Asymptotic Distortion Research Articles

Detection of Spatially Modulated Signals via RLS: Theoretical Bounds and Applications

This paper characterizes the performance of massive multiuser spatial modulation MIMO systems, when a regularized form of the least-squares method is used for detection. For a generic distortion function and right unitarily invariant channel matrices, the per-antenna transmit rate and the asymptotic distortion achieved by this class of detectors are derived. Invoking an asymptotic characterization, we address two particular applications. Namely, we derive the error rate achieved by the computationally-intractable optimal Bayesian detector, and we propose an efficient approach to tune LASSO-type detectors. We further validate our derivations through various numerical experiments.

On Conjugates and the Asymptotic Distortion of One-Dimensional C1+bv Diffeomorphisms

Abstract We show that a $C^{1+bv}$ circle diffeomorphism with absolutely continuous derivative and irrational rotation number can be conjugated to diffeomorphisms that are $C^{1+bv}$ arbitrary close to the corresponding rotation. This improves a theorem of M. Herman, who established the same result but starting with a $C^2$ diffeomorphism. We prove that the same holds for countable Abelian groups of circle diffeomorphisms acting freely, a result that is new even in the $C^{\infty }$ context. Related results and examples concerning the asymptotic distortion of diffeomorphisms are presented. Along this path, we provide a straightened version of the classical Denjoy–Kocsma inequality for absolutely continuous potentials.$\hfill$In memory of Jean-Christophe Yoccoz

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C1+bv conjugacy class if and only if it is the time-1 map of a C1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval diffeomorphisms. Several related results in other regularity classes are obtained, and many open questions are addressed.

Statistical Mechanics of MAP Estimation: General Replica Ansatz

The large-system performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear system with additive white Gaussian noise. The analysis considers the system matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-system analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with $b$ steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued system decouples into a bank of equivalent decoupled scalar systems followed by maximum-a-posterior estimators. The structure of the decoupled system is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For $b$ steps of symmetry breaking, the decoupled system is found to be an additive system with a non-Gaussian noise term given as the sum of an independent Gaussian random variable with $b$ non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition system described by its corresponding decoupled system. As an application of our study, we investigate large compressive sensing systems by considering the $\ell _{p}$ norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for $\ell _{0}$ norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansatze are needed in order to assess the performance precisely.

On asymptotic size distortions in the random coefficients logit model

We show that, in the random coefficients logit model, standard inference procedures can suffer from asymptotic size distortions. The problem arises due to boundary issues and is aggravated by the standard parameterization of the model, in terms of standard deviations. For example, in case of a single random coefficient, the asymptotic size of the nominal 95% confidence interval obtained by inverting the two-sided t-test for the standard deviation equals 83.65%. In seeming contradiction, we also show that standard error estimates for the estimator of the standard deviation can be unreasonably large. This problem is alleviated if the model is reparameterized in terms of variances. Furthermore, a numerical evaluation of a conjectured lower bound suggests that the asymptotic size of the nominal 95% confidence interval obtained by inverting the two-sided t-test for variances (means) is within 0.5 percentage points of the nominal level as long as there are no more than five (four) random coefficients and as long as an optimal weighting matrix is employed.

Least Square Error Precoders for Massive MIMO With Signal Constraints: Fundamental Limits

This paper proposes nonlinear least square error (LSE) precoders for multiuser MIMO broadcast channels. The LSE precoders are designed such that the discrete output signals are from a predefined set. This predefined set allows us to model several signal constraints such as peak power constraint, constant envelope, and discrete constellations. We study the large-system performance of these precoders via the replica method from statistical physics, and derive a closed-form expression for the asymptotic distortion. Our results demonstrate that an LSE precoder with the output peak-to-average power ratio of 3 dB can perform similar to the regularized zero forcing (RZF) precoder. As the peak-to-average power ratio reduces to one, the constant envelope precoder is recovered. The investigations show that the performance of the RZF precoder is achieved by a constant envelope precoder with 20% additional transmit antennas. For $M$ -phase shift keying constellations, our analysis gives a lower bound on the asymptotic distortion which is tight for moderate antenna-to-user ratios and deviates as the ratio grows. We improve this bound by deriving the replica solution under one-step of replica symmetry breaking. Our numerical investigations for this case show that the bound is tight for antenna-to-user ratios less than 5.

