This paper presents novel analytical results on the successful decoding probability for random linear network coding in acyclic networks. The results consist of a tight lower bound on the successful decoding probability, its convergence, and its application in constructing a practical algorithm to identify the minimum field size for random linear network coding subject to a target on the successful decoding probability. From the two characterizations of random linear network coding, namely the set of local encoding kernels and the set of global encoding kernels, we first show that choosing randomly and uniformly the coefficients of the local encoding kernels results in uniform and independent coefficients for the global encoding kernels. The set of global encoding kernels for an arbitrary destination is thus a random matrix whose invertibility is equivalent to decodability. The lower bound on the successful decoding probability is then derived in terms of the probability that this random matrix is non-singular. The derived bound is a function of the field size and the dimension of global encoding kernels. The convergence rates of the bound over these two parameters are provided. Compared to the mathematical expression of the exact probability, the derived bound provides a more compact expression and is close to the exact value. As a benefit of the bound, we construct a practical algorithm to identify the minimum field size in order to achieve a target on the successful decoding probability. Simulation and numerical results verify the validity of the derived bound as well as its higher precision than previously established bounds.

Let (X,Y) be a Rd R-valued random vector. In regression analysis one wants to estimate the regression function m(x):=E(Y|X=x) from a data set. In this paper we consider the convergence rate of the error for the k nearest neighbor estimators in case that m is (p,C)-smooth. It is known that the minimax rate is unachievable by any k nearest neighbor estimator for p > 1.5 and d=1. We generalize this result to any d ≥ 1. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L2 error.

The electronically steerable passive array radiator (ESPAR) antenna is one kind of the parasitic elements based single-port output antennas with several variable reactances. It performs analog aerial beamforming and none of the signals on its passive elements can be observed. This fact and one that is more important--the nonlinear dependence of the output of the antenna from adjustable reactances--makes the problem substantially new and not resolvable by means of conventional adaptive array beamforming techniques. A novel approach based on stochastic approximation theory is proposed for the adaptive beamforming of the ESPAR antenna as a nonlinear spatial filter by variable parameters, thus forming both beam and nulls. Two learning rate schedule were examined about output SINR, stability, convergence, misadjustment, noise effect, bias term, etc., and the optimal one was proposed. Further development was traced. Our theoretic study, simulation results and performance analysis show that the ESPAR antenna can be controlled effectively, has strong potential for use in mobile terminals and seems to be very perspective.