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Register a new userLimits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices
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One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy-Widom distribution.
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This paper studies a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrary signal distribution, and forward error control (FEC) coding. We develop a universal Gram-Schmidt orthogonalization for orthogonal approximate message passing (OAMP). Numerous area properties are established based on the state evolution and minimum mean squared error (MMSE) property of OAMP in an un-coded LUIS. As a byproduct, we provide an alternative derivation for the constrained capacity of a LUIS. Under the assumption that the state evolution for OAMP is correct for the coded system, the achievable rate of OAMP is analyzed. We prove that OAMP achieves the constrained capacity of the LUIS with an arbitrary signal distribution provided that a matching condition is satisfied. Meanwhile, we elaborate a capacity-achieving coding principle for LUIS, based on which irregular low-density parity-check (LDPC) codes are optimized for binary signaling in the numerical results. We show that OAMP with the optimized codes has significant performance improvement over the un-optimized ones and the well-known Turbo linear MMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, capacity-approaching bit error rate (BER) performances are observed under various channel conditions.
The integrated sensing and communication (ISAC), in which the sensing and communication share the same frequency band and hardware, has emerged as a key technology in future wireless systems. Early works on ISAC have been focused on the design, analysis and optimization of practical ISAC technologies for various ISAC systems. While this line of works are necessary, it is equally important to study the fundamental limits of ISAC in order to understand the gap between the current state-of-the-art technologies and the performance limits, and provide useful insights and guidance for the development of better ISAC technologies that can approach the performance limits. In this paper, we aim to provide a comprehensive survey for the current research progress on the fundamental limits of ISAC. Particularly, we first propose a systematic classification method for both traditional radio sensing (such as radar sensing and wireless localization) and ISAC so that they can be naturally incorporated into a unified framework. Then we summarize the major performance metrics and bounds used in sensing, communications and ISAC, respectively. After that, we present the current research progresses on fundamental limits of each class of the traditional sensing and ISAC systems. Finally, the open problems and future research directions are discussed.
In this paper, a sparse Kronecker-product (SKP) coding scheme is proposed for unsourced multiple access. Specifically, the data of each active user is encoded as the Kronecker product of two component codewords with one being sparse and the other being forward-error-correction (FEC) coded. At the receiver, an iterative decoding algorithm is developed, consisting of matrix factorization for the decomposition of the Kronecker product and soft-in soft-out decoding for the component sparse code and the FEC code. The cyclic redundancy check (CRC) aided interference cancellation technique is further incorporated for performance improvement. Numerical results show that the proposed scheme outperforms the state-of-the-art counterparts, and approaches the random coding bound within a gap of only 0.1 dB at the code length of 30000 when the number of active users is less than 75, and the error rate can be made very small even if the number of active users is relatively large.
We consider the problem of recovering $n$ i.i.d samples from a zero mean multivariate Gaussian distribution with an unknown covariance matrix, from their modulo wrapped measurements, i.e., measurement where each coordinate is reduced modulo $Delta$, for some $Delta>0$. For this setup, which is motivated by quantization and analog-to-digital conversion, we develop a low-complexity iterative decoding algorithm. We show that if a benchmark informed decoder that knows the covariance matrix can recover each sample with small error probability, and $n$ is large enough, the performance of the proposed blind recovery algorithm closely follows that of the informed one. We complement the analysis with numeric results that show that the algorithm performs well even in non-asymptotic conditions.
This paper considers the problem of recovering an unknown sparse ptimes p matrix X from an mtimes m matrix Y=AXB^T, where A and B are known m times p matrices with m << p. The main result shows that there exist constructions of the sketching matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and efficiently using a convex program as long as these non-zeros are not concentrated in any single row/column of X. Furthermore, it suffices for the size of Y (the sketch dimension) to scale as m = O(sqrt{# nonzeros in X} times log p). The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require non-standard techniques. Indeed our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a ptimes n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X:=DD^{T} is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S=AD with A in R^{mtimes p}.
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