Distributed Synchronization on Weakly Connected Networks

Distributed synchronization for wireless networks is based on the mutual exchange of the same chirp-signature by nodes. Collisions of these signatures drive the system toward time and (carrier) frequency synchronization using distributed consensus algorithms. This letter investigates the convergence and the asymptotic distortion properties on noisy networks when neighboring clusters of nodes are weakly connected to each other only through a subset of nodes and bridging links. These heavily connected clusters act as macroagents, and the consensus properties of the ensemble depend on the number of bridge links between them. The convergence rate and mean square synchronization deviation are derived as functions of the number of bridge links for different examples of weakly connected noisy networks via the analytic calculation of Laplacian spectra. Our approach facilitates the study of network topology optimization for distributed synchronization.

An Improved Delay-Limited Source and Channel Coding of Quasy-Stationary Sources over Block Fading Channels: Design and Scaling Laws

In this paper, delay-limited transmission of Quasi-stationary sources over block fading channels is considered. Considering distortion outage probability as the performance measure, two source and channel coding schemes with power adaptive transmission are presented. The delay limited transmission of a quasi-stationary source over a block fading channel is from the perspective of source and channel coding design and it utilizes performance scaling laws. Because In a Quasi -stationary sources time utilization takes a major role. For that the first one is optimized for fixed rate transmission, and hence enjoys simplicity of implementation. The second one is a high performance scheme, which also benefits from optimized rate adaptation with respect to source and channel states. In high SNR regime, the performance scaling laws in terms of outage distortion exponent and asymptotic outage distortion gain are derived, where two schemes with fixed transmission power and adaptive or optimized fixed rates are considered as benchmarks for comparisons. Various analytical and numerical results are provided which demonstrate a superior performance for source and channel optimized rate and power adaptive scheme.

On GLS -- Detrending for Deterministic Seasonality Testing

In this paper we propose tests based on GLS-detrending for testing the null hypothesis of deterministic seasonality. Unlike existing tests for deterministic seasonality, our tests do not suffer from asymptotic size distortions under near integration. We also investigate the behavior of the proposed tests when the initial condition is not asymptotically negligible.

On Transmission of Multiresolution Gaussian Sources over Noisy Relay Networks

This paper investigates joint source-channel coding (JSCC) in a decode-and-forward three-node relay network, in which scalable source coding (SSC) is coupled with superposition coding (SPC) to form a layered coding architecture of SSC-SPC. In contrast to any previously reported research using asymptotic capacity-based distortion (CBD) measure, we derive the mean-squared error end-to-end distortion (EED) of such JSCC system based on a real-valued Gaussian source, aiming to achieve better precision and practicality for applications in which channels are subject to large error probabilities. The EED evaluation is formulated and applied to demonstrate achievable gains of the SSC-SPC architecture versus a number of conventional approaches. Power allocation optimization is performed based on the developed non-asymptotic EED model and compared to that by using an asymptotic CBD measure in which symbol losses caused by channel error cannot be considered. We demonstrate the performance gaps between results solved from EED versus CBD in our numerical example, and conclude that the optimization using CBD behaves awkwardly in computing proper power allocation configurations in the considered SSC-SPC architecture.

Scalar Quantization With Random Thresholds

The distortion-rate performance of certain randomly-designed scalar quantizers is determined. The central results are the mean-squared error distortion and output entropy for quantizing a uniform random variable with thresholds drawn independently from a uniform distribution. The distortion is at most 6 times that of an optimal (deterministically-designed) quantizer, and for a large number of levels the output entropy is reduced by approximately (1-gamma)/(ln 2) bits, where gamma is the Euler-Mascheroni constant. This shows that the high-rate asymptotic distortion of these quantizers in an entropy-constrained context is worse than the optimal quantizer by at most a factor of 6 exp(-2(1-gamma)).

Optimal Network Size and Encoding Rate for Wireless Sensor Network-Based Decentralized Estimation under Power and Bandwidth Constraints

In this paper, we address the problem of decentralized parameter estimation via Wireless Sensor Networks (WSNs). We consider two different encoding strategies, namely, Quantize-and-Estimate (Q&E) and Compress-and-Estimate (C&E) and assume that sensor observations are conveyed to the Fusion Center (FC) over a number of orthogonal Gaussian or Rayleigh-fading channels. We constrain both power and bandwidth to be constant irrespectively of the network size and find approximate closed-form expressions of the optimal number of sensors for a number of cases of interest. Besides, we derive the optimal encoding rate for the Q&E scheme when, in the absence of Transmit Channel State Information (CSIT), sensors must encode their observations at a common and constant rate. For the (successive) C&E strategy, we also determine the encoding order that minimizes the resulting distortion in the FC estimates. We complement the analysis by deriving an expression of the asymptotic distortion when the number of sensors grows without bound, and the rate at which distortion decreases in the high-SNR regime. Finally, we close the paper by presenting some computer simulation results.

