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6385 results in Communications and signal processing

Page 127 of 320

  • 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.

  • […]

  • 4 - Digital modulation schemes

  • Svilen Dimitrov, Harald Haas, University of Edinburgh
  • Book:
    Principles of LED Light Communications
    Published online:
    05 March 2015
    Print publication:
    12 March 2015, pp 72-94

  • Summary

    Introduction

    The data transmission in optical wireless communication (OWC) with incoherent light sources is realized through intensity modulation and direct detection (IM/DD). For this purpose, the transmitted signal needs to be real-valued and non-negative. In practice, this is achieved by single-carrier modulation techniques, such as multi-level pulse position modulation (M-PPM) and multi-level pulse amplitude modulation (M-PAM), and through multi-carrier modulation such as multi-level quadrature amplitude modulation (M-QAM) optical orthogonal frequency division multiplexing (O-OFDM). Conventionally, the average optical power is defined as the first moment of the transmitted signal, while the average electrical power is defined as the second moment of the transmitted signal. In practice, the dynamic range can be linearized through pre-distortion only between levels of minimum and maximum radiated optical power. In addition, eye safety regulations [44] and/or design requirements also impose an average optical power constraint. Because of these constraints, there is a fixed relation between average electrical power and the average optical power of the single-carrier and multi-carrier signals which varies with the change in the biasing setup, i.e. a combination of direct current (DC) bias and signal variance. The optical-to-electrical (O/E) conversion of the optical signals is investigated in this chapter. In this study, the electrical energy consumption of the OWC system is considered, and therefore the average electrical power carries the information [67]. Here, the received electrical signal-to-noise ratio (SNR) is presented as a function of the channel equalization penalty, the DC-bias penalty, and the non-linear distortion parameters. The analytical framework is verified by means of a Monte Carlo bit-error ratio (BER) simulation [131, 133, 134].

    Optical signals

    As opposed to a radio frequency (RF) system, where the data-carrying signal modulates the complex-valued bipolar electric field radiated by an antenna, in an OWC system the signal modulates the intensity of the optical emitter, and therefore it needs to be real-valued and unipolar non-negative. Since light emitting diodes (LEDs) are incoherent light sources, it is difficult to collect signal power in a single electromagnetic mode and to provide a stable carrier in an indoor OWC scenario.

  • 2 - Optical wireless communication

  • Svilen Dimitrov, Harald Haas, University of Edinburgh
  • Book:
    Principles of LED Light Communications
    Published online:
    05 March 2015
    Print publication:
    12 March 2015, pp 12-56

  • Summary

    Introduction

    In optical wireless communication (OWC), the light intensity of a light emitting diode (LED) is modulated by a message signal. After propagating through the optical wireless channel, the light message is detected by a photodiode (PD). Key characteristics of the optical transmitter (Tx) and receiver (Rx) include their optical spectral response, electrical modulation bandwidth, radiation/detection patterns, optical power output of the LED, the photosensitive area, and the noise figure of the PD. The optical wireless channel has been shown to be a linear, time-invariant, memoryless system with an impulse response of a finite duration [21]. The primary characteristic of the channel is the path loss. An accurate representation of the light distribution in an indoor setup can be obtained by means of a Monte Carlo ray-tracing (MCRT) simulation. Channel modeling confirms that the path loss in dB is linear over logarithmic distance, and it ranges between 27 dB and 80 dB for line-of-sight (LOS) and non-line-of-sight (NLOS) communication scenarios in the considered setup [16, 66]. At high data rates, where the signal bandwidth exceeds the channel coherence bandwidth, the channel can be characterized as a frequency selective channel due to dispersion [67]. There is no fast fading, but the signal undergoes slow fading. Ray-tracing simulations show that slow fading can be modeled as a log-normally distributed random variable [16]. The delay spread of dispersive optical wireless channels can be accurately modeled by a rapidly decaying exponential impulse response function. Root-mean-squared (RMS) delay spreads between 1.3 ns and 12 ns are reported for LOS links, whereas RMS delay spreads between 7 ns and 13 ns are reported for NLOS links [21]. In order to counter the channel effect, single-carrier pulse modulation techniques, e.g. multi-level pulse position modulation (M-PPM) and multi-level pulse amplitude modulation (M-PAM), employ a linear feed-forward equalizer (FFE) or a non-linear decision-feedback equalizer (DFE) with zero forcing (ZF) or minimum mean-squared error (MMSE) criteria at the expense of increased computational complexity.

