Abstract not found

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, Hilbert-space formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shift-invariant” 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 (anti-aliasing) prefilters that are not necessarily ideal low-pass. 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—Band-limited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,

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 time-varying nature of the process. This need has given rise to the idea of instantaneous frequency. The instantaneous frequency (IF) of a signal is a parame-ter 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 mathe-matical 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 group-delay, analytic signal, and bandwidth-time (BT) product are explored, as well as the relationship with time-frequency distribu-tions. 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. I.

Abstract—An ultra-wide 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 maximum-energy-capture 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 maximum-likelihood detector. Energy capture as a function of the number of single-path signal correlators used in UWB SRake receiver provides a complexity versus performance tradeoff. Bit-error-probability performance of a UWB SRake receiver, based on measured channels, is given as a function of signal-to-noise ratio and the number of correlators implemented in the receiver. Index Terms—All Rake receiver (ARake), bit-error probability (BEP), energy capture, propagation channel, selective Rake (SRake) receiver, spread-spectrum, ultra-wide bandwidth (UWB). I.

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 q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as two-dimensional subspaces. The parameter space contains a "time-frequency-scale volume ", and thus encompasses both the short-time 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: shear-in-time (obtained through convolution with a q-chirp) and shearin -frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform which we call the "q-chirplet transform", or simply the "chiplet transform". ...

The feedback capacity of additive stationary Gaussian noise channels 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 first-order autoregressive moving average noise spectrum, this variational characterization yields a closed-form expression for the feedback capacity. In particular, this result shows that the celebrated Schalkwijk–Kailath coding scheme achieves the feedback capacity for the first-order autoregressive moving average Gaussian channel, positively answering a long-standing open problem studied by Butman, Schalkwijk–Tiernan, Wolfowitz, Ozarow, Ordentlich, Yang–Kavčić– Tatikonda, and others. More generally, it is shown that a k-dimensional generalization of the Schalkwijk– Kailath coding scheme achieves the feedback capacity for any autoregressive moving average noise spectrum of order k. Simply put, the optimal transmitter iteratively refines the receiver’s knowledge of the intended message. I.

A canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent of the true channel parameters. The basis captures the essential degrees of freedom in the received signal using discrete multipath delays, Doppler shifts, and directions of arrival (DOA). The canonical representation provides a robust representation of the propagation dynamics and eliminates the need for estimating delay, Doppler and DOA parameters of different multipaths. Furthermore, it furnishes a natural framework for designing low-complexity space-time receivers. Single-user receivers based on the canonical channel representation are developed and analyzed. It is demonstrated that the resulting canonical space-time receivers deliver near-optimal performance at substantially reduced complexity compared to existing designs. Keywords: Diversity methods, Time-varying channels, Antenna Arrays, Multipath, RAKE receiver. 0 1 Introduction The use of antenna ...

A novel definition of a spread-spectrum system as a communications system in which the Fourier bandwidth is much greater than the Shannon bandwith (the number of dimensions of signal space used per second) is proposed. Six different communication systems are analyzed in terms of this definition. It is shown that there is a fundamental difference between the bandwidth expansion due to coding and due to "spectrum spreading". It is further shown that spectrum spreading plays no role in increasing the capacity of the system, but can perform other useful roles such as providing low probability of interception of the signal, good electromagnetic compatibility, and a multiple-access capability. I. INTRODUCTION The main purpose of this paper is to consider spreadspectrum systems from the fundamental viewpoint of Shannon's information theory [1]. To do this requires that we carefully define what we mean by a spread-spectrum system. This is done in Section II in which we give a rather unconve...

In pattern recognition tasks, it is often convenient to alter the sampling rate of the signal, either to convert to a more appropriate scale/resolution, or to produce an image pyramid and perform the task in an multiresolution fashion. In many of these applications, the mse criteria optimized by the Shannon Sampling Theorem is not appropriate, and other sampling strategies should be considered. In this paper we present a new Critical Sampling Theorem, and extends the results to the case where the connection between several parts of the signal (i.e., the homotopy of the set) is of primary interest. The results are presented for binary signals in an hexagonal grid. Extension for square grids with some specific cases of connectivity criteria is also presented. The results show that it is possible to preserve homotopy while using a sampling density 3 to 4 times smaller than required by previous results. This can be used to reduce the sample density or to improve the detail preservation, a...

In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.