Large deviation asymptotics for occupancy problems

In the standard formulation of the occupancy problem one considers the distribution of r balls in n cells, with each ball assigned independently to a given cell with probability 1/n. Although closed form expressions can be given for the distribution of various interesting quantities (such as the fraction of cells that contain a given number of balls), these expressions are often of limited practical use. Approximations provide an attractive alternative, and in the present paper we consider a large deviation approximation as r and n tend to infinity. In order to analyze the problem we first consider a dynamical model, where the balls are placed in the cells sequentially and ``time'' corresponds to the number of balls that have already been thrown. A complete large deviation analysis of this ``process level'' problem is carried out, and the rate function for the original problem is then obtained via the contraction principle. The variational problem that characterizes this rate function is analyzed, and a fairly complete and explicit solution is obtained. The minimizing trajectories and minimal cost are identified up to two constants, and the constants are characterized as the unique solution to an elementary fixed point problem. These results are then used to solve a number of interesting problems, including an overflow problem and the partial coupon collector's problem.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000013

On convergence and optimality of maximum-likelihood APA

Affine projection algorithm (APA) is a well-known algorithm in adaptive filtering applications such as audio echo cancellation. APA relies on three parameters: $P$ (projection order), $\mu$ (step size) and $\delta$ (regularization parameter). It is known that running APA for a fixed set of parameters leads to a tradeoff between convergence speed and accuracy. Therefore, various methods for adaptively setting the parameters have been proposed in the literature. Inspired by maximum likelihood (ML) estimation, we derive a new ML-based approach for adaptively setting the parameters of APA, which we refer to as ML-APA. For memoryless Gaussian inputs, we fully characterize the expected misalignment error of ML-APA as a function of iteration number and show that it converges to zero as $O({1\over t})$. We further prove that the achieved error is asymptotically optimal. ML-APA updates its estimate once every $P$ samples. We also propose incremental ML-APA (IML-APA), which updates the coefficients at every time step and outperforms ML-APA in our simulations results. Our simulation results verify the analytical conclusions for memoryless inputs and show that the new algorithms also perform well for strongly correlated input signals

Methods and systems for determining crosstalk for a joining line in a vectored system

At least one example embodiment discloses a method of determining crosstalk for a joining line in a communication system having a plurality of current active lines. The method includes obtaining a number of disturber lines among the plurality of current active lines, the number of disturber lines being less than a number of the plurality of current active lines, obtaining a pilot matrix, a first dimension of the pilot matrix being based on the number of disturber lines, the pilot matrix representing a sequence of pilots to be transmitted across the plurality of current active lines and the joining line, the first dimension being a number of time instances, the number of time instances being less than the number of the plurality of active lines and determining a crosstalk coupling vector for the joining line based on the pilot matrix

Performance of digital subscriber line spectrum optimization algorithms

A management standard for digital subscriber line (DSL) systems has been defined in order to ensure the spectral compatibility of the signals, services, and technologies that are deployed. Compatibility is typically accomplished by using static spectral masks. This conservative approach may lead to suboptimal performance of DSL systems. Recently, a number of dynamic spectrum management (DSM) solutions were proposed that improve DSL performance by adaptive application of spectral masks, also known as DSM level 2. The masks are calculated taking into account actual performance requirements and impairments, such as crosstalk. We present improved DSM level 2 algorithms that are faster than existing algorithms and demonstrate potential rate/reach gains by numerical simulations for some DSL technologies, such as asymmetric digital subscriber line transceivers. We also present a solution for flexible deployment of DSM spectrum optimization algorithms in the field.18 page(s