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36
Sampling signals with finite rate of innovation
 IEEE Transactions on Signal Processing
, 2002
"... Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials ..."
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Cited by 350 (67 self)
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Abstract—Consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic “bandlimited and sinc kernel ” case. In particular, we show how to sample and reconstruct periodic and finitelength streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinitelength signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and errorcorrection coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems. Index Terms—Analogtodigital conversion, annihilating filters, generalized sampling, nonbandlimited signals, nonuniform splines, piecewise polynomials, poisson processes, sampling. I.
Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
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Cited by 104 (16 self)
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
Time Delay and Spatial Signature Estimation Using Known Asynchronous Signals
 IEEE Trans. Signal Processing
, 1997
"... This paper considers the problem of estimating the parameters of known signals received asynchronously by an array of antennas. The parameters of primary interest are the time delays of the signals and their spatial signatures at the array. Estimates of the signal directions of arrival are also cons ..."
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Cited by 48 (11 self)
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This paper considers the problem of estimating the parameters of known signals received asynchronously by an array of antennas. The parameters of primary interest are the time delays of the signals and their spatial signatures at the array. Estimates of the signal directions of arrival are also considered, but are of secondary importance in this work. Maximum likelihood algorithms and more computationally efficient approximations are developed for both the case where all received signals are identical (the channel estimation/overlapping echo problem), and where they are all distinct. Conditions are also derived under which the standard matched filter approach yields consistent and statistically efficient parameter estimates. The issue of solution uniqueness is addressed, and in particular an upper bound on the number of signals whose parameters may be uniquely identified is derived for a number of different cases. Typically, the bound is far greater than the number of sensors, and is l...
Timedelay estimation from lowrate samples: a union of subspaces approach
 IEEE TRANS. SIGNAL PROCESS
, 2010
"... Timedelay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In ..."
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Cited by 33 (19 self)
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Timedelay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In this paper, we develop a unified approach to time delay estimation from lowrate samples. This problem can be formulated in the broader context of sampling over an infinite union of subspaces. Although sampling over unions of subspaces has been receiving growing interest, previous results either focus on unions of finitedimensional subspaces, or finite unions. The framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses, and allows for a variety of different sampling methods. The sampling rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. This result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces. By properly manipulating the lowrate samples, we show that the time delays can be recovered using the wellknown ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation, we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.
Detection of Signals by Information Theoretic Criteria: General Asymptotic Performance Analysis
, 2002
"... Detecting the number of sources is a well known and a well investigated problem. In this problem, the number of sources impinging on an array of sensors is to be estimated. The common approach for solving this problem is to use an information theoretic criterion like the Minimum Description Length ( ..."
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Cited by 26 (4 self)
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Detecting the number of sources is a well known and a well investigated problem. In this problem, the number of sources impinging on an array of sensors is to be estimated. The common approach for solving this problem is to use an information theoretic criterion like the Minimum Description Length (MDL), or the Akaike Information Criterion. Although gaining much popularity and being used in a variety of problems, the performance of information theoretic criteria based estimators for the unknown number of sources has not been su#ciently studied, yet. In the context of array processing, the performance of such estimators were analyzed only for the special case of Gaussian sources where no prior knowledge of the array structure, if given, is used. Based on the theory of misspecified models, this paper presents a general asymptotic analysis of the performance of any information theoretic criterion based estimator, and especially of the MDL estimator. In particular, the performance of the MDL estimator which assumes Gaussian sources and structured array when applied to Gaussian sources is analyzed. Also, it is shown that the performance of a certain MDL estimator is not very sensitive to the actual distribution of the source signals. However, appropriate use of prior knowledge about the array geometry can lead to significant improvement in the performance of the MDL estimator. Simulation results show good fit between the empirical and the theoretical results.
Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach
, 2010
"... Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In ..."
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Cited by 24 (22 self)
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Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Previous methods for time delay recovery either operate on the analog received signal, or require sampling at the Nyquist rate of the transmitted pulse. In this paper, we develop a unified approach to time delay estimation from low rate samples. This problem can be formulated in the broader context of sampling over an infinite union of subspaces. Although sampling over unions of subspaces has been receiving growing interest, previous results either focus on unions of finitedimensional subspaces, or finite unions. The framework we develop here leads to perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses, and allows for a variety of different sampling methods. The sampling rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. This result can be viewed as a sampling theorem over an infinite union of infinite dimensional subspaces. By properly manipulating the lowrate samples, we show that the time delays can be recovered using the wellknown ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation, we develop sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.
Performance Bounds and Design Criteria for Estimating Finite Rate of Innovation Signals
"... Abstract—In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainabl ..."
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Cited by 15 (6 self)
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Abstract—In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific samplingapproach. Essential differences between the noisy and noisefree cases arise from this analysis. In particular, we identify settings in which noisefree recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related toaspecific typeofstructure,which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels for linear reconstruction, based on a generalization of the Karhunen–Loève transform. The results are illustrated for several types of timedelay estimation problems. Index Terms—Cramér–Rao bound (CRB), finite rate of innovation (FRI), sampling, timedelay estimation, union of subspaces. I.
Low Rate Sampling Schemes for Time Delay Estimation
, 2009
"... Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Various approaches have been proposed in the literature to identify time delays introduced by multipath environments. However, these methods either operate ..."
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Cited by 8 (8 self)
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Time delay estimation arises in many applications in which a multipath medium has to be identified from pulses transmitted through the channel. Various approaches have been proposed in the literature to identify time delays introduced by multipath environments. However, these methods either operate on the analog received signal, or require high sampling rates in order to achieve reasonable time resolution. In this paper, our goal is to develop a unified approach to time delay estimation from low rate samples of the output of a multipath channel. Our methods result in perfect recovery of the multipath delays from samples of the channel output at the lowest possible rate, even in the presence of overlapping transmitted pulses. This rate depends only on the number of multipath components and the transmission rate, but not on the bandwidth of the probing signal. In addition, our development allows for a variety of different sampling methods. By properly manipulating the lowrate samples, we show that the time delays can be recovered using the wellknown ESPRIT algorithm. Combining results from sampling theory with those obtained in the context of direction of arrival estimation methods, we develop necessary and sufficient conditions on the transmitted pulse and the sampling functions in order to ensure perfect recovery of the channel parameters at the minimal possible rate.
Parametric Synchronization and Spatial Signature Estimation for Multiple Known CoChannel Signals
 In Proc. 29th Asilomar Conf. on Signals, Systems, and Computers
, 1995
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