# The Finite Harmonic Oscillator and Its Applications to Sequences, Communication, and Radar

@article{Gurevich2008TheFH, title={The Finite Harmonic Oscillator and Its Applications to Sequences, Communication, and Radar}, author={Shamgar Gurevich and Ronny Hadani and Nir A. Sochen}, journal={IEEE Transactions on Information Theory}, year={2008}, volume={54}, pages={4239-4253} }

A novel system, called the oscillator system, consisting of order of p3 functions (signals) on the finite field Fp, with p an odd prime, is described and studied. The new functions are proved to satisfy good autocorrelation, cross-correlation, and low peak-to- average power ratio properties. Moreover, the oscillator system is closed under the operation of discrete Fourier transform. Applications of the oscillator system for discrete radar and digital communication theory are explained. Finally…

## Figures from this paper

## 41 Citations

### Group Representation Design of Digital Signals and Sequences

- Computer ScienceSETA
- 2008

The oscillator system, consisting of order of p3functions (signals) on the finite field F, is described and studied and the new functions are proved to satisfy good auto-correlation, cross-cor correlation and low peak-to-average power ratio properties.

### A BRIEF SURVEY OF THE FINITE HARMONIC OSCILLATOR AND ITS APPLICATIONS

- Mathematics
- 2015

A finite oscillator system was introduced by Gurevich, Hadani and Sochen. This is a survey of how the system is constructed using Weil representation on the group SL(2,Fp) and its applications on…

### The finite harmonic oscillator and its associated sequences

- Computer ScienceProceedings of the National Academy of Sciences
- 2008

A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are…

### Computing the Matched Filter in Linear Time

- Computer ScienceArXiv
- 2011

A novel approach of designing new waveforms that allow faster matched filter algorithm, called the flag algorithm, and the cross algorithm are introduced, which solve the TFS problem in O(p\cdot log (p) operations).

### The Incidence and Cross methods for efficient radar detection

- Computer Science2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2013

The Incidence and Cross methods for radar detection are introduced and both methods employ chirp sequences, which are commonly used by radar systems, and hence are attractive for real world applications.

### Performance estimates of the pseudo-random method for radar detection

- Computer Science, Mathematics2014 IEEE International Symposium on Information Theory
- 2014

A performance of the pseudo-random method for the radar detection is analyzed, and the natural assumptions of uniformity on the channel and of the square root cancellation on the noise are assumed.

### Almost Linear Complexity Methods for Delay-Doppler Channel Estimation

- Computer ScienceArXiv
- 2013

The flag method was introduced to provide a faster algorithm for delay-Doppler channel estimation and its complexity is O(rN logN) for channels of sparsity r, and the incidence and cross methods for channel estimation are introduced.

### On the Construction of Finite Oscillator Dictionary

- MathematicsArXiv
- 2010

Close formulae of the finite split oscillator dictionary are revisited by a simple proof, the structure of nonsplit tori of the group SL(2, 𝔽 p ) is studied, and an explicit algorithm for computing the finite nonsplIt oscillator Dictionary is described.

### Unit time-phase signal sets with the explicit maximum cross ambiguity amplitudes

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2014

Gauss sums are used to determine the explicit maximum cross ambiguity amplitudes of some infinite series of unit time-phase signal sets which were constructed by Ding et al. (2013).

### New Sequences Design From Weil Representation With Low Two-Dimensional Correlation in Both Time and Phase Shifts

- Computer ScienceIEEE Transactions on Information Theory
- 2011

A new elementary expression of the construction first proposed by Gurevich, Hadani, and Sochen is given, which avoids the explicit use of the Weil representation, and a proof is given through finding a newexpression of the sequences in the finite harmonic oscillator system.

## References

SHOWING 1-10 OF 33 REFERENCES

### Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar

- Computer Science
- 2005

This paper presents a meta-analyses of correlation in cyclic Hadamard sequences and its applications to radar, sonar, and synchronization, and describes the properties of correlation as well as applications to Boolean functions.

### Synthesis of band-limited orthogonal signals for multichannel data transmission

- Computer Science
- 1966

This paper presents a principle of orthogonal multiplexing for transmitting a number of data messages simultaneously through a linear band-limited transmission medium at a maximum data rate without…

### CDMA: Principles of Spread Spectrum Communication

- Computer Science
- 1995

Generating Pseudorandom Signals (Pseudonoise) from PseudOrandom Sequences by Modulation and Demodulation of Spread Spectrum Signals in Multipath and Multiple Access Interference.

### On the existence and construction of good codes with low peak-to-average power ratios

- Computer Science2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)
- 2000

A lower bound on PAPR is proved in terms of R and d and it is shown that there exist asymptotically good codes whose PAPr is at most 8logn.

### The classical groups : their invariants and representations

- Mathematics
- 1940

In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from…

### The Classical Groups

- Mathematics
- 1939

: We consider the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan. These generalize the doubling method of Piatetski-Shapiro and Rallis and represent the standard L-function for…

### Probability and information theory II

- Mathematics
- 1969

Uniform flows in cascade graphs.- An approach to the theory of pressure broadening of spectral lines.- Moments of point processes.- Additive and non-additive entropies of finite measurable…

### Gauss Sums, Kloosterman Sums, And Monodromy Groups

- Mathematics
- 1987

The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's…

### Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation

- Mathematics
- 2006

We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility…