25,764 research outputs found
Almost Linear Complexity Methods for Delay-Doppler Channel Estimation
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 delay-Doppler
parameters is the pseudo-random method. 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^2logN) arithmetic operations. In
[1] the flag method was introduced to provide a faster algorithm for
delay-Doppler channel estimation. It uses specially designed flag sequences and
its complexity is O(rNlogN) for channels of sparsity r. In these notes, we
introduce the incidence and cross methods for channel estimation. They use
triple-chirp and double-chirp sequences of length N, correspondingly. These
sequences are closely related to chirp sequences widely used in radar systems.
The arithmetic complexity of the incidence and cross methods is O(NlogN + r^3),
and O(NlogN + r^2), respectively.Comment: 4 double column pages. arXiv admin note: substantial text overlap
with arXiv:1309.372
Digital demodulator-correlator
An apparatus for demodulation and correlation of a code modulated 10 MHz signal is presented. The apparatus is comprised of a sample and hold analog-to-digital converter synchronized by a frequency coherent 40 MHz pulse to obtain four evenly spaced samples of each of the signal. Each sample is added or subtracted to or from one of four accumulators to or from the separate sums. The correlation functions are then computed. As a further feature of the invention, multipliers are each multiplied by a squarewave chopper signal having a period that is long relative to the period of the received signal to foreclose contamination of the received signal by leakage from either of the other two terms of the multipliers
Point compression for the trace zero subgroup over a small degree extension field
Using Semaev's summation polynomials, we derive a new equation for the
-rational points of the trace zero variety of an elliptic curve
defined over . Using this equation, we produce an optimal-size
representation for such points. Our representation is compatible with scalar
multiplication. We give a point compression algorithm to compute the
representation and a decompression algorithm to recover the original point (up
to some small ambiguity). The algorithms are efficient for trace zero varieties
coming from small degree extension fields. We give explicit equations and
discuss in detail the practically relevant cases of cubic and quintic field
extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph
Exact and Efficient Simulation of Concordant Computation
Concordant computation is a circuit-based model of quantum computation for
mixed states, that assumes that all correlations within the register are
discord-free (i.e. the correlations are essentially classical) at every step of
the computation. The question of whether concordant computation always admits
efficient simulation by a classical computer was first considered by B. Eastin
in quant-ph/1006.4402v1, where an answer in the affirmative was given for
circuits consisting only of one- and two-qubit gates. Building on this work, we
develop the theory of classical simulation of concordant computation. We
present a new framework for understanding such computations, argue that a
larger class of concordant computations admit efficient simulation, and provide
alternative proofs for the main results of quant-ph/1006.4402v1 with an
emphasis on the exactness of simulation which is crucial for this model. We
include detailed analysis of the arithmetic complexity for solving equations in
the simulation, as well as extensions to larger gates and qudits. We explore
the limitations of our approach, and discuss the challenges faced in developing
efficient classical simulation algorithms for all concordant computations.Comment: 16 page
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
Dynamical forcing of circular groups
In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set X ā R/Z consisting of rotation numbers Īø which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number Īø is forced by a pair (G_Īø, Ī±), where G_Īø is a finitely presented group G_Īø and Ī± ā G_Īø is some element, if the set of rotation numbers of Ļ(Ī±) as Ļ ā Hom(G_Īø, Homeo^(+)(S^1)) is precisely the set {0,Ā±Īø}.
We show that the set of subsets of R/Z which are of the
form rot(X(G, Ī±)) = {r ā R/Z | r = rot(Ļ(Ī±)), Ļ ā Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under xāāx. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, Ī±_i)) for finitely presented G_i.
As another application, we construct a finitely generated group Ī which acts faithfully on the circle, but which does not admit any faithful C^1 action, thus answering in the negative a question of John Franks
A Near-Optimal Algorithm for Computing Real Roots of Sparse Polynomials
Let be an arbitrary polynomial of degree with
non-zero integer coefficients of absolute value less than . In this
paper, we answer the open question whether the real roots of can be
computed with a number of arithmetic operations over the rational numbers that
is polynomial in the input size of the sparse representation of . More
precisely, we give a deterministic, complete, and certified algorithm that
determines isolating intervals for all real roots of with
many exact arithmetic operations over the
rational numbers.
When using approximate but certified arithmetic, the bit complexity of our
algorithm is bounded by , where
means that we ignore logarithmic. Hence, for sufficiently sparse polynomials
(i.e. for a positive constant ), the bit complexity is
. We also prove that the latter bound is optimal up to
logarithmic factors
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