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100
Splines: A Perfect Fit for Signal/Image Processing
 IEEE SIGNAL PROCESSING MAGAZINE
, 1999
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Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 340 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Estimating and Interpreting the Instantaneous Frequency of a Signal  Part 1: Fundamentals
 PROCEEDINGS OF THE IEEE
, 1992
"... The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, t ..."
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Cited by 263 (9 self)
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The frequency of a sinusoidal signal is a well defined quantity. However, often in practice, signals are not truly sinusoidal, or even aggregates of sinusoidal components. Nonstationary signals in particular do not lend themselves well to decomposition into sinusoidal components. For such signals, the notion of frequency loses its effectiveness, and one needs to use a parameter which accounts for the timevarying nature of the process. This need has given rise to the idea of instantaneous frequency. The instantaneous frequency (IF) of a signal is a parameter which is often of significant practical importance. In many situations such as seismic, radar, sonar, communications, and biomedical applications, the IF is a good descriptor of some physical phenomenon. This paper discusses the concept of instantaneous frequency, its definitions, and the correspondence between the various mathematical models formulated for representation of IF. The paper also considers the extent to which the IF corresponds to our intuitive expectation of reality. A historical review of the successive attempts to define the IF is presented. Then the relationships between the IF and the groupdelay, analytic signal, and bandwidthtime (BT) product are explored, as well as the relationship with timefrequency distributions. Finally, the notions of monocomponent and multicomponent signals, and instantaneous bandwidth are discussed. It is shown that all these notions are well described in the context of the theory presented.
Characterization of ultrawide bandwidth wireless indoor channels: a communicationtheoretic view
 IEEE Journal on Selected Areas in Communications
, 2002
"... Abstract—An ultrawide bandwidth (UWB) signal propagation experiment is performed in a typical modern laboratory/office building. The bandwidth of the signal used in this experiment is in excess of 1 GHz, which results in a differential path delay resolution of less than a nanosecond, without specia ..."
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Cited by 120 (12 self)
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Abstract—An ultrawide bandwidth (UWB) signal propagation experiment is performed in a typical modern laboratory/office building. The bandwidth of the signal used in this experiment is in excess of 1 GHz, which results in a differential path delay resolution of less than a nanosecond, without special processing. Based on the experimental results, a characterization of the propagation channel from a communications theoretic view point is described, and its implications for the design of a UWB radio receiver are presented. Robustness of the UWB signal to multipath fading is quantified through histograms and cumulative distributions. The all Rake (ARake) receiver and maximumenergycapture selective Rake (SRake) receiver are introduced. The ARake receiver serves as the best case (bench mark) for Rake receiver design and lower bounds the performance degradation caused by multipath. Multipath components of measured waveforms are detected using a maximumlikelihood detector. Energy capture as a function of the number of singlepath signal correlators used in UWB SRake receiver provides a complexity versus performance tradeoff. Biterrorprobability performance of a UWB SRake receiver, based on measured channels, is given as a function of signaltonoise ratio and the number of correlators implemented in the receiver. Index Terms—All Rake receiver (ARake), biterror probability (BEP), energy capture, propagation channel, selective Rake (SRake) receiver, spreadspectrum, ultrawide bandwidth (UWB). I.
Compressed Channel Sensing: A New Approach to Estimating Sparse Multipath Channels
"... Highrate data communication over a multipath wireless channel often requires that the channel response be known at the receiver. Trainingbased methods, which probe the channel in time, frequency, and space with known signals and reconstruct the channel response from the output signals, are most co ..."
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Cited by 84 (9 self)
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Highrate data communication over a multipath wireless channel often requires that the channel response be known at the receiver. Trainingbased methods, which probe the channel in time, frequency, and space with known signals and reconstruct the channel response from the output signals, are most commonly used to accomplish this task. Traditional trainingbased channel estimation methods, typically comprising of linear reconstruction techniques, are known to be optimal for rich multipath channels. However, physical arguments and growing experimental evidence suggest that many wireless channels encountered in practice tend to exhibit a sparse multipath structure that gets pronounced as the signal space dimension gets large (e.g., due to large bandwidth or large number of antennas). In this paper, we formalize the notion of multipath sparsity and present a new approach to estimating sparse (or effectively sparse) multipath channels that is based on some of the recent advances in the theory of compressed sensing. In particular, it is shown in the paper that the proposed approach, which is termed as compressed channel sensing, can potentially achieve a target reconstruction error using far less energy and, in many instances, latency and bandwidth than that dictated by the traditional leastsquaresbased training methods.
