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625 results in Statistics for physical sciences and engineering

Page 12 of 32

  • 7 - Smoothness priors

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 237-282

  • Summary

    Until now we have assumed that the signal x being sampled lies in a known subspace. We saw that under an appropriate direct-sum condition, perfect recovery from the samples is possible. We now treat a less restrictive formulation of the sampling problem in which the only prior knowledge on the signal is that it is smooth in some sense. In our development, we consider two models of smoothness. In the first, the weighted norm ∥Lx∥ of x is bounded where L is an appropriate weighting operator, such as a differential operator. In the SI setting this corresponds to the assumption that the L2 signal-norm at the output of a continuous-time filter L(ω) with x(t) as its input is bounded. An alternative formulation is to assume that x(t) is a wide-sense stationary (WSS) process with known spectrum. We will see that under appropriate objective functions, these two priors lead to the same recovery methods with the weighting function L chosen as the whitening filter of the process x(t). Relevant background on WSS signals and whitening is summarized in Appendix B.

    Unlike subspace priors, a one-to-one correspondence between smooth signals and their samples does not exist, since smoothness is a far less restrictive constraint than confining the signal to a subspace. Perfect recovery, or even error-norm minimization, is therefore generally impossible. Instead, we focus on LS and minimax approaches for designing the reconstruction system in both the unconstrained and constrained recovery scenarios.

  • 3 - Fourier analysis

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 67-93

  • Summary

    In the previous chapter we considered expansions of signals in appropriate bases, which can be interpreted as sampling and reconstruction. However, the expansions we discussed were written in abstract form so that it is not immediately obvious how to translate them to concrete sampling methods and practical reconstruction algorithms. From an engineering perspective, our interest is in sampling theories that can be implemented in practice using standard analog components such as filters and modulators. In the next chapter we move away from the abstract formulation and concentrate on concrete signal classes for which the representations we are interested in can be implemented efficiently using common engineering building blocks. The advantage of these settings is that much of the analysis can be carried out efficiently and in a simple manner in the Fourier domain. Therefore, a large part of the tools and methods used in the rest of the book for developing sampling theorems will rely on Fourier analysis.

    In this chapter we provide a self-contained presentation of the results needed in the context of linear time-invariant systems and Fourier representations. We review both the continuous-time and discrete-time Fourier representations, and discuss the relationship between the two in the context of sampled signals. There are many excellent textbooks on Fourier analysis; our goal here is to summarize the properties needed in our derivations and the key relations we exploit between continuous-time and discrete-time transforms.

  • 14 - Multiband sampling

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 569-641

  • Summary

    In this chapter we consider sampling of multiband signals, namely, signals whose Fourier transform comprises a small number of bands, spread over a possibly wide frequency range. We begin by discussing the case in which the carrier frequencies, or band positions, are known. We review both classical methods and techniques based on interleaved ADC structures to sample such signals at a rate proportional to the actual band occupancy, and not to the high Nyquist rate associated with the highest frequency. The more challenging scenario in which the carrier frequencies are unknown, resulting in a union of subspace model, is treated next. Combining sampling approaches for the case in which the carriers are known, with methods for compressed sampling of analog signals developed in the previous chapter, leads to several sub-Nyquist sampling techniques for multiband signals with unknown band positions. Along with developing the theoretical concepts, we address practical constraints and present an example of a sub-Nyquist hardware realization for this class of signals.

    Sampling of multiband signals

    Consider the scenario of a multiband input x(t), which has sparse spectra, such that its CTFT X(ω) is supported on N frequency intervals, or bands, with individual widths not exceeding B. In addition, the largest frequency component is below ωmax. Figure 14.1 illustrates a typical multiband spectrum. Multiband models are common in communication channels, when a receiver intercepts several RF transmissions, each modulated onto a different (possibly high) carrier frequency.

  • 8 - Nonlinear sampling

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 283-322

  • Summary

    In this chapter we depart from the linear sampling model treated until now and consider recovering signals that lie in a subspace, from their nonlinear samples. Surprisingly, we will see that many types of nonlinearities that are encountered in practice can be completely compensated for without having to increase the sampling rate. Thus, even though typically nonlinearities result in bandwidth expansion, using the algorithms discussed in this chapter we will be able to recover an input from its nonlinear samples at the rate associated with the input space.

    Nonlinearities appear in a variety of setups and applications of digital signal processing, including power electronics and wireless communications where an amplifier introduces nonlinear distortions to its input signal, radiometric photography, and charge coupled device (CCD) image sensors. For example, in radiometric photography, the radiometric response M(x) of a camera relates the radiance x at the input of a sensor to intensity y = M(x) at its output. This mapping is monotonically increasing (which also means that it is invertible), but may exhibit nonlinearity. The intensity signal is sampled by an ADC, and the radiance then needs to be estimated from the samples. Similar nonlinear distortions of amplitude occur in many CCD image sensors, owing to excessive light intensity which causes saturation. More generally, amplifier saturation introduces nonlinear distortion to the input signal, and occurs in many applications such as satellite communications, and power amplifiers in hand-held devices.

