Mutual Incoherence Property Research Articles

Non-orthogonal pilot pattern design and sparse channel estimation for underwater acoustic MIMO-OFDM systems

This paper proposes a novel non-orthogonal pilot pattern design method for underwater acoustic (UWA) multiple-input-multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) systems, leveraging compressive sensing (CS) channel estimation. This approach adopts the mutual incoherence property (MIP) and enhances pilot density without increasing pilot overhead. We conducted rigorous mathematical derivation to generate the explicit formula for the non-orthogonal pilot pattern with the mutual coherence of zero. In addition, we estimate multipath delay and amplitude by employing parallel orthogonal matching pursuit (OMP) channel estimation. The co-channel interference (CCI) is eliminated from the systems with the proposed non-orthogonal pilot pattern to estimate the channel coefficients from two transmitters without interference in a single iteration. Simulations and lake trial results demonstrate the superior performance of the proposed non-orthogonal pilot pattern over the orthogonal and existing non-orthogonal pilot patterns in terms of normalized mean square error (NMSE) of channel estimation and bit error rate (BER). The 2×8 UWA MIMO-OFDM system achieves a data rate of 78.11 kbps with a BER of 1.42×10−4 in the lake trial over the communication distance of 773 m.

Joint channel estimation and data detection for high rate orthogonal time frequency space systems

SummaryThis paper proposes a new pilot pattern in the delay‐Doppler (DD) domain for the orthogonal time frequency space (OTFS) system. In contrast to the embedded‐pilot schemes, guard intervals are not used so as increase the spectral efficiency. Also, compared to the superimposed design where data symbols and pilots are arranged on the entire DD grid, in the proposed rearrangement, the number of pilots used is only spread over a subgrid of the DD grid. Hence, the interference of pilots with data symbols is reduced. Afterwards, an algorithm for channel estimation (CE) and symbol detection in the DD domain benefiting from the sparsity of the DD channel is designed. The sparse CE step is formulated as a specific marginalization of the maximum a posteriori (MAP) criterion by providing a Bayesian approach via the mean‐field approximation that involves the variational Bayesian expectation maximization (VB‐EM) algorithm. Detection of data symbols is done using a low complexity MP algorithm. We also propose an interference cancellation (IC) scheme to mitigate contamination of data by pilots that is run after each CE step. To achieve a high CE accuracy, based on the mean mutual incoherence property (MIP), a pilot optimization problem for OTFS is formulated and develop a simulated annealing‐based algorithm to solve it. Finally, simulation results show that the proposed scheme achieves a good compromise between spectral efficiency, complexity, and performance in terms of bit error rate (BER) and normalized mean square error (NMSE) when compared to literature benchmarks.

Recovery analysis of log-sum minimization under mutual incoherence property

The log-sum minimization has been widely employed in signal and image processing. In this paper, in order to analyze its applicable conditions and achievable boundary for compressed sensing (CS) scenarios, some theoretical derivations of the log-sum minimization are studied based on the mutual incoherence property (MIP) of a measurement matrix. Firstly, the existing condition of the global minimum is demonstrated for the log-sum minimization without noise perturbation. Secondly, for the iterative threshold log-sum (ITL) algorithm, the theoretical performance guarantee of recovering sparse signals disturbed by noise is proposed. Additionally, comprehensive experimental results verify the theoretical analysis for the log-sum item and display the advantages and disadvantages of five sparse estimation algorithms, i.e., the ITL algorithm, ℓ1/2 regularization, ℓ1 regularization, ℓ0 regularization and orthogonal matching pursuit (OMP).

Recovery Conditions of Sparse Signals Using Orthogonal Least Squares-Type Algorithms

Orthogonal least squares (OLS)-type algorithms are efficient in reconstructing sparse signals, which include the well-known OLS, multiple OLS (MOLS) and block OLS (BOLS). In this paper, we first investigate the noiseless exact recovery conditions of these algorithms. Specifically, based on mutual incoherence property (MIP), we provide theoretical analysis of OLS and MOLS to ensure that the correct nonzero support can be selected during the iterative procedure. Nevertheless, theoretical analysis for BOLS utilizes the block-MIP to deal with the block sparsity. Furthermore, the noiseless MIP-based analyses are extended to the noisy scenario. Our results indicate that for K-sparse signals, when MIP or SNR satisfies certain conditions, OLS and MOLS obtain reliable reconstruction in at most K iterations, while BOLS succeeds in at most (K/d) iterations where d is the block length. It is shown that our derived theoretical results improve the existing ones, which are verified by simulation tests.

Signal recovery under mutual incoherence property and oracle inequalities

We consider the signal recovery through an unconstrained minimization in the framework of mutual incoherence property. A sufficient condition is provided to guarantee the stable recovery in the noisy case. Furthermore, oracle inequalities of both sparse signals and non-sparse signals are derived under the mutual incoherence condition in the case of Gaussian noises. Finally, we investigate the relationship between mutual incoherence property and robust null space property and find that robust null space property can be deduced from the mutual incoherence property.

