Chapter 9 - Discrete-Time Signals and Systems

Chapter 1 - 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).

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.

The basic concepts and relationships of the theory of discrete-time signals and systems are analogous to those for continuous-time signals and systems. In some respects, however, they are more simply derived and perhaps easier to visualize in the discrete-time case. In this chapter, we will introduce these basic concepts and relationships, developing them further in subsequent chapters.

Master the basic concepts and methodologies of digital signal processing with this systematic introduction, without the need for an extensive mathematical background. The authors lead the reader through the fundamental mathematical principles underlying the operation of key signal processing techniques, providing simple arguments and cases rather than detailed general proofs. Coverage of practical implementation, discussion of the limitations of particular methods and plentiful MATLAB illustrations allow readers to better connect theory and practice. A focus on algorithms that are of theoretical importance or useful in real-world applications ensures that students cover material relevant to engineering practice, and equips students and practitioners alike with the basic principles necessary to apply DSP techniques to a variety of applications. Chapters include worked examples, problems and computer experiments, helping students to absorb the material they have just read. Lecture slides for all figures and solutions to the numerous problems are available to instructors.

Techniques are outlined for synthesizing continuous-time systems and signals which are covariance-invariant (CI) with corresponding discrete-time systems and signals. The property of covariance-invariance (CI) insures that the covariance sequence characterizing the response of a stable, linear discrete-time system to white-noise equals the sampled covariance function that characterizes the sampled output of a stable, linear continuous-time system excited by white-noise. Applications of inverse covariance-invariant (ICI) signal processing are suggested, and numerical examples of inverse covariance-invariant (ICI) synthesis are presented.

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.

This chapter presents time-domain analysis of continuous-time systems. It develops representation of signals in terms of impulses. The notions of linearity, time-invariance, causality, stability, memorability, and invertibility are introduced. It has shown that the input-output relationship for linear time-invariant (LTI) continuous systems is described in terms of a convolution integral. The differential equation representation of LTI continuous systems and classical solutions of differential equations are also presented. Next, block-diagram representation of LTI continuous-time systems is introduced. Furthermore, a brief discussion on singularity functions is provided. Finally, the state-space representation of continuous-time LTI systems is described.

The Signals and Systems Concept Inventory (SSCI) is a set of multiple-choice questions that measures students' understanding of fundamental concepts in continuous-time (CT) and discrete-time (DT) signals and systems. In this paper, we discuss and analyze statistically the results of these assessment tests in our undergraduate courses. We use the results to assess the students' performance from year to year and determine evidence of learning outcomes. We offer useful suggestions for future offerings of the courses based on our findings. Some conclusions are made on whether we meet our assessment goals and on the efficacy of the SSCI CT Tests and the impact it has had on our pedagogy.

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

Chapter 2 - Continuous-Time Systems

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.

A discrete time signal is one that has a value only for a finite or infinite number of time instants whereas a continuous time signal has a value for every (real) time instant. For example, consider: The sound pressure wave of a speech signal The electrical that is the output of a transducer that is used to mesdsure the cound pressure wave of a speech signal The sequence of numbers that is obtained from connecting an analg to digital converter (ADC) to the electrical signal sound pressure wave of a speech signal KeywordsSpeech SignalAnalog SignalAnalog InputLinear Predictive CodeDiscrete Time SignalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Aggregated research evidence suggests that active learning and student-centered instruction are positively associated with student achievement; however, context-specific studies in engineering classes indicated inconsistent results. To further investigate the effects of classroom environment and the amount of active learning on student achievement, we conducted a quasi-experimental study. We compared student achievement in two Discrete-Time Signals and Systems classes: one taught in an active and collaborative environment, and one taught in a lecture classroom. To assess student learning, we used three exams and the Signals and Systems Concept Inventory. The results showed that students in the lecture classroom outperformed their peers in the active classroom on midterm 1 but scored lower on midterm 2. No difference was found in performance on the final exam and the concept inventory. Additionally, for the active classroom, we examined students' attitudes towards the classroom environment, as the literature suggests that this factor may contribute to achievement. However, no relationship was found between attitudes and student achievement.