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Register a new userThe importance of phase in complex compressive sensing
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Added by
Laurent Jacques
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider the question of estimating a real low-complexity signal (such as a sparse vector or a low-rank matrix) from the phase of complex random measurements. We show that in this phase-only compressive sensing (PO-CS) scenario, we can perfectly recover such a signal with high probability and up to global unknown amplitude if the sensing matrix is a complex Gaussian random matrix and if the number of measurements is large compared to the complexity level of the signal space. Our approach proceeds by recasting the (non-linear) PO-CS scheme as a linear compressive sensing model built from a signal normalization constraint, and a phase-consistency constraint imposing any signal estimate to match the observed phases in the measurement domain. Practically, stable and robust signal direction estimation is achieved from any instance optimal algorithm of the compressive sensing literature (such as basis pursuit denoising). This is ensured by proving that the matrix associated with this equivalent linear model satisfies with high probability the restricted isometry property under the above condition on the number of measurements. We finally observe experimentally that robust signal direction recovery is reached at about twice the number of measurements needed for signal recovery in compressive sensing.
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We demonstrate that a sparse signal can be estimated from the phase of complex random measurements, in a phase-only compressive sensing (PO-CS) scenario. With high probability and up to a global unknown amplitude, we can perfectly recover such a signal if the sensing matrix is a complex Gaussian random matrix and the number of measurements is large compared to the signal sparsity. Our approach consists in recasting the (non-linear) PO-CS scheme as a linear compressive sensing model. We built it from a signal normalization constraint and a phase-consistency constraint. Practically, we achieve stable and robust signal direction estimation from the basis pursuit denoising program. Numerically, robust signal direction estimation is reached at about twice the number of measurements needed for signal recovery in compressive sensing.
Distributed Compressive Sensing (DCS) improves the signal recovery performance of multi signal ensembles by exploiting both intra- and inter-signal correlation and sparsity structure. However, the existing DCS was proposed for a very limited ensemble of signals that has single common information cite{Baron:2009vd}. In this paper, we propose a generalized DCS (GDCS) which can improve sparse signal detection performance given arbitrary types of common information which are classified into not just full common information but also a variety of partial common information. The theoretical bound on the required number of measurements using the GDCS is obtained. Unfortunately, the GDCS may require much a priori-knowledge on various inter common information of ensemble of signals to enhance the performance over the existing DCS. To deal with this problem, we propose a novel algorithm that can search for the correlation structure among the signals, with which the proposed GDCS improves detection performance even without a priori-knowledge on correlation structure for the case of arbitrarily correlated multi signal ensembles.
In most compressive sensing problems l1 norm is used during the signal reconstruction process. In this article the use of entropy functional is proposed to approximate the l1 norm. A modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregmans row action D-projection method for compressive sensing applications. Simulation examples are presented.
Recently, it was observed that spatially-coupled LDPC code ensembles approach the Shannon capacity for a class of binary-input memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived by Kudekar, Richardson and Urbanke. In particular, it was shown that the belief propagation (BP) threshold of the spatially coupled codes is equal to the maximum a posteriori (MAP) decoding threshold of the underlying constituent codes. In this sense, the BP threshold is saturated to its maximum value. Moreover, it has been empirically observed that the same phenomena also occurs when transmitting over more general classes of BMS channels. In this paper, we show that the effect of spatial coupling is not restricted to the realm of channel coding. The effect of coupling also manifests itself in compressed sensing. Specifically, we show that spatially-coupled measurement matrices have an improved sparsity to sampling threshold for reconstruction algorithms based on verification decoding. For BP-based reconstruction algorithms, this phenomenon is also tested empirically via simulation. At the block lengths accessible via simulation, the effect is quite small and it seems that spatial coupling is not providing the gains one might expect. Based on the threshold analysis, however, we believe this warrants further study.
Quantized compressive sensing (QCS) deals with the problem of coding compressive measurements of low-complexity signals with quantized, finite precision representations, i.e., a mandatory process involved in any practical sensing model. While the resolution of this quantization clearly impacts the quality of signal reconstruction, there actually exist incompatible combinations of quantization functions and sensing matrices that proscribe arbitrarily low reconstruction error when the number of measurements increases. This work shows that a large class of random matrix constructions known to respect the restricted isometry property (RIP) is compatible with a simple scalar and uniform quantization if a uniform random vector, or a random dither, is added to the compressive signal measurements before quantization. In the context of estimating low-complexity signals (e.g., sparse or compressible signals, low-rank matrices) from their quantized observations, this compatibility is demonstrated by the existence of (at least) one signal reconstruction method, the projected back projection (PBP), whose reconstruction error decays when the number of measurements increases. Interestingly, given one RIP matrix and a single realization of the dither, a small reconstruction error can be proved to hold uniformly for all signals in the considered low-complexity set. We confirm these observations numerically in several scenarios involving sparse signals, low-rank matrices, and compressible signals, with various RIP matrix constructions such as sub-Gaussian random matrices and random partial discrete cosine transform (DCT) matrices.
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