  • 8 - Random processes in linear systems and spectral analysis

  • Bruce Hajek, University of Illinois, Urbana-Champaign
  • Book:
    Random Processes for Engineers
    Published online:
    11 May 2018
    Print publication:
    12 March 2015, pp 248-279

  • Summary

    Random processes can be passed through linear systems in much the same way as deterministic signals can. A time-invariant linear system is described in the time domain by an impulse response function, and in the frequency domain by the Fourier transform of the impulse response function. In a sense we shall see that Fourier transforms provide a diagonalization of WSS random processes, just as the Karhunen–Loève expansion allows for the diagonalization of a random process defined on a finite interval. While a m.s. continuous random process on a finite interval has a finite average energy, a WSS random process has a finite mean average energy per unit time, called the power.

    Nearly all the definitions and results of this chapter can be carried through in either discrete time or continuous time. The set of frequencies relevant for continuous-time random processes is all of ℝ, while the set of frequencies relevant for discrete-time random processes is the interval [−π, π]. For ease of notation we shall primarily concentrate on continuous-time processes and systems in the first two sections, and give the corresponding definition for discrete time in the third section.

    Representations of baseband random processes and narrowband random processes are discussed in Sections 8.4 and 8.5. Roughly speaking, baseband random processes are those which have power only in low frequencies. A baseband random process can be recovered from samples taken at a sampling frequency that is at least twice as large as the largest frequency component of the process. Thus, operations and statistical calculations for a continuous-time baseband process can be reduced to considerations for the discrete-time sampled process. Roughly speaking, narrowband random processes are those processes which have power only in a band (i.e. interval) of frequencies. A narrowband random process can be represented as a baseband random process that is modulated by a deterministic sinusoid. Complex random processes naturally arise as baseband equivalent processes for real-valued narrowband random processes. A related discussion of complex random processes is given in the last section of the chapter.

  • 7 - Throughput of cellular OWC networks

  • Svilen Dimitrov, Harald Haas, University of Edinburgh
  • Book:
    Principles of LED Light Communications
    Published online:
    05 March 2015
    Print publication:
    12 March 2015, pp 151-182

  • Summary

    Introduction

    An optical wireless communication (OWC) multiple access network can deliver wireless data to multiple mobile users. In order to maximize the coverage area, a cellular network approach can be deployed. The time and frequency resources can be allocated to the cells in a static fashion using resource partitioning with wavelength/frequency reuse or by means of a dynamic interference-aware scheduling. An aircraft cabin has been considered as a representative indoor scenario since: (a) it provides a very dense distribution of users, and hence results in an interference-limited environment; (b) it is a scenario where radio frequency (RF) communication is not preferred; and (c) it allows very small cells. It is well known that the introduction of small cells is one of the key reasons for the significant network spectral efficiency increase in RF cellular communications.

    In this chapter, first, a cellular OWC system based on orthogonal frequency division multiplexing (OFDM), i.e. optical OFDM (O-OFDM), with static resource partitioning is considered inside an aircraft cabin. Asymmetrically clipped optical OFDM (ACO-OFDM) and direct-current-biased optical OFDM (DCO-OFDM) transmission schemes are compared in terms of throughput and cell coverage. Wavelength reuse factors of 1 (full reuse), 2, and 3 are studied, and off-the-shelf optical front-end components are assumed. The signal and interference distributions in an indoor cellular network can be accurately modeled by means of a Monte Carlo ray-tracing (MCRT) global irradiation simulation in a computer-aided design (CAD) cabin model. Due to the use of OFDM, adaptive modulation and coding (AMC) can be applied to the signal-to-interference-and-noise ratio (SINR) maps to obtain the spatial throughput distribution inside the cabin [16, 17]. It is shown that throughput within the range of 1.56–30 Mbps is achievable with the considered ACO-OFDM system, while DCO-OFDM doubles the throughput to 3.13–60 Mbps. In addition, wavelength reuse factors of 3 or higher are needed to ensure full cell support for both transmission schemes.

Page 127 of 320