Asymptotic distortion exponents for the estimation of time-varying channels in multihop sensor networks

The problem of time-varying channel estimation in multihop sensor networks is examined. Two relay processing methods are explored: amplify-and-forward and encode-and-forward. Bounds on the end-to-end distortion for all internode channel estimates are computed for these two relay processing schemes. Performance is analyzed via the asymptotic limit of the decay rate of the end-to-end distortion with respect to SNR at high SNR. It is also established that asymptotically in SNR, amplify-and-forward can outperform encode-and-forward and in fact can achieve the maximum possible distortion exponent (distortion decay rate) order of unity. Linear and many-to-one topologies are then examined and it is shown that orthogonal access in the many-to-one network is optimal.

Asymptotic Distortion Performance of Source-Channel Diversity over Multihop and Relay Channels

A key challenge in the design of real-time wireless multimedia systems is the presence of fading coupled with strict delay constraints. A very effective answer to this problem is the use of diversity achieving techniques to overcome the fading nature of the wireless channels caused by the mobility of the nodes. The mobility of the nodes gives rise to the need of cooperation among the nodes to enhance the system performance. This paper focuses on comparing systems that exhibit diversity of three forms: source coding diversity, channel coding diversity, and user cooperation diversity implemented through multihop or relay channels with amplify-and-forward or decode-and-forward protocols. Commonly used in multimedia communications, performance is measured in terms of the distortion exponent, which measures the rate of decay of the end-to-end distortion at asymptotically high signal-to-noise ratio (SNR). For the case of repetition coding at the relay nodes, we prove that having more relays is not always beneficial. For the general case of having a large number of relays that can help the source using repetition coding, the optimum number of relay nodes that maximizes the distortion exponent is determined in this paper. This optimum number of relay nodes will depend on the system bandwidth as well as the channel quality. The derived result shows a trade-off between the quality (resolution) of the source encoder and the amount of cooperation (number of relay nodes). Also, the performances of the channel coding diversity-based scheme and the source coding diversity-based scheme are compared. The results show that for both relay and multihop channels, channel coding diversity provides the best performance, followed by the source coding diversity.

Dimensionality Reduction for Distributed Estimation in the Infinite Dimensional Regime

Distributed estimation of an unknown signal is a common task in sensor networks. The scenario usually envisioned consists of several nodes, each making an observation correlated with the signal of interest. The acquired data is then wirelessly transmitted to a fusion center that aims at estimating the desired signal within a prescribed accuracy. Motivated by the obvious processing limitations inherent to such distributed infrastructures, we seek to find efficient compression schemes that account for limited available power and communication bandwidth. In this paper, we propose a transform-based approach to this problem where each sensor provides the fusion center with a low-dimensional approximation of its local observation by means of a suitable linear transform. Under the mean squared error criterion, we derive the optimal solution to apply at one sensor assuming all else being fixed. This naturally leads to an iterative algorithm whose optimality properties are exemplified using a simple though illustrative correlation model. The stationarity issue is also investigated. Under restrictive assumptions, we then provide an asymptotic distortion analysis, as the size of the observed vectors becomes large. Our derivation relies on a variation of the Toeplitz distribution theorem, which allows us to provide a reverse ldquowater-fillingrdquo perspective to the problem of optimal dimensionality reduction. We illustrate, with a first-order Gauss-Markov model, how our findings allow for the computation of analytical closed-form distortion formulas that provide an accurate estimation of the reconstruction error obtained in the finite-dimensional regime.

