Javier Villares

The Gaussian Assumption in Second-Order Estimation Problems in Digital Communications

This paper deals with the goodness of the Gaussian assumption when designing second-order blind estimation methods in the context of digital communications. The low- and high-signal-to-noise ratio (SNR) asymptotic performance of the maximum likelihood estimator - derived assuming Gaussian transmitted symbols - is compared with the performance of the optimal second-order estimator, which exploits the actual distribution of the discrete constellation. The asymptotic study concludes that the Gaussian assumption leads to the optimal second-order solution if the SNR is very low or if the symbols belong to a multilevel constellation such as quadrature-amplitude modulation (QAM) or amplitude-phase-shift keying (APSK). On the other hand, the Gaussian assumption can yield important losses at high SNR if the transmitted symbols are drawn from a constant modulus constellation such as phase-shift keying (PSK) or continuous-phase modulations (CPM). These conclusions are illustrated for the problem of direction-of-arrival (DOA) estimation of multiple digitally-modulated signals.

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting any approximation about the observation statistics or the a priori distribution of the parameters. The proposed solution is obtained minimizing the estimator variance subject to some constraints on the estimator bias. The resulting optimal estimator is found to depend on the observation fourth-order moments that can be calculated analytically from the known signal model. Unfortunately, in most cases, the performance of this estimator is severely limited by the residual bias inherent to nonlinear estimation problems. To overcome this limitation, the second-order minimum variance unbiased estimator is deduced from the general solution by assuming accurate prior information on the vector of parameters. This small-error approximation is adopted to design iterative estimators or trackers. It is shown that the associated variance constitutes the lower bound for the variance of any unbiased estimator based on the sample covariance matrix. The paper formulation is then applied to track the angle-of-arrival (AoA) of multiple digitally-modulated sources by means of a uniform linear array. The optimal second-order tracker is compared with the classical maximum likelihood (ML) blind methods that are shown to be quadratic in the observed data as well. Simulations have confirmed that the discrete nature of the transmitted symbols can be exploited to improve considerably the discrimination of near sources in medium-to-high SNR scenarios.

Frequency-Domain GLR Detection of a Second-Order Cyclostationary Signal Over Fading Channels

Cyclostationary processes exhibit a form of frequency diversity. Based on that, we show that a digital waveform with symbol period T can be asymptotically represented as a rank-1 frequency-domain vector process which exhibits uncorrelation at different frequencies inside the Nyquist spectral support of 1/T. By resorting to the fast Fourier transform (FFT), this formulation obviates the need of estimating a cumbersome covariance matrix to characterize the likelihood function. We then derive the generalized likelihood ratio test (GLRT) for the detection of a cyclostationary signal in unknown white noise without the need of a assuming a synchronized receiver. This provides a sound theoretical basis for the exploitation of the cyclostationary feature and highlights an explicit link with classical square timing recovery schemes, which appear implicitly in the core of the GLRT. Moreover, to avoid the well-known sensitivity of cyclostationary-based detection schemes to frequency-selective fading channels, a parametric channel model based on a lower bound on the coherence bandwidth is adopted and incorporated into the GLRT. By exploiting the rank-1 structure of small spectral covariance matrices, the obtained detector outperforms the classical spectral correlation magnitude detector.

A Nondata-Aided SNR Estimation Technique for Multilevel Modulations Exploiting Signal Cyclostationarity

Signal-to-noise ratio (SNR) estimators of linear modulation schemes usually operate at one sample per symbol at the matched filter output. In this paper we propose a new method for estimating the SNR in the complex additive white Gaussian noise (AWGN) channel that operates directly on the oversampled cyclostationary signal at the matched filter input. Exploiting cyclostationarity proves to be advantageous due to the fact that a signal-free Euclidean noise subspace can be identified such that only second order moments of the received waveform need to be computed. The proposed method is nondata-aided (NDA), as well as constellation and phase independent, and only requires prior timing synchronization to fully exploit the cyclostationarity property. The estimator can also be applied to nonconstant modulus constellations without requiring any tuning, which is a feature not found in existing approaches. Implementation aspects and simpler suboptimal solutions are also provided.

Cramér-Rao Bounds for SNR Estimation of Oversampled Linearly Modulated Signals

Most signal-to-noise ratio (SNR) estimators use the receiver matched filter output sampled at the symbol rate, an approach which does not preserve all information in the analog waveform due to aliasing. Thus, it is relevant to ask whether avoiding aliasing could improve SNR estimation. To this end, we compute the corresponding data-aided (DA) and nondata-aided (NDA) Cramér-Rao bounds (CRBs). We adopt a novel dual filter framework, which is shown to be information-preserving under suitable conditions and considerably simplifies the analysis. It is shown that the CRB can be substantially reduced by exploiting any available excess bandwidth, depending on the modulation scheme, the SNR range, and the estimator type (DA or NDA).

Sample covariance matrix based parameter estimation for digital synchronization

In this paper we develop a new, versatile framework for the design of optimal non-data-aided (NDA) parameter estimators based on the exploitation of the received signal sample covariance matrix. The estimator coefficients are optimized in. order to yield minimum mean squared error (MSE) estimates of the parameter. Some linear constraints are introduced into the optimization process allowing the designer to have control over the estimator characteristic response. For those scenarios where bias is forbidden, as it happens in ranging applications, we provide the optimal solution minimizing the estimates bias within the range of the received parameter. The adopted approach is Bayesian as we treat the wanted parameter as a random variable with a known a priori probability distribution (prior). This modeling allows us to unify the design of both open- and closed-loop estimators. The proposed formulation encompasses all the linear modulations as well as the binary continuous phase modulation (CPM). The new approach supplies optimal estimation schemes without the need of assuming a given statistics for the unknown symbols, that is, avoiding the common adoption of the Gaussian assumption, which does not apply in digital communications. Special attention is paid to those low-complexity implementations for which the maximum likelihood efficiency is not guaranteed.

Quadratic sphericity test for blind detection over time-varying frequency-selective fading channels

This paper addresses the problem of blind detection of a wide-sense stationary (WSS) signal over fading channels. We propose a test statistic which is optimal from a correlation-matching perspective that shows invariance with respect to the noise power and the channel gain. In the blind scenario, we derive the quadratic sphericity test (QST) which exploits the structure of the fading channel as a squared mean to arithmetic mean ratio of the eigenvalues of the autocorrelation matrix of the observations. We provide numerical results to assess the performance of the QST in several fading scenarios, as well as the benchmarking to other blind and non-blind detectors.

Binary graphs and message passing strategies for Compressed Sensing in the noiseless setting

We propose a scheme for Compressed Sensing in the noiseless setting that reconstructs the original signal operating on a binary graph where the samples are obtained sequentially. The proposed scheme has an affordable computational complexity and a large performance enhancement with respect to similar schemes in the literature, thanks to the proposed measurement matrix structure and enhanced decoding based on a message passing algorithm.

Asymptotic and Finite User PER Analysis of Successive Interference Cancellation for DS-CDMA

An expression is derived for the average Packet Error Rate (PER) of a Successive Interference Canceller (SIC) for DS-CDMA when the number of users asymptotically tends to infinity. The asymptotic probability density function of the interference power is governed by a Fokker-Planck differential equation with drift and (asymptotically vanishing) diffusion depending on the PER function of the adopted forward error correcting code (FEC). In addition to the asymptotic solution for the PER, a particle-based algorithm is also developed for computing efficiently the PER in the finite user case.

SINR profile for spectral efficiency optimization of SIC receivers in the many-user regime

In dense wireless scenarios, and particularly under high traffic loads, the design of efficient random access protocols is necessary. Some candidate solutions are based on Direct-Sequence Spread Spectrum (DS-SS) combined with a Successive Interference Cancellation (SIC) demodulator, but the performance of these techniques is highly related to the distribution of the users received power. In that context, this paper presents a theoretical analysis to calculate the optimum user SINR profile at the decoder maximizing the spectral efficiency in bps/Hz for a specific modulation and practical Forward Error Correction (FEC) code. This solution is achieved by means of Variational Calculus operating in the asymptotic large-user case. Although a constant SINR function has been typically assumed in the literature (the one maximizing capacity), the theoretical results evidence that the optimum SINR profile must be an increasing function of the users received power. Its performance is compared with that of the uniform profile for two representative scenarios with different channel codes in a slightly overloaded system. The numerical results show that the optimum solution regulates the network load preventing the aggregate throughput from collapsing when the system is overloaded. In scenarios with a large number of transmitters, this optimum solution can be implemented in an uncoordinated manner with the knowledge of a few public system parameters.