This paper studies the problem of recovering a non-negative sparse signal x ∈ Rn from highly corrupted linear measurements y = Ax + e ∈ Rm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any non-negative, sufficiently sparse signal x can be recovered by solving an ℓ1-minimization problem: min ‖x‖1 + ‖e‖1 subject to y = Ax + e. More precisely, if the fraction ρ of errors is bounded away from one and the support of x grows sublinearly in the dimension m of the observation, then as m goes to infinity, the above ℓ1-minimization succeeds for all signals x and almost all sign-and-support patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 % of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of iid Gaussian vectors with nonzero mean and small variance, which we call the “cross-and-bouquet ” model. Simulations and experimental results corroborate the findings, and suggest extensions to the result.

Abstract—Coherent data communication over doubly-selective channels requires that the channel response be known at the receiver. Training-based schemes, which involve probing of the channel with known signaling waveforms and processing of the corresponding channel output to estimate the channel parameters, are commonly employed to learn the channel response in practice. Conventional training-based methods, often comprising of linear least squares channel estimators, are known to be optimal under the assumption of rich multipath channels. Numerous measurement campaigns have shown, however, that physical multipath channels tend to exhibit a sparse structure at high signal space dimension (time-bandwidth product), and can be characterized with significantly fewer parameters compared to the maximum number dictated by the delay-Doppler spread of the channel. In this paper, it is established that traditional training-based channel learning techniques are ill-suited to fully exploiting the inherent low-dimensionality of sparse channels. In contrast, key ideas from the emerging theory of compressed sensing are leveraged to propose sparse channel learning methods for both single-carrier and multicarrier probing waveforms that employ reconstruction algorithms based on convex/linear programming. In particular, it is shown that the performance of the proposed schemes come within a logarithmic factor of that of an ideal channel estimator, leading to significant reductions in the training energy and the loss in spectral efficiency associated with conventional training-based methods. I.

In this paper we consider a mobile cooperative network that is tasked with building a map of the received signal strength to a fixed station. By using the recent results in the area of compressed sensing, we show how the nodes can exploit the sparse representation of the channel’s spatial variations to build a map of the signal strength with minimal sensing. We furthermore propose a successive interference cancellation method for signal reconstruction based on a considerably incomplete set of measurements. The proposed method is an extension of the existing signal reconstruction strategies but with a considerably better performance. Finally, we present simulation results that show the performance of the proposed framework.

Abstract — In this paper, we derive concentration of measure inequalities for compressive Toeplitz matrices (having fewer rows than columns) with entries drawn from an independent and identically distributed (i.i.d.) Gaussian random sequence. These inequalities show that the norm of a vector mapped by a Toeplitz matrix to a lower dimensional space concentrates around its mean with a tail probability bound that decays exponentially in the dimension of the range space divided by a factor that is a function of the sample covariance of the vector. Motivated by the emerging field of Compressive Sensing (CS), we apply these inequalities to problems involving the analysis of high-dimensional systems from convolution-based compressive measurements. We discuss applications such as system identification, namely the estimation of the impulse response of a system, in cases where one can assume that the impulse response is high-dimensional, but sparse. We also consider the problem of detecting a change in the dynamic behavior of a system, where the change itself can be modeled by a system with a sparse impulse response. I.

We consider the estimation of doubly selective wireless channels within pulseshaping multicarrier systems (which include OFDM systems as a special case). A pilot-assisted channel estimation technique using the methodology of compressed sensing (CS) is proposed. By exploiting a channel’s delay-Doppler sparsity, CS-based channel estimation allows an increase in spectral efficiency through a reduction of the number of pilot symbols that have to be transmitted. We also present an extension of our basic channel estimator that employs a sparsity-improving basis expansion. We propose a framework for optimizing the basis and an iterative approximate basis optimization algorithm. Simulation results using three different CS recovery algorithms demonstrate significant performance gains (in terms of improved estimation accuracy or reduction of the number of pilots) relative to conventional least-squares estimation, as well as substantial advantages of using an optimized basis.

Abstract — Compressive sensing is a revolutionary idea proposed recently to achieve much lower sampling rate for sparse signals. For large wireless sensor networks, the events are relatively sparse compared with the number of sources. Because of deployment cost, the number of sensors is limited, and due to energy constraint, not all the sensors are turned on all the time. In this paper, the first contribution is to formulate the problem for sparse event detection in wireless sensor networks as a compressive sensing problem. The number of (wake-up) sensors can be greatly reduced to the similar level of the number of sparse events, which is much smaller than the total number of sources. Second, we suppose the event has the binary nature, and employ the Bayesian detection using this prior information. Finally, we analyze the performance of the compressive sensing algorithms under the Gaussian noise. From the simulation results, we show that the sampling rate can reduce to 25 % without sacrificing performance. With further decreasing the sampling rate, the performance is gradually reduced until 10 % of sampling rate. Our proposed detection algorithm has much better performance than the l1-magic algorithm proposed in the literature. I.

Abstract—In this paper, we present various channel estimators that exploit the channel sparsity in a multicarrier underwater acoustic system, including subspace algorithms from the array precessing literature, namely root-MUSIC and ESPRIT, and recent compressed sensing algorithms in form of Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP). Numerical simulation and experimental data of an OFDM block-by-block receiver are used to evaluate the proposed algorithms in comparison to the conventional least-squares (LS) channel estimator. We observe that subspace methods can tolerate small to moderate Doppler effects, and outperform the LS approach when the channel is indeed sparse. On the other hand, compressed sensing algorithms uniformly outperform the LS and subspace methods. Coupled with a channel equalizer mitigating intercarrier interference, the compressed sensing algorithms can handle channels with significant Doppler spread.

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimise the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε −2 log 3 n) instead of the classical bound k = O(ε −2 log n). AMS Classification: 52C99, 68Q01 Keywords and phrases: Johnson-Lindenstrauss lemma, circulant matrices, decoupling lemma

In this paper, we consider the estimation of sparse nonlinear communication channels. Transmission over the channels is represented by sparse Volterra models that incorporate the effect of Power Amplifiers. Channel estimation is performed by compressive sensing methods. Efficient algorithms are proposed based on Kalman filtering and Expectation Maximization. Simulation studies confirm that the proposed algorithms achieve significant performance gains in comparison to the conventional non-sparse methods.

Abstract—Multipath interference is an ubiquitous phenomenon in modern communication systems. The conventional way to compensate for this effect is to equalize the channel by estimating its impulse response by transmitting a set of training symbols. The primary drawback to this type of approach is that it can be unreliable if the channel is changing rapidly. In this paper, we show that randomly encoding the signal can protect it against channel uncertainty when the channel is sparse. Before transmission, the signal is mapped into a slightly longer codeword using a random matrix. From the received signal, we are able to simultaneously estimate the channel and recover the transmitted signal. We discuss two schemes for the recovery. Both of them exploit the sparsity of the underlying channel. We show that if the channel impulse response is sufficiently sparse, the transmitted signal can be recovered reliably. I.