Analysis of Multiple Antenna Systems With Finite-Rate Channel Information Feedback Over Spatially Correlated Fading Channels

This paper employs a high resolution quantization framework to study the effects of finite-rate quantization of the channel state information (CSI) on the performance of MISO systems over correlated fading channels. The contributions of this paper are twofold. First, as an application of the general distortion analysis, tight lower bounds on the capacity loss of correlated MISO systems due to the finite-rate channel quantization are provided. Closed-form expressions for the capacity loss in high-signal-to-noise ratio (SNR) and low-SNR regimes are also provided, and their analysis reveals that the capacity loss of correlated MISO channels is related to that of i.i.d. fading channels by a simple multiplicative factor which is given by the ratio of the geometric mean to the arithmetic mean of the eigenvalues of the channel covariance matrix. Second, this paper extends the general asymptotic distortion analysis to the important practical problem of suboptimal quantizers resulting from mismatches in the distortion functions, source statistics, and quantization criteria. As a specific application, two types of mismatched MISO CSI quantizers are investigated: quantizers whose codebooks are designed with minimum mean square error (MMSE) criterion but the distortion measure is the ergodic capacity loss (i.e., mismatched design criterion), and quantizers with codebook designed with a mismatched channel covariance matrix (i.e., mismatched statistics). Bounds on the channel capacity loss of the mismatched codebooks are provided and compared to that of the optimal quantizers. Finally, numerical and simulation results are presented and they confirm the tightness of theoretical distortion bounds.

Analysis of Multiple-Antenna Systems With Finite-Rate Feedback Using High-Resolution Quantization Theory

This paper considers the development of a general framework for the analysis of transmit beamforming methods in multiple-antenna systems with finite-rate feedback. Inspired by the results of classical high-resolution quantization theory, the problem of finite-rate quantized communication system is formulated as a general fixed-rate vector quantization problem with side information available at the encoder (or the quantizer) but unavailable at the decoder. The framework of the quantization problem is sufficiently general to include quantization schemes with general non-mean-squared distortion functions and constrained source vectors. Asymptotic distortion analysis of the proposed general quantization problem is provided by extending the vector version of the Bennett's integral. Specifically, tight lower and upper bounds of the average asymptotic distortion are proposed. Sufficient conditions for the achievability of the distortion bounds are also provided and related to corresponding classical fixed-rate quantization problems. The proposed general methodology provides a powerful analytical tool to study a wide range of finite-rate feedback systems. To illustrate the utility of the framework, we consider the analysis of a finite-rate feedback multiple-input single-output (MISO) beamforming system over independent and identically distributed (i.i.d.) Rayleigh flat-fading channels. Numerical and simulation results are presented that further confirm the accuracy of the analytical results

Structural change tests under regression misspecifications

This paper considers structural change tests under regression misspecifications. The asymptotic distortions of regression misspecifications on the Sup-Wald statistic are quantified for a class of drifting data generating processes. It is found that there is no asymptotic distortion caused by regression misspecifications when the misspecification term possesses no shifts.

Sample-adaptive product quantization: asymptotic analysis and examples

Vector quantization (VQ) is an efficient data compression technique for low bit rate applications. However the major disadvantage of VQ is that its encoding complexity increases dramatically with bit rate and vector dimension. Even though one can use a modified VQ, such as the tree-structured VQ, to reduce the encoding complexity, it is practically infeasible to implement such a VQ at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement. To overcome this difficulty, a structurally constrained VQ called the sample-adaptive product quantizer (SAPQ) has recently been proposed. We extensively study the SAPQ that is based on scalar quantizers in order to exploit the simplicity of scalar quantization. Through an asymptotic distortion result, we discuss the achievable performance and the relationship between distortion and encoding complexity. We illustrate that even when SAPQ is based on scalar quantizers, it can provide VQ-level performance. We also provide numerical results that show a 2-3 dB improvement over the Lloyd-Max (1982, 1960) quantizers for data rates above 4 b/point.

Asymptotic performance of vector quantizers with a perceptual distortion measure

Gersho's (1979) bounds on the asymptotic performance of vector quantizers are valid for vector distortions which are powers of the Euclidean norm. Yamada, Tazaki, and Gray (1980) generalized the results to distortion measures that are increasing functions of the norm of their argument. In both cases, the distortion is uniquely determined by the vector quantization error, i.e., the Euclidean difference between the original vector and the codeword into which it is quantized. We generalize these asymptotic bounds to input-weighted quadratic distortion measures and measures that are approximately output-weighted-quadratic when the distortion is small, a class of distortion measures often claimed to be perceptually meaningful. An approximation of the asymptotic distortion based on Gersho's conjecture is derived as well. We also consider the problem of source mismatch, where the quantizer is designed using a probability density different from the true source density. The resulting asymptotic performance in terms of distortion increase in decibels is shown to be linear in the relative entropy between the true and estimated probability densities.