  • 9 - Resampling

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 323-367

  • Summary

    In previous chapters we considered the problem of reconstructing a continuous-time signal from its discrete set of samples. Consequently, the recovery systems we designed were of hybrid nature with discrete-time input and continuous-time output. In this chapter we demonstrate that sampling theory also plays a crucial role in the design of fully discrete-time algorithms. In particular, we treat applications in which one would like to obtain a sequence of samples of a signal from a different set of samples of the same signal. In such settings, both the input and the output are digital.

    One example is sampling rate conversion. Consider, for instance, digital audio files, which nowadays populate personal computers and media devices in enormous amounts. The sources of these files are diverse, and consequently a very common situation is that files are recorded at different sampling rates. When an audio file is played, the samples representing the recording are passed through a DAC. The analog output is then amplified and fed into a speaker. A typical DAC supports only a small number of sampling (and reconstruction) rates. If the audio samples do not correspond to one of these rates, then a preprocessing step is required in order to change their rate. This operation is performed by appropriate digital signal processing and is termed sampling rate conversion. Rate conversion is always required if two audio files with different sampling rates are played simultaneously, even when both sampling rates are supported by the DAC, since a standard DAC operates at a single rate.

  • 10 - Union of subspaces

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 368-389

  • Summary

    Our primary focus in the book thus far has been on sampling and recovery of signals that lie in a single subspace. Subspace models lead to powerful sampling theorems, resulting in perfect recovery of the signal from its linear and nonlinear samples under very broad conditions. An appealing feature of the recovery algorithms is that they can often be implemented using simple digital and analog filtering, generalizing the Shannon-Nyquist theorem to a much broader set of input classes. The subspace viewpoint also leads to nice geometrical interpretations of classic and new sampling theorems in terms of orthogonal and oblique projections.

    To a large extent, the notion that any possible vector in a given subspace is a valid signal has driven the explosion in the dimensionality of the data we must sample and process. Despite the simplicity and geometric appeal of subspace modeling, there are many classes of signals whose structure is not captured well by a single subspace. This is particularly true in applications in which the sampler does not have full knowledge of the received signal. In response to these challenges, there has been a surge of interest in recent years, across many fields, in a variety of low-dimensional signal representations that quantify the notion that the number of degrees of freedom in high-dimensional signals is often quite small compared with their ambient dimension. Low-dimensional modeling has been used extensively in machine learning, parameter estimation, and detection techniques.

  • 15 - Finite rate of innovation sampling

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 642-744

  • Summary

    In previous chapters we have seen that the UoS model can pave the way to sub-Nyquist sampling of certain categories of structured analog signals. In this chapter we consider an alternative model that relies on parametric representations: finite rate of innovation (FRI) signals [105]. This class corresponds to families of functions defined by a finite number of parameters per unit time, a quantity referred to as the rate of innovation. More specifically, a FRI signal x(t) is characterized by the fact that any finite duration segment of length r is completely determined by no more than k parameters. In this case, the function x(t) is said to have a local rate of innovation equal to k/r. The FRI viewpoint complements the UoS framework: a signal may lie in a UoS and have FRI; however, not all FRI signals can be described by a UoS model and vice versa, as we will show in examples below.

    Interest in this class of signals emerges from the observation that several commonly encountered FRI signals can be perfectly recovered from samples taken at their rate of innovation. The advantage of this result is self-evident: FRI signals need not be bandlimited, and even if they are, the Nyquist frequency may be much higher than their rate of innovation. Thus, by using FRI techniques, the sampling rate required for perfect reconstruction can be reduced substantially. However, exploiting these capabilities requires careful design of the sampling mechanism and of the digital postprocessing.

  • 6 - Subspace priors

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 177-236

  • Summary

    We now begin applying the machinery developed in previous chapters to generalized sampling. In this chapter we focus on subspace priors and linear sampling. Our goal is to recover a signal x from its samples, when we know a priori that x lies in some subspace A of a Hilbert space H. For example, x may be a bandlimited signal, a piecewise polynomial signal, or a pulse amplitude modulation (PAM) signal with known pulse shape. We focus on SI subspaces, which were treated in the previous chapter, and uniform sampling grids. In this setting, we will show that sampling and reconstruction can be performed by filtering operations. However, all the results herein hold in more abstract Hilbert space settings including finite-dimensional spaces and spaces that are not SI, and for sampling grids that are not necessarily uniform.

