Technology & Engineering
Discretization
Discretization is the process of converting continuous data or functions into discrete form. In the context of technology and engineering, it is commonly used in numerical analysis and computer simulations to represent continuous systems with a finite set of values or intervals. This method allows for easier computation and analysis of complex systems and phenomena.
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5 Key excerpts on "Discretization"
- Bohdan T. Kulakowski, John F. Gardner, J. Lowen Shearer(Authors)
- 2007(Publication Date)
- Cambridge University Press(Publisher)
The number of digital computer applications in data acquisition and control of continuous processes has been growing rapidly over the past two decades; thus the knowledge of basic methods available for analysis and design of discrete-time systems is an increasingly important element of engineering education. This chapter and the next provide introductory material on analysis and control of linear discrete-time systems. In Section 15.2, a problem of mathematical modeling of discrete-time systems is presented. Both input–output and state forms of system models are introduced. The process of Discretization of continuous systems as a result of sampling at discrete-time 389 390 Analysis of Discrete-Time Systems intervals is described in Section 15.3. Theoretical and practical criteria for selecting the sampling frequency to ensure that no information relevant to the dynamics of the continuous process is lost as a result of sampling are discussed. In Section 15.4, the z transform is introduced. The concept of the pulse transfer function of discrete-time systems defined in the domain of complex variable z is presented in Section 15.5. A procedure for calculation of a response of a discrete-time system to an arbitrary input is also outlined. 15.2 MATHEMATICAL MODELING A discretized model of a continuous system uses a sequence of values of each con-tinuous variable taken only at carefully chosen distinct increments of time 1 (usu-ally equal increments). A continuous variable x ( t ), for instance, is represented in a discretized model by a sequence { x ( k ) } , k = 0 , 1 , 2 ,... , consisting of the values x (0) , x ( T ) , x (2 T ), or, simply, x (0) , x (1) , x (2), . . . . Furthermore, the amplitude of a signal in a discrete-time system may be quantized; that is, it may take only a finite number of values, and in such a case the signal is called a digital signal. If the sig-nal amplitude is not quantized, such a signal is referred to as a sampled-data signal.
- Richard J. Tervo(Author)
- 2013(Publication Date)
- Wiley(Publisher)
The processing of discrete signals by computer, or digital signal processing (DSP), is increasingly employed to replace traditional analog approaches in modern systems. For example, long-distance telephone calls are typically transmitted over digital links as discrete time values of amplitude measured at 8000 samples per second. Similarly, an audio compact disk (CD) stores music digitally at 44,100 samples per second, and a CD player reconstructs the original signal waveforms from the sample values stored on the disk. Generally, digital signal processing refers to the real-time processing of signals as in the above examples. In this context, the term real-time implies that a very fast specialized computer is used to performed operations on samples of a continuous signal as quickly as those samples are acquired. Digital signal processing provides an attractive alternative to the use of analog system components. It has already been 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... -2 -1 0 1 2 n s[n] 0 2.5 5 7.5 10 -2 -1 0 1 2 Time (ms) s(t) FIGURE 8.1 Sampling and Reconstruction Graphs produced by hand or by computer generally begin with a set of points from a table after which a smooth curve is drawn through the points to complete a continuous shape. The intent of drawing a graph in this way is that the final curve should represent an underlying continuous function s ðt Þ. 302 Chapter 8 Discrete Signals observed that the overall behavior of a system is fully described by its impulse response function. Thus, just as a first-order lowpass filter might be built using a resistor and a capacitor (RC) or a resistor and an inductor (RL), it is also possible to perform this operation numerically using digital signal processing. After sampling an input signal, the discrete points can be manipulated mathematically to perform first-order lowpass filtering, and the filtered signal can be reconstructed from the resulting discrete signal.
- Bernd Jahne(Author)
- 2004(Publication Date)
- CRC Press(Publisher)
Consequently, we lose the horizontal sampling again, as the signal from a line of sensors is converted back into an analog signal. Mathematically, digitization is described as the mapping from a continuous func-tion in R 2 onto an array with a finite number of elements: g(x 1 , x 2 ) D −→ G m,n x 1 , x 2 ∈ R and m, n ∈ Z . (6.1) The array of discrete points can take different geometries. Solid state physicists, min-eralogists, and chemists are familiar with problems of this kind. Crystals show periodic 3-D patterns of the arrangements of their atoms, ions, or molecules which can be clas-sified due to their symmetries. In 3-D this problem is quite complex. In 2-D we have fewer choices. For a two-dimensional grid of the image matrix, a rectangular basis cell is almost exclusively chosen. 6.3.1a Pixel or Pel. A point on the 2-D grid is called a pixel or pel . Both words are abbreviations of the word picture element. A pixel represents the gray value at the corresponding grid position. The position of the pixel is given in the common notation for matrices. The first index, m , denotes the position of the row, the second, n , the position of the column (Fig. 6.1a). If the image is represented by an M × N matrix, the 6.3 Concepts 209 a columns rows 0 1 m M 0 1 n N G m,n Discrete MxN image G x y b l m n Figure 6.1: Representation of digital images by arrays of discrete points with rectangular cells: a 2-D image, b 3-D image. index n runs from 0 to N − 1, and the index m from 0 to M − 1. M gives the number of rows, N the number of columns. Note also that the vertical axis runs from top to bottom. A point on a 3-D grid is called a voxel , an abbreviation of volume element. On a rectangular grid, each voxel represents the mean gray value of a cuboid. The position of a voxel is given by three indices. The first, l , denotes the depth, m the row, and n the column (Fig. 6.1b).
eBook - ePub- Chris Bissell(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Fig. 9.1 , is that the latter involves the following thrn distinct, although related, effects.The sampled signal of Fig. 9.2 (a) is a discrete signal, but not a digital one, because it is not restricted to one of a finite set of values. The distinction between digital and discrete signals is often glossed over in the technical literature. Most of the theory of the next three chapters is concerned with discrete models: the effects of digitization have to be modelled separately if necessary.(1) Sampling. When a continuous signal is sampled, its value is measured a particular instants of time only. Samples are usually equally spaced in time. If th sampling interval is T seconds, then the number of samples per second, generall: referred to as the sampling frequency f s , is 1/T , as illustrated in Fig. 9.2 (a). Th behaviour of the signal between sampling instants is not detected. Because th1 sampled signal is defined at specific instants only, it is known as a discrete-tim signal. Note, however, that the samples of the figure are still (in theory) precis1 measures of the analogue waveform at the sampling instants.(2) Quantization. Once a .signal has been sampled, its continuously variable valm must be coded using a finite number of bits. The number of bits available to cod1 a sample represents a fundamental limitation on resolution: for example, if eacl sample is coded using 4 bits, then a maximum of 24 = 16 different signal levels ar1 available to cover the range of the signal. Signal values lying between thes1 quantization levels must be coded by rounding to the nearest level (Fig. 9.2 (b)) introducing a certain unavoidable error in the representation of the signal. Thi quantized sample values are limited to a finite set of numerical values and thu constitute a digital
- S. Bingulac(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
Chapter 2 System