Compressed Sensing Performance Analysis via Replica Method using Bayesian framework

Compressive sensing (CS) is a new methodology to capture signals at lower rate than the Nyquist sampling rate when the signals are sparse or sparse in some domain. The performance of CS estimators is analyzed in this paper using tools from statistical mechanics, especially called replica method. This method has been used to analyze communication systems like Code Division Multiple Access (CDMA) and multiple input multi- ple output (MIMO) systems with large size. Replica analysis, now days rigorously proved, is an efficient tool to analyze large systems in general. Specifically, we analyze the performance of some of the estimators used in CS like LASSO (the Least Absolute Shrinkage and Selection Operator) estimator and Zero-Norm regularizing estimator as a special case of maximum a posteriori (MAP) estimator by using Bayesian framework to connect the CS estimators and replica method. We use both replica symmetric (RS) ansatz and one-step replica symmetry breaking (1RSB) ansatz, clamming the latter is efficient when the problem is not convex. This work is more analytical in its form. It is deferred for next step to focus on the numerical results.Comment: The analytical work and results were presented at the 2012 IEEE European School of Information Theory in Antalya, Turkey between the 16th and the 20th of Apri

RSB Decoupling Property of MAP Estimators

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector $\boldsymbol{x}$ from the observation $\boldsymbol{y}=\mathbf{A}\boldsymbol{x}+\boldsymbol{z}$ with $\mathbf{A}$ being chosen from a wide range of matrix ensembles, and the noise vector $\boldsymbol{z}$ being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the $b$RSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and $b$ correlated interference terms. As the $b$RSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.Comment: 5 pages, presented in Information Theory Workshop 201

Seventy Years of Radar and Communications: The road from separation to integration

Radar and communications (R&C) as key utilities of electromagnetic (EM) waves have fundamentally shaped human society and triggered the modern information age. Although R&C had been historically progressing separately, in recent decades, they have been converging toward integration, forming integrated sensing and communication (ISAC) systems, giving rise to new highly desirable capabilities in next-generation wireless networks and future radars. To better understand the essence of ISAC, this article provides a systematic overview of the historical development of R&C from a signal processing (SP) perspective. We first interpret the duality between R&C as signals and systems, followed by an introduction of their fundamental principles. We then elaborate on the two main trends in their technological evolution, namely, the increase of frequencies and bandwidths and the expansion of antenna arrays. We then show how the intertwined narratives of R&C evolved into ISAC and discuss the resultant SP framework. Finally, we overview future research directions in this field