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Ronny Hadani

Publications and source records attributed to Ronny Hadani.

At least 37 records · Page 2Linked to original sources
arXiv2018

OTFS: A New Generation of Modulation Addressing the Challenges of 5G

In this paper, we introduce a new 2D modulation scheme referred to as OTFS (Orthogonal Time Frequency & Space) that multiplexes information QAM symbols over new class of carrier waveforms that correspond to localized pulses in a signal representation called the delay-Doppler representation. OTFS constitutes a far reaching generalization of conventional time and frequency modulations such as TDM and FDM and, from a broader perspective, it establishes a conceptual link between Radar and communication. The OTFS waveforms couple with the wireless channel in a way that directly captures the underlying physics, yielding a high-resolution delay-Doppler Radar image of the constituent reflectors. As a result, the time-frequency selective channel is converted into an invariant, separable and orthogonal interaction, where all received QAM symbols experience the same localized impairment and all the delay-Doppler diversity branches are coherently combined. The high resolution delay-Doppler separation of the reflectors enables OTFS to approach channel capacity with optimal performance-complexity tradeoff through linear scaling of spectral efficiency with the MIMO order and robustness to Doppler and multipath channel conditions. OTFS is an enabler for realizing the full promise of MUMIMO gains even in challenging 5G deployment settings where adaptation is unrealistic.

cs.IT↗
arXiv2016

OTFS - Orthogonal Time Frequency Space

In this paper we introduce a new 2D modulation technique called OTFS (Orthogonal Time Frequency & Space) that transforms information carried in the Delay-Doppler coordinate system to the familiar time-frequency domain utilized by traditional modulation schemes such as OFDM, CDMA and TDMA. OTFS converts the fading, time-varying wireless channel into a non-fading, time-independent interaction revealing the underlying geometry of the wireless channel. In this new formulation, all QAM symbols experience the same channel and all Delay-Doppler diversity branches of the channel are coherently combined. Reference signal multiplexing is done in the time-independent Delay-Doppler domain, achieving high density pilot packing, which is a crucial requirement for Massive MIMO. Regardless of the Doppler scenario, OTFS enables approaching channel capacity through linear scaling of throughput with MIMO order, thus realizing the full promise of Massive MIMO throughput gains even in challenging 5G deployment settings.

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arXiv2012

Delay-Doppler Channel Estimation with 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, a widely used method 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(N^{2}log(N)). 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 the channel parameters in O(mNlog(N)) operations, for channel of sparsity m. We discuss applications of the flag method to GPS, radar system, and mobile communication as well.

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arXiv2011

Computing the Matched Filter in Linear Time

A fundamental problem in wireless communication is the time-frequency shift (TFS) problem: Find the time-frequency shift of a signal in a noisy environment. The shift is the result of time asynchronization of a sender with a receiver, and of non-zero speed of a sender with respect to a receiver. A classical solution of a discrete analog of the TFS problem is called the matched filter algorithm. It uses a pseudo-random waveform S(t) of the length p, and its arithmetic complexity is O(p^{2} \cdot log (p)), using fast Fourier transform. In these notes we introduce a novel approach of designing new waveforms that allow faster matched filter algorithm. We use techniques from group representation theory to design waveforms S(t), which enable us to introduce two fast matched filter (FMF) algorithms, called the flag algorithm, and the cross algorithm. These methods solve the TFS problem in O(p\cdot log (p)) operations. We discuss applications of the algorithms to mobile communication, GPS, and radar.

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arXiv2011

The categorical Weil representation

In a previous work the authors gave a conceptual explanation for the linearity of the Weil representation over a finite field k of odd characteristic: There exists a canonical system of intertwining operators between the Lagrangian models of the Heisenberg representation. This defines a canonical vector space H(V) associated with a symplectic vector space V over k. In this paper we prove a general theorem about idempotents in categories, and we use it to solve the sign problem, formulated by Bernstein and Deligne, on the compatibility between the associativity constraint and the convolution structure of the l-adic sheaf of canonical intertwining kernels. This sheaf governs---via the sheaf-to-function correspondence---the function theoretic system of intertwiners. As an application we define a canonical category C(V) associated with the symplectic vector space variety V, and we obtain the canonical model of the categorical Weil representation.

math.RT↗
arXiv2011

Representation theoretic patterns in three dimensional cryo-electron microscopy II - The class averaging problem

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced by Hadani, Shkolnisky, Singer and Zhao, for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability, thus putting it on firm mathematical grounds. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two dimensional sphere, called the localized parallel transport operator. Along the way, we develop further the representation theoretic set-up for three-dimensional cryo-EM.

math.RT↗
arXiv2010

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

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 property of the Weil representation. As a result, we obtain a sharp form of the Hecke quantum unique ergodicity theorem for generic linear symplectomorphisms of the 2N-dimensional torus.

math.RT↗
arXiv2009

Representation theoretic patterns in three dimensional Cryo-Electron Microscopy I - The intrinsic reconstitution algorithm

In this paper, we describe and study a mathematical framework for cryo-elecron microscopy. The main result, is a a proof of the admissability (correctness) and the numerical stability of the intrinsic reconstitution algorithm which was introduced by Singer and Shkolnisky in [7]. In addition, we explain how the various numerical observations reported in that work, follow from basic representation theoretic principles.

math.RT↗
arXiv2009

Quantization of symplectic vector spaces over finite fields

In this paper, we construct a quantization functor, associating a complex vector space H(V) to a finite dimensional symplectic vector space V over a finite field of odd characteristic. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main new technical result is a proof of a stronger form of the Stone-von Neumann property for the Heisenberg group. Our result answers, for the case of the Heisenberg group, a question of Kazhdan about the possible existence of a canonical vector space attached to a coadjoint orbit of a general unipotent group over finite field.

math.RT↗
arXiv2009

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

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 generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.

math-ph↗
arXiv2009

Notes on quantization of symplectic vector spaces over finite fields

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the symplectic group Sp(V). The main technical result is a proof of a stronger form of the Stone-von Neumann theorem for the Heisenberg group over 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 F_q.

math-ph↗
arXiv2009

Statistical RIP and Semi-Circle Distribution of Incoherent Dictionaries

In this paper we formulate and prove a statistical version of the Candes-Tao restricted isometry property (SRIP for short) which holds in general for any incoherent dictionary which is a disjoint union of orthonormal bases. In addition, we prove that, under appropriate normalization, the eigenvalues of the associated Gram matrix fluctuate around 1 according to the Wigner semicircle distribution. The result is then applied to various dictionaries that arise naturally in the setting of finite harmonic analysis, giving, in particular, a better understanding on a remark of Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg dictionary of chirp like functions.

cs.IT↗
arXiv2009

Incoherent dictionaries and the statistical restricted isometry property

In this article we present a statistical version of the Candes-Tao restricted isometry property (SRIP for short) which holds in general for any incoherent dictionary which is a disjoint union of orthonormal bases. In addition, under appropriate normalization, the eigenvalues of the associated Gram matrix fluctuate around 1 according to the Wigner semicircle distribution. The result is then applied to various dictionaries that arise naturally in the setting of finite harmonic analysis, giving, in particular, a better understanding on a remark of Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg dictionary of chirp like functions.

cs.IT↗
arXiv2009

Application of the Weil representation: diagonalization of the discrete Fourier transform

We survey a new application of the Weil representation to construct a canonical basis of eigenvectors for the discrete Fourier transform (DFT). The transition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). In addition, we describe a fast algorithm for computing the DOT in certain cases.

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arXiv2008

Group representation design of digital signals and sequences

In this survey a novel system, called the oscillator system, consisting of order of p^3 functions (signals) on the finite field F_{p}, 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.

cs.IT↗
arXiv2008

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. Keywords: Weil representation, commutative subgroups, eigenfunctions, random behavior, deterministic construction

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arXiv2008

The finite harmonic oscillator and its applications to sequences, communication and radar

A novel system, called the oscillator system, consisting of order of p^3 functions (signals) on the finite field F_p; with p an odd prime, 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.

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