Abstract-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(m · N log N ) operations, for channel of sparsity m. We discuss applications of the flag method to GPS, radar system, and mobile communication as well.
Abstract-A novel system, called the oscillator system, consisting of order of p 3 functions (signals) on the finite field Fp, 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.
We construct a geometric analogue of the Weil representation over a finite field. Our construction is principally invariant, not choosing any specific realization. This eliminates most of the unpleasant formulas that appear in the traditional (non-invariant) approaches, and puts in the forefront some delicate geometric phenomena which underlie this representation.
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 H(V ). 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.
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.
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