The Chirplet Transform: Physical Considerations
, 1995
"... We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timef ..."
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Cited by 49 (3 self)
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We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call qchirps for short), giving rise to a parameter space that includes both the timefrequency plane and the timescale plane as twodimensional subspaces. The parameter space contains a "timefrequencyscale volume ", and thus encompasses both the shorttime Fourier transform (as a slice along the time and frequency axes), and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shearintime (obtained through convolution with a qchirp) and shearin frequency (obtained through multiplication by a qchirp). Signals in this multidimensional space can be obtained by a new transform which we call the "qchirplet transform", or simply the "chiplet transform". ...
Fifty Years of Shannon Theory
, 1998
"... A brief chronicle is given of the historical development of the central problems in the theory of fundamental limits of data compression and reliable communication. ..."
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Cited by 49 (1 self)
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A brief chronicle is given of the historical development of the central problems in the theory of fundamental limits of data compression and reliable communication.
Feedback capacity of stationary Gaussian channels
"... The capacity of stationary additive Gaussian noise channels with feedback is characterized as the solution to a variational problem. Toward this end, it is proved that the optimal feedback coding scheme is stationary. When specialized to the firstorder autoregressive movingaverage noise spectrum, ..."
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Cited by 47 (10 self)
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The capacity of stationary additive Gaussian noise channels with feedback is characterized as the solution to a variational problem. Toward this end, it is proved that the optimal feedback coding scheme is stationary. When specialized to the firstorder autoregressive movingaverage noise spectrum, this variational characterization yields a closedform expression for the feedback capacity. In particular, this result shows that the celebrated Schalkwijk–Kailath coding scheme achieves the feedback capacity for the firstorder autoregressive movingaverage Gaussian channel, resolving a longstanding open problem studied by Butman, Schalkwijk– Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang–Kavčić–Tatikonda, and others. 1 Introduction and
ON THE IDENTIFICATION OF PARAMETRIC UNDERSPREAD LINEAR SYSTEMS
"... Identification of timevarying linear systems, which introduce both timeshifts (delays) and frequencyshifts (Dopplershifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as timevarying linear sy ..."
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Cited by 29 (10 self)
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Identification of timevarying linear systems, which introduce both timeshifts (delays) and frequencyshifts (Dopplershifts), is a central task in many engineering applications. This paper studies the problem of identification of underspread linear systems (ULSs), defined as timevarying linear systems whose responses lie within a unitarea region in the delay–Doppler space, by probing them with a known input signal. The main contribution of the paper is that it characterizes conditions on the bandwidth and temporal support of the input signal that ensure identification of ULSs described by a finite set of delays and Dopplershifts, and referred to as parametric ULSs, from single observations. In particular, the paper establishes that sufficientlyunderspread parametric linear systems are identifiable as long as the time–bandwidth product of the input signal is proportional to the square of the total number of delay–Doppler pairs in the system. In addition, the paper describes a procedure that enables identification of parametric ULSs from an input train of pulses in polynomial time by exploiting recent results on subNyquist sampling for time delay estimation and classical results on recovery of frequencies from a sum of complex exponentials. 1.
Approaches to informationtheoretic analysis of neural activity
 Biol. Theory
, 2006
"... Understanding how neurons represent, process, and manipulate information is one of the main goals of neuroscience. These issues are fundamentally abstract, and information theory plays a key role in formalizing and addressing them. However, application of information theory to experimental data i ..."
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Cited by 22 (0 self)
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Understanding how neurons represent, process, and manipulate information is one of the main goals of neuroscience. These issues are fundamentally abstract, and information theory plays a key role in formalizing and addressing them. However, application of information theory to experimental data is fraught with many challenges. Meeting these challenges has led to a variety of innovative analytical techniques, with complementary domains of applicability, assumptions, and goals.