Introduction

Chapter 9 - Discrete-Time Signals and Systems

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, the nonstationarity kind of a continuous-time GACS signal cannot be deducted from that of the discrete-time signal of its samples. However, in the paper it is shown how, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.

Chapter 1 - Continuous-Time Signals

A valuable introduction to the fundamentals of continuous and discrete time signal processing, this book is intended for the reader with little or no background in this subject. The emphasis is on development from basic principles. With this book the reader can become knowledgeable about both the theoretical and practical aspects of digital signal processing.Some special features of this book are: (1) gradual and step-by-step development of the mathematics for signal processing, (2) numerous examples and homework problems, (3) evolutionary development of Fourier series, Discrete Fourier Transform, Fourier Transform, Laplace Transform, and Z-Transform, (4) emphasis on the relationship between continuous and discrete time signal processing, (5) many examples of using the computer for applying the theory, (6) computer based assignments to gain practical insight, (7) a set of computer programs to aid the reader in applying the theory.

Hybrid systems modelers like Simulink come with a rich collection of discrete-time and continuous-time blocks. Most blocks are not defined in terms of more elementary ones—and some cannot be—but are instead written in imperative code and explained informally in a reference manual. This raises the question of defining a minimal set of orthogonal programming constructs such that most blocks can be programmed directly and thereby given a specification that is mathematically precise, and whose compiled version performs comparably to handwritten code. In this paper, we show that a fairly large set of blocks of a standard library like the one provided by Simulink can be programmed in a precise, purely functional language using stream equations, hierarchical automata, Ordinary Differential Equations (ODEs), and deterministic synchronous parallel composition. Some blocks cannot be expressed in our setting as they mix discrete-time and continuous-time signals in unprincipled ways that are statically forbidden by the type checker. The experiment is conducted in Zélus, a synchronous language that conservatively extends L ustre with ODEs to program systems that mix discrete-time and continuous-time signals.

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

We discuss the basic definitions of analog/continuous time (CT) and discrete time (DT) signals in this chapter. We need to understand the basics of discrete time signals, i.e., the sampled signals. The theory of sampled signals is introduced in this chapter. The analog signal is first interfaced to a digital computer via analog to digital converter (ADC). ADC consists of a sampler and a quantizer. We will mainly discuss the sampler in this chapter. The analog signal, when sampled, gets converted to discrete time (DT) signal. Here, the time axis is digitized with a constant sampling interval T . The inverse of T is the sampling frequency. The sampling frequency must be properly selected for faithful reconstruction of the analog signal. Introduction to Signals Any physical quantity that carries some information can be called a signal. The physical quantities like temperature, pressure, humidity, etc. are continuously monitored in a process. Usually, the information carried by a signal is a function of some independent variable, for example, time. The actual value of the signal at any instant of time is called its amplitude. These signals are normally plotted as amplitude vs. time graph. This graph is termed as the waveform of the signal. The signal can be a function of one or more independent variables. Let us now define a signal. Definition of a signal A signal can be defined as any physical quantity that varies with one or more independent variables. Let us consider temperature measurement in a plant. The measured value of temperature will be its amplitude. This temperature changes from one instant of time to another. Hence, it is a function of time, which is an independent variable, as it does not depend on anything else. Temperature can be measured at two different locations in a plant. The values of temperature at two different locations may be different. Hence, the temperature measured depends on the time instant and also on the location. We can say that temperature is a function of two independent variables, namely time and location in a plant. This temperature can take on any continuous value, like 30.1, 30.001, 31.3212, etc. The time axis is also continuous i.e. the temperature is noted at each time value. The signal is then called continuous time continuous amplitude (CTCA) signal.

