Modulated Unit-Norm Tight Frames for Compressed Sensing

In this paper, we propose a compressed sensing (CS) framework that consists of three parts: a unit-norm tight frame (UTF), a random diagonal matrix and a column-wise orthonormal matrix. We prove that this structure satisfies the restricted isometry property (RIP) with high probability if the number of measurements $m = O(s \log^2s \log^2n)$ for $s$-sparse signals of length $n$ and if the column-wise orthonormal matrix is bounded. Some existing structured sensing models can be studied under this framework, which then gives tighter bounds on the required number of measurements to satisfy the RIP. More importantly, we propose several structured sensing models by appealing to this unified framework, such as a general sensing model with arbitrary/determinisic subsamplers, a fast and efficient block compressed sensing scheme, and structured sensing matrices with deterministic phase modulations, all of which can lead to improvements on practical applications. In particular, one of the constructions is applied to simplify the transceiver design of CS-based channel estimation for orthogonal frequency division multiplexing (OFDM) systems.Comment: submitted to IEEE Transactions on Signal Processin

Compressed Sensing and Parallel Acquisition

Parallel acquisition systems arise in various applications in order to moderate problems caused by insufficient measurements in single-sensor systems. These systems allow simultaneous data acquisition in multiple sensors, thus alleviating such problems by providing more overall measurements. In this work we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing recovery guarantees for which, subject to appropriate conditions, the number of measurements required per sensor decreases linearly with the total number of sensors. Throughout, we consider two different sampling scenarios -- distinct (corresponding to independent sampling in each sensor) and identical (corresponding to dependent sampling between sensors) -- and a general mathematical framework that allows for a wide range of sensing matrices (e.g., subgaussian random matrices, subsampled isometries, random convolutions and random Toeplitz matrices). We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. This model includes both sparse and distributed signals and clustered sparse signals. As our results show, optimal recovery guarantees for both distinct and identical sampling are possible under much broader conditions on the so-called sensor profile matrices (which characterize environmental conditions between a source and the sensors) for the sparse in levels model than for the sparse model. To verify our recovery guarantees we provide numerical results showing phase transitions for a number of different multi-sensor environments.Comment: 43 pages, 4 figure

Compressive Signal Processing with Circulant Sensing Matrices

Compressive sensing achieves effective dimensionality reduction of signals, under a sparsity constraint, by means of a small number of random measurements acquired through a sensing matrix. In a signal processing system, the problem arises of processing the random projections directly, without first reconstructing the signal. In this paper, we show that circulant sensing matrices allow to perform a variety of classical signal processing tasks such as filtering, interpolation, registration, transforms, and so forth, directly in the compressed domain and in an exact fashion, \emph{i.e.}, without relying on estimators as proposed in the existing literature. The advantage of the techniques presented in this paper is to enable direct measurement-to-measurement transformations, without the need of costly recovery procedures

Uniform Recovery from Subgaussian Multi-Sensor Measurements

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.Comment: 37 pages, 5 figure