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.

Concentration of measure inequalities are at the heart of much theoretical analysis of randomized compressive operators. Though commonly studied for dense matrices, in this paper we derive a concentration of measure bound for block diagonal matrices where the nonzero entries along the main diagonal blocks are i.i.d. subgaussian random variables. Our main result states that the concentration exponent, in the best case, scales as that for a fully dense matrix. We also identify the role that the energy distribution of the signal plays in distinguishing the best case from the worst. We illustrate these phenomena with a series of experiments. Index Terms — Compressive Sensing, concentration of measure, Johnson-Lindenstrauss lemma, block diagonal matrices 1.

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Abstract — Recovering or estimating the initial state of a highdimensional system can require a potentially large number of measurements. In this paper, we explain how this burden can be significantly reduced for certain linear systems when randomized measurement operators are employed. Our work builds upon recent results from the field of Compressive Sensing (CS), in which a high-dimensional signal containing few nonzero entries can be efficiently recovered from a small number of random measurements. In particular, we develop concentration of measure bounds for the observability matrix and explain circumstances under which this matrix can satisfy the Restricted Isometry Property (RIP), which is central to much analysis in CS. We also illustrate our results with a simple case study of a diffusion system. Aside from permitting recovery of sparse initial states, our analysis has potential applications in solving inference problems such as detection and classification of more general initial states. I.

This short note studies a variation of the Compressed Sensing paradigm introduced recently by Vaswani et al., i.e. the recovery of sparse signals from a certain number of linear measurements when the signal support is partially known. The reconstruction method is based on a convex minimization program coined innovative Basis Pursuit DeNoise (or i BPDN). Under the common ℓ2-fidelity constraint made on the available measurements, this optimization promotes the (ℓ1) sparsity of the candidate signal over the complement of this known part. In particular, this paper extends the results of Vaswani et al. to the cases of compressible signals and noisy measurements. Our proof relies on a small adaption of the results of Candes in 2008 for characterizing the stability of the Basis Pursuit DeNoise (BPDN) program. We emphasize also an interesting link between our method and the recent work of Davenport et al. on the δ-stable embeddings and the cancel-then-recover strategy applied to our problem. For both approaches, reconstructions are indeed stabilized when the sensing matrix respects the Restricted Isometry Property for the same sparsity order. We conclude by sketching an easy numerical method relying on monotone operator splitting and proximal methods that iteratively solves i BPDN.

In this paper, we study a simple correlation-based strategy for estimating the unknown delay and amplitude of a signal based on a small number of noisy, randomly chosen frequency-domain samples. We model the output of this “compressive matched filter ” as a random process whose mean equals the scaled, shifted autocorrelation function of the template signal. Using tools from the theory of empirical processes, we prove that the expected maximum deviation of this process from its mean decreases sharply as the number of measurements increases, and we also derive a probabilistic tail bound on the maximum deviation. Putting all of this together, we bound the minimum number of measurements required to guarantee that the empirical maximum of this random process occurs sufficiently close to the true peak of its mean function. We conclude that for broad classes of signals, this compressive matched filter will successfully estimate the unknown delay (with high probability, and within a prescribed tolerance) using a number of random frequency-domain samples that scales inversely with the signal-to-noise ratio and only logarithmically in the in the observation bandwidth and the possible range of delays.

Abstract—Compressive sensing (CS) is an alternative to Shannon/Nyquist sampling for the acquisition of sparse or compressible signals. Instead of taking periodic samples, CS measures inner products with M random vectors, where M is much smaller than the number of Nyquist-rate samples. The implications of CS are promising for many applications and enable the design of new kinds of analog-to-digital converters, imaging systems, and sensor networks. In this paper, we propose and study a wideband compressive radio receiver (WCRR) architecture that can efficiently acquire and track FM and other narrowband signals that live within a wide frequency bandwidth. The receiver operates below the Nyquist rate and has much lower complexity than either a traditional sampling system or CS recovery system. Our methods differ from most standard approaches to the problem of CS recovery in that we do not assume that the signals of interest are confined to a discrete set of frequencies, and we do not rely on traditional recovery methods such as ℓ1minimization. Instead, we develop a simple detection system that identifies the support of the narrowband FM signals and then applies compressive filtering techniques based on discrete prolate spheroidal sequences to cancel interference and isolate the signals. Lastly, a compressive phase-locked loop (PLL) directly recovers the FM message signals. I.

Achieving computer vision on micro-scale devices is a challenge. On these platforms, the power and mass constraints are severe enough for even the most common computations (matrix manipulations, convolution, etc.) to be difficult. This paper proposes and analyzes a class of miniature vision sensors that can help overcome these constraints. These sensors reduce power requirements through template-based optical convolution, and they enable a wide field-of-view within a small form through a novel optical design. We describe the trade-offs between the field of view, volume, and mass of these sensors and we provide analytic tools to navigate the design space. We also demonstrate milli-scale prototypes for computer vision tasks such as locating edges, tracking targets, and detecting faces. 1.

Theoretical analysis of randomized, compressive operators often depends on a concentration of measure inequality for the operator in question. Typically, such inequalities quantify the likelihood that a random matrix will preserve the norm of a signal after multiplication. When this likelihood is very high for any signal, the random matrices have a variety of known uses in dimensionality reduction and Compressive Sensing. Concentration of measure results are well-established for unstructured compressive matrices, populated with independent and identically distributed (i.i.d.) random entries. Many real-world acquisition systems, however, are subject to architectural constraints that make such matrices impractical. In this paper we derive concentration of measure bounds for two types of block diagonal compressive matrices, one in which the blocks along the main diagonal are random and independent, and one in which the blocks are random but equal. For both types of matrices, we show that the likelihood of norm preservation depends on certain properties of the signal being measured, but that for the best case signals, both types of block diagonal matrices can offer concentration performance on par with their unstructured, i.i.d. counterparts. We support our theoretical results with illustrative simulations as well as (analytical and empirical) investigations of several signal classes that are highly amenable to measurement using block diagonal matrices. Finally, we discuss applications of these results in establishing performance guarantees for solving signal processing tasks in the compressed domain (e.g., signal detection), and in establishing the Restricted Isometry Property for the Toeplitz matrices that arise in compressive channel sensing.

Abstract—We present the first custom integrated circuit implementation of the compressed sensing based non-uniform sampler (NUS). By sampling signals non-uniformly, the average sample rate can be more than a magnitude lower than the Nyquist rate, provided that these signals have a relatively low information content as measured by the sparsity of their spectrum. The hardware design combines a wideband Indium-Phosphide (InP) heterojunction bipolar transistor (HBT) sample-and-hold with a commercial off-the-shelf (COTS) analog-to-digital converter (ADC) to digitize an 800 MHz to 2 GHz band (having 100 MHz of non-contiguous spectral content) at an average sample rate of 236 Msps. Signal reconstruction is performed via a non-linear compressed sensing algorithm, and an efficient GPU implementation is discussed. Measured bit-error-rate (BER) data for a GSM channel is presented, and comparisons to a conventional wideband 4.4 Gsps ADC are made. Index Terms—Non-uniform sampler, compressed sensing, wideband ADC, indium-phosphide HBT, sample-and-hold. I.