Central limit theorem, modern extreme value theory and the theory that the discrete orthogonal frequency division multiplexing signals converge weakly to a Gaussian random process are generally employed to derive the best approximation of peak-to-average power ratio distributions of discrete-time and continuous-time signals. In this paper, we arrive at a simple, rigorously justified, and accurate expression of the upper bound of the probability density function of the PAPR in context of orthogonal frequency division multiplexing systems. Since this bound has a complex expression not convenient to deal with in practice, we show also the demonstration of an accurate estimation of the proposed bound.

One of the problems encountered by users of standard frequency and time (SFT) signals in the high-frequency (hf) band is the interference between stations transmitting signals in allocated standard frequency bands. This problem has been dealt with in detail by the Seventh Investigation Commission of the International Advisory Con~nittee on Radio Communication. In particular, the IA-I/7 research program foresees an investigation of the possibility of reducing interstation interferency by shifting transmission within the allocated bands and using the most suitable form of modulation. Reduction of interstation interference between signals on the basis of time division (intermittent operation or continuous operation with pulse signals transmitted at different instants of time by each station) are mostly ineffective. In the first case as a result of the considerable time needed for the signals to arrive and in the second case because of the difficulties involved in station recognition.

This paper proposes an adaptive digital filtering method which improves continuous-time performance. To take account of continuous-time performance, we make an assumption that the adaptive filter can exploit the continuous-time error signal, and define the integral of the square error signal as a performance index. Using lifting, the performance index can be equivalently reduced to a summation of discrete-time signals. Therefore it is shown that the recursive formula for the filtering problem with the performance index is given by the well-known RLS algorithm. Next, a more implementable case is considered where the output of the unknown system is approximated by the oversampled discrete-time signal and the summation of the squared discrete-time signals is defined as an approximate performance index. It is shown that the approximate performance index can be directly applied to the first result. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method.

Basics of Multirate Digital Signal Processing

Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

In order to avoid large scale failure, it is necessary to monitor the widespread and complex performance of interconnected power grid system in real time. In case of bulk power system, the operation of Protective Relays and the Circuit Breakers are fast enough for secured performance but tracking of the system parameters during an event is beyond the scope of such devices. So, real time tracking before an event and quick analysis for potential failure of the system at the time of disturbances is quite difficult. Modernization of technology provides us with instruments to sample the continuous-time analog signal into discrete-time digital signal with the help of a device called Phasor Measuring Unit (PMU). As a result, it is also possible to estimate the dynamic phasor of the system and to determine the frequency and the Rate of Change of Frequency (ROCOF) of the system within a specific frequency zone. The purpose of this paper is to present a mechanism for continuous observation of the interconnected power system within a specific frequency zone and detection of its nature with improved accuracy using Phasor Measuring Unit (PMU). Having the Rate of Change of Frequency as the prime detector of disturbances at the interconnected grid system in Gaussian plane, this should facilitate the continuous observation and a system protection scheme which is more efficient. This should minimize the sudden damages of the system equipments due to abnormal behavior of the bulk interconnected electrical power system.

When discretizing continuous-time systems or signals, one is often interested in preserving a property termed covariance-invariance. In this paper a technique is outlined for synthesizing discrete-time systems and signals which are covariance-invariant with corresponding continuous-time systems and signals. Applications of the technique to process simulation, minimum mean-squared error estimation, and digital filter synthesis are outlined, with example designs presented for covariance-invariant Butterworth and Chebychev digital filters. Based on the frequency response of these designs it is argued that the method of covariance-invariance is superior to the methods of impulse-invariance and bilinear-z as a response matching design technique for the synthesis of digital filters. This superiority is especially apparent at sampling rates that are marginal with respect to filter critical frequencies.

Analytical background and techniques: discrete-time signals, systems and transforms analysis of discrete-time speech signals probability and random processes linear model and dynamic system model optimization methods and estimation theory statistical pattern recognition. Fundamentals of speech science: phonetic process phonological process. Computational phonology and phonetics: computational phonology computational models for speech production computational models for auditory speechprocessing. Speech technology in selected areas: speech recognition speech enhancement speech synthesis.