Joint Design of Pilot Power and Pilot Pattern for Sparse Cognitive Radio Systems

Existing works design the pilot pattern for sparse channel estimation, assuming that the power of all pilots is equal. However, equal power allocation is not optimal in cognitive radio (CR) systems. In this correspondence, we jointly design the pilot power and pilot pattern for sparse channel estimation in orthogonal-frequency-division-multiplexing-based CR systems, based on the rule of mutual incoherence property that minimizes the coherence of the measurement matrix used for the sparse recovery. Under the sum power constraint and peak power constraint, the pilot design is formulated as a joint optimization problem, which is then decoupled into tractable sequential formations. Given a pilot pattern, we formulate the design of pilot power as a second-order cone programming. Then, we propose a joint design algorithm, which includes discrete optimization for pilot pattern and continuous optimization for pilot power. Simulation results show that the proposed algorithm can achieve better channel estimation performance in terms of mean square error and bit error rate and can further improve the spectrum efficiency by 2.4%, compared with existing algorithms assuming equal pilot power.

Pilot Design Schemes for Sparse Channel Estimation in OFDM Systems

In this paper, we consider the pilot design based on the mutual incoherence property (MIP) for sparse channel estimation in orthogonal frequency-division multiplexing (OFDM) systems. With respect to the length of channel impulse response (CIR), we first derive a sufficient condition for the optimal pilot pattern generated from the cyclic different set (CDS). Since the CDS does not exist for most practical OFDM systems, we propose three pilot design schemes to obtain a near-optimal pilot pattern. The first two schemes, including stochastic sequential search (SSS) and stochastic parallel search (SPS), are based on the stochastic search. The third scheme called iterative group shrinkage (IGS) employs a tree-based searching structure and removes rows in a group instead of removing a single row at each step. We later extend our work to multiple-input-multiple-output (MIMO) systems and propose two schemes, i.e., sequential design scheme and joint design scheme. We also combine them to design the multiple orthogonal pilot patterns, i.e., using the sequential scheme for the first several transmit antennas and using the joint scheme to design the pilot pattern for the remaining transmit antennas. Simulation results show that the proposed SSS, SPS, and IGS converge much faster than the cross-entropy optimization and the exhaustive search and are thus more efficient. Moreover, SSS and SPS outperform IGS in terms of channel estimation performance.

Randomized pick-freeze for sparse Sobol indices estimation in high dimension

This article investigates selection of variables in high-dimension from a non-parametric regression model. In many concrete situations, we are concerned with estimating a non-parametric regression function f that may depend on a large number p of inputs variables. Unlike standard procedures, we do not assume that f belongs to a class of regular functions (Holder, Sobolev, ...), yet we assume that f is a square-integrable function with respect to a known product measure. Furthermore, observe that, in some situations, only a small number s of the coordinates actually affects f in an additive manner. In this context, we prove that, with only 𝒪(s log p ) random evaluations of f , one can find which are the relevant input variables with overwhelming probability. Our proposed method is an unconstrained l 1 -minimization procedure based on the Sobol’s method. One step of this procedure relies on support recovery using l 1 -minimization and thresholding. More precisely, we use a thresholded-LASSO to faithfully uncover the significant input variables. In this frame, we prove that one can relax the mutual incoherence property (known to require 𝒪(s 2 log p ) observations) and still ensure faithful recovery from 𝒪(s α log p ) observations for any 1 ≤ α ≤ 2.

A primal dual active set with continuation algorithm for the [formula omitted]-regularized optimization problem

We develop a primal dual active set with continuation algorithm for solving the ℓ0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, a finite step global convergence of the overall algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and its convergence behavior.

Nonuniform support recovery from noisy random measurements by Orthogonal Matching Pursuit

This article considers nonuniform support recovery via Orthogonal Matching Pursuit (OMP) from noisy random measurements. Given m admissible random measurements (of which Subgaussian measurements is a special case) of a fixed s-sparse signal x in Rn corrupted with additive noise, we show that under a condition on the minimum magnitude of the nonzero components of x, OMP can recover the support of x exactly after s iterations with overwhelming probability provided that m=O(slogn). This extends the results of Tropp and Gilbert (2007) [53] to the case with noise. It is a real improvement over previous results in the noisy case, which are based on mutual incoherence property or restricted isometry property analysis and require O(s2logn) random measurements. In addition, this article also considers sparse recovery from noisy random frequency measurements via OMP. Similar results can be obtained for the partial random Fourier matrix via OMP provided that m=O(s(s+log(n−s))). Thus, for some special cases, this answers the open question raised by Kunis and Rauhut (2008) [34], and Tropp and Gilbert (2007) [53].

On Recovery of Sparse Signals Via $\ell _{1}$ Minimization

This paper considers constrained lscr1 minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr1 minimization with an lscrinfin constraint (Dantzig selector) and lscr1 minimization under an llscr2 constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between lscr1 minimization with an llscr2 constraint and lscr1 minimization with an lscrinfin constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended.