The finite harmonic oscillator and its associated sequences

Abstract

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.Weil representation ͉ commutative subgroups ͉ eigenfunctions ͉ random behavior ͉ deterministic construction O ne-dimensional analog signals are complex valued functions on the real line ‫.ޒ‬ In the same spirit, one-dimensional digital signals, also called sequences, might be considered as complex valued functions on the finite line ‫ކ‬ p , i.e., the finite field with p elements. In both situations the parameter of the line is denoted by t and is referred to as time. In this work, we will consider digital signals only, which will be simply referred to as signals. The space of signals H ϭ ‫ކ(ރ‬ p ) is a Hilbert space with the Hermitian product given byA central problem is to construct interesting and useful systems of signals. Given a system ᑭ, there are various desired properties that appear in the engineering wish list. For example, in various situations (1, 2), one requires that the signals will be weakly correlated, i.e., that for every ʦᑭThis property is trivially satisfied if ᑭ is an orthonormal basis. Such a system cannot consist of more than dim(H) signals; however, for certain applications, e.g., code division multiple access (CDMA) (3) a larger number of signals is desired; in that case, the orthogonality condition is relaxed.During the transmission process, a signal might be distorted in various ways. Two basic types of distortions are time shift (t) ‫ۋ‬ L (t) ϭ (t ϩ ) and phase shift (t) ‫ۋ‬ M w (t) ϭ e 2i p wt (t), where , w ʦ ‫ކ‬ p . The first type appears in asynchronous communication and the second type is a Doppler effect due to relative velocity between the transmitting and receiving antennas. In conclusion, a general distortion is of the type ‫ۋ‬ M w L , suggesting that for every ʦᑭ, it is natural to require (1) the following stronger condition ͉͗, M w L ͉͘ Ͻ Ͻ 1.

Delay-Doppler Channel Estimation in Almost Linear Complexity

“…The group G = SL 2 (Z N ) is non-commutative, but contains a special class of maximal commutative subgroups called tori 9 [8], [9], [3]. Each torus T ⊂ G acts via the Weil operators…”

“…This is a commutative collection of diagonalizable operators, and it admits [8], [9] a natural orthonormal basis B T for H, consisting of common eigensequences for all the operators (III-B.1)…”

“…In order to extract the time-frequency shift (τ 0 , ω 0 ), it is "standard" 5 (see [6], [8], [9], [11], [16], [17], [18]) to use pseudorandom sequence S ∈ H of norm one. In this case M(S, S)[τ − τ 0 , ω − ω 0 ] = 1 for (τ , ω) = (τ 0 , ω 0 ), and of order O(…”

“…The design of these sequences is done using group representation theory. The pseudorandom sequences are designed [8], [9], [18] using the Weil representation operators (II-B.3), and will be called Weil (spike) sequences 8 . The structural sequences are designed [10], [11] using the Heisenberg representation operators (II-A.1), and will be called Heisenberg (line) sequences.…”

“…A classical solution [6], [8], [9], [11], [16], [17], [18] to Problem I-C.1, is the matched filter algorithm. We define the following matched filter (MF) matrix of R and S:…”