Asymptotic behavior of the pairwise error probability with applications to transmit beamforming and rake receivers

We derive the precise asymptote of the pairwise error probability for high signal-to-noise ratio (SNR) and apply it to obtain new results concerning transmit beamforming and selective Rake receivers. For downlink beamforming (with N transmit antennas and independently identically distributed (i.i.d.) Rayleigh fading) based on quantized feedback from the mobile, we show that at least /spl lceil/log/sub 2/(N)/spl rceil/ bits of feedback (per coherence time) is required to obtain full diversity, and among all beamforming schemes using /spl lceil/log/sub 2/(N)/spl rceil/ bits of feedback, selection diversity is optimal. We give the exact expression for the SNR loss of selection diversity with respect to ideal beamforming based on perfect knowledge of fading coefficients. Further, we study selective Rake receivers for independent arbitrary fading distribution and arbitrary power delay profile (PDP). In particular, we show that the SNR loss of the SRake receiver with respect to the all-Rake receiver does not depend on the PDP, and we also propose a transformation to adapt the expressions known for the symbol error probability for the case of i.i.d. Rayleigh fading to the general case.     

Convergence analysis of the constant modulus algorithm

We study the global convergence of the stochastic gradient constant modulus algorithm (CMA) in the absence of channel noise as well as in the presence of channel noise. The case of fractionally spaced equalizer and/or multiple antenna at the receiver is considered. For the noiseless case, we show that with proper initialization, and with small step size, the algorithm converges to a zero-forcing filter with probability close to one. In the presence of channel noise such as additive Gaussian noise, we prove that the algorithm diverges almost surely on the infinite-time horizon. However, under suitable conditions, the algorithm visits a small neighborhood of the Wiener filters a large number of times before ultimately diverging.     

Multivariate Signal Parameter Estimation Under Dependent Noise From 1-Bit Dithered Quantized Data

Motivated by applications in sensor networks and communications, we consider multivariate signal parameter estimation when only dithered 1-bit quantized samples are available. The observation noise is taken to be a stationary, strongly mixing process, which covers a wide range of processes including autoregressive moving average (ARMA) models. The noise is allowed to be Gaussian or to have a heavy-tail (with possibly infinite variance). An estimate of the signal parameters is proposed and is shown to be weakly consistent. Joint asymptotic normality of the parameters estimate is also established and the asymptotic mean and covariance matrices are identified.     

Analysis of mean-square error and transient speed of the LMSadaptive algorithm

For the least mean square (LMS) algorithm, we analyze the correlation matrix of the filter coefficient estimation error and the signal estimation error in the transient phase as well as in steady state. We establish the convergence of the second-order statistics as the number of iterations increases, and we derive the exact asymptotic expressions for the mean square errors. In particular, the result for the excess signal estimation error gives conditions under which the LMS algorithm outperforms the Wiener filter with the same number of taps. We also analyze a new measure of transient speed. We do not assume a linear regression model: the desired signal and the data process are allowed to be nonlinearly related. The data is assumed to be an instantaneous transformation of a stationary Markov process satisfying certain ergodic conditions     

Signal Parameter Estimation Using 1-Bit Dithered Quantization

Motivated by the estimation of spatio-temporal events with cheap, simple sensors, we consider the problem of estimation of a parameter thetas of a signal s(x;thetas) corrupted by noise assuming that only 1-bit precision dithered quantized samples are available. An estimate that does not require the knowledge of the dither signal and the noise distribution is proposed, and it is analyzed in detail under variety of nonidealities. The consistency and asymptotic normality of the estimate is established for deterministic and random sampling, imprecise knowledge of sampling locations, Gaussian and non-Gaussian noise (with possibly infinite variance), a wide class of dither distributions, and under erroneous transmission of the binary observations via binary-symmetric channels (BSCs). It is also shown that if approximation to the log-likelihood equation in the full precision case yields a good estimate, then there is a corresponding good estimate based on 1-bit dithered samples. The proposed estimate requires no more computation than the maximum-likelihood estimate for the full precision case and suffers only a logarithmic rate loss compared to the full precision case when uniform dithering is used. It is shown that uniform dithering leads to the best rate among a broad class of dither distributions. A condition under which no dithering leads to a better estimate is also given