Space-time autocoding

Prior treatments of space-time communications in Rayleigh flat fading generally assume that channel coding covers either one fading interval-in which case there is a nonzero “outage capacity”-or multiple fading intervals-in which case there is a nonzero Shannon capacity. However, we establish conditions under which channel codes span only one fading interval and yet are arbitrarily reliable. In short, space-time signals are their own channel codes. We call this phenomenon space-time autocoding, and the accompanying capacity the space-time autocapacity. Let an M-transmitter antenna, N-receiver antenna Rayleigh flat fading channel be characterized by an M×N matrix of independent propagation coefficients, distributed as zero-mean, unit-variance complex Gaussian random variables. This propagation matrix is unknown to the transmitter, it remains constant during a T-symbol coherence interval, and there is a fixed total transmit power. Let the coherence interval and number of transmitter antennas be related as T=βM for some constant β. A T×M matrix-valued signal, associated with R·T bits of information for some rate R is transmitted during the T-symbol coherence interval. Then there is a positive space-time autocapacity Ca such that for all R<Ca, the block probability of error goes to zero as the pair (T, M)→∞ such that T/M=β. The autocoding effect occurs whether or not the propagation matrix is known to the receiver, and Ca=Nlog(1+ρ) in either case, independently of β, where ρ is the expected signal-to-noise ratio (SNR) at each receiver antenna. Lower bounds on the cutoff rate derived from random unitary space-time signals suggest that the autocoding effect manifests itself for relatively small values of T and M. For example, within a single coherence interval of duration T=16, for M=7 transmitter antennas and N=4 receiver antennas, and an 18-dB expected SNR, a total of 80 bits (corresponding to rate R=5) can theoretically be transmitted with a block probability of error less than 10^-9, all without any training or knowledge of the propagation matrix

The distribution of the zeroes of random trigonometric polynomials

We study the asymptotic distribution of the number $Z_{N}$ of zeros of random trigonometric polynomials of degree $N$ as $N\to\infty$. It is known that as $N$ grows to infinity, the expected number of the zeros is asymptotic to $\frac{2}{\sqrt{3}}\cdot N$. The asymptotic form of the variance was predicted by Bogomolny, Bohigas and Leboeuf to be $cN$ for some $c>0$. We prove that \frac{Z_{N}-\E Z_{N}}{\sqrt{cN}} converges to the standard Gaussian. In addition, we find that the analogous result is applicable for the number of zeros in short intervals.Comment: 51 pages. We cut the size of the paper to better suit publication. In particular, all the results of empirical experiments were cut off. Some standard results in probability and stochastic processes were also omitted. Numerous typos and mistakes were corrected following the suggestions of referees. This paper was accepted for publication in the American Journal of Mathematics

Bernstein-Szego Polynomials Associated with Root Systems

We introduce multivariate generalizations of the Bernstein-Szego polynomials, which are associated to the root systems of the complex simple Lie algebras. The multivariate polynomials in question generalize Macdonald's Hall-Littlewood polynomials associated with root systems. For the root system of type A1 (corresponding to the Lie algebra SL (2;C)) the classic Bernstein-Szego polynomials are recovered.Comment: LaTeX, 12 page

Limits of elliptic hypergeometric integrals

In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.Comment: 41 pages LaTeX. Minor stylistic changes, statement of Theorem 4.7 fixe