    Sampling and reconstruction processes

    Sampling setups

    Although our focus in this chapter is on subspace priors, many of the essential ideas will also be used in Chapter 7 when treating smoothness and stochastic priors. Therefore, in the next section we provide an overview of the different setups and criteria that will be used in both chapters. We then focus on the subspace setting and defer the discussion on other signal classes to the next chapter. In Chapter 8 we revisit subspace priors and generalize the sampling mechanism to include nonlinear sampling. Subsequent chapters are devoted to nonlinear signal priors taking the form of unions of subspaces.

  • 12 - Sampling over finite unions

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp 472-533

  • Summary

    In the previous chapter we considered sampling of sparse finite-dimensional vectors. As we have seen, this setting can be viewed as a special case of a union model, where the subspaces Ui comprising the union are a direct sum of k one-dimensional subspaces, spanned by columns of the identity matrix. Thus, for each Ui = U, we have that

    where ∈j, 1 ≤ j ≤ N is the subspace spanned by the jth column of the N × N identity matrix, and ij, 1 ≤ j ≤ k are indices between 1 and N.

    Viewing the sparsity model in the form of a UoS such as (12.1) leads to an immediate extension which allows formuchmore general signal classes. Specifically, we may replace each of the one-dimensional subspaces ∈j of ℝN by a subspace Aj. These subspaces have arbitrary dimension, and are described over an arbitrary Hilbert space H. In particular, they can represent subspaces of analog signals. In this chapter we focus on finite- dimensional unions of this form, on methods for recovering signals that lie in the union from few measurements, and on performance guarantees. We also consider learning the subspaces from training data, when the possible subspaces are not known in advance. Finally, we treat the more difficult scenario of subspace learning from compressed data, leading to an extension of compressed sensing referred to as blind compressed sensing.

  • Preface

  • Yonina C. Eldar, Weizmann Institute of Science, Israel
  • Book:
    Sampling Theory
    Published online:
    05 August 2014
    Print publication:
    09 April 2015, pp xvii-xxiv

  • Summary

    Digital signal processing (DSP) is one of the most prominent areas in engineering, including subfields such as speech and image processing, statistical data processing, spectral estimation, biomedical applications, and many others. As the name suggests, the goal is to perform various signal processing tasks (e.g., filtering, amplification, and more) in the digital domain where design, verification, and implementation are considerably simplified compared with analog signal processing. DSP is the basis of many areas of technology, and is one of the most powerful technologies that have shaped science and engineering in the past century.

    In order to represent and process analog signals on a computer the signals must be sampled with an analog-to-digital converter (ADC) which converts the signal to a sequence of numbers. After processing, the samples are converted back to the analog domain via a digital-to-analog converter (DAC). Consequently, the theory and practice of sampling are at the heart of DSP. Evidently, any technology advances in ADCs and DACs have a huge impact on a vast array of applications.

    The goal of this book is to provide a comprehensive treatment of the theory and practice of sampling from an engineering perspective. Although there are many excellent mathematical textbooks on signal expansions and harmonic analysis, our aim is to present an up-to-date engineering textbook on the topic by combining the fundamental mathematical underpinnings of sampling with practical engineering applications and principles.

  • Preface

  • Bruce Hajek, University of Illinois, Urbana-Champaign
  • Book:
    Random Processes for Engineers
    Published online:
    11 May 2018
    Print publication:
    12 March 2015, pp xi-xiv

  • Summary

    From an applications viewpoint, the main reason to study the subject of this book is to help deal with the complexity of describing random, time-varying functions. A random variable can be interpreted as the result of a single measurement. The distribution of a single random variable is fairly simple to describe. It is completely specified by the cumulative distribution function F(x), a function of one variable. It is relatively easy to approximately represent a cumulative distribution function on a computer. The joint distribution of several random variables is much more complex, for in general it is described by a joint cumulative probability distribution function, F(x1, x2, …, xn), which is much more complicated than n functions of one variable. A random process, for example a model of time-varying fading in a communication channel, involves many, possibly infinitely many (one for each time instant t within an observation interval) random variables. Woe the complexity!

    This book helps prepare the reader to understand and use the following methods for dealing with the complexity of random processes:

  • • Work with moments, such as means and covariances.

  • • Use extensively processes with special properties. Most notably, Gaussian processes are characterized entirely by means and covariances, Markov processes are characterized by one-step transition probabilities or transition rates, and initial distributions. Independent increment processes are characterized by the distributions of single increments.

  • • Appeal to models or approximations based on limit theorems for reduced complexity descriptions, especially in connection with averages of independent, identically distributed random variables. The law of large numbers tells us, in a certain sense, that a probability distribution can be characterized by its mean alone. The central limit theorem similarly tells us that a probability distribution can be characterized by its mean and variance. These limit theorems are analogous to, and in fact examples of, perhaps the most powerful tool ever discovered for dealing with the complexity of functions: Taylor's theorem, in which a function in a small interval can be approximated using its value and a small number of derivatives at a single point.

  • […]

Page 12 of 32