Shamgar Gurevich

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

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, an explicit algorithm to construct the oscillator system is presented.

Delay-Doppler Channel Estimation in Almost Linear Complexity

A fundamental task in wireless communication is channel estimation: Compute the channel parameters a signal undergoes while traveling from a transmitter to a receiver. In the case of delay-Doppler channel, i.e., a signal undergoes only delay and Doppler shifts, a widely used method to compute the delay-Doppler parameters is the matched filter algorithm. It uses a pseudo-random sequence of length N, and, in case of non-trivial relative velocity between transmitter and receiver, its computational complexity is O(N2logN). In this paper we introduce a novel approach of designing sequences that allow faster channel estimation. Using group representation techniques we construct sequences, which enable us to introduce a new algorithm, called the flag method, that significantly improves the matched filter algorithm. The flag method finds m delay-Doppler parameters in O(mNlogN) operations. We discuss applications of the flag method to GPS, and radar systems.

The finite harmonic oscillator and its associated sequences

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 explained.

Group Representation Design of Digital Signals and Sequences

In this survey a novel system, called the oscillator system, consisting of order of p3functions (signals) on the finite field

Small Representations of Finite Classical Groups

\mathbb{F}_{p},

Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos

is described and studied. The new functions are proved to satisfy good auto-correlation, 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, an explicit algorithm to construct the oscillator system is presented.

Notes on Canonical Quantization of Symplectic Vector Spaces over Finite Fields

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the “small” representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the “eta correspondence”, to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.

Performance estimates of the pseudo-random method for radar detection

In these notes we discuss the self-reducibility property of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for a generic linear symplectomorphism A of the torus \({\mathbb{T} = \mathbb{R}}^{2N}/{\mathbb{Z}}^{2N}.\)

The Incidence and Cross methods for efficient radar detection

In these notes we construct a quantization functor, associating a Hilbert space \(\mathcal{H}(V )\) to a finite dimensional symplectic vector space V over a finite field \({\mathbb{F}}_{q}\). As a result, we obtain a canonical model for the Weil representation of the symplectic group SpV . The main technical result is a proof of a stronger form of the Stone–von Neumann theorem for the Heisenberg group over \({\mathbb{F}}_{q}\). Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical Hilbert space attached to a coadjoint orbit of a general unipotent group over \({\mathbb{F}}_{q}\).

The geometric Weil representation

A performance of the pseudo-random method for the radar detection is analyzed. The radar sends a pseudo-random sequence of length N, and receives echo from r targets. We assume the natural assumptions of uniformity on the channel and of the square root cancellation on the noise. Then for r ≤ N1-δ, where δ > 0, the following holds: (i) the probability of detection goes to one, and (ii) the expected number of false targets goes to zero, as N goes to infinity.