[I] G. S . Bloom and S . W. Golomb, Applications of ntfmbered undirected graphs, Proceedings of the IEEE, 65:4 (1977), pp. 562570. [2] S. W . Golomb and H. Taylor, Two -Dimensional Synchronization Patterns f o r M m i m u m Ambiguity, IEEE Transactions on Information Theory, 28:4 (1982), pp. 600-604.

‘Work partially supported by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico Grant and the ARO grant for Cornell MSI.

R.A. Games and H. Taylor, Sonar Sequences from Gostas arrays and the best known sonar sequences with up to 100 symbols, IEEE (1992)

[3] 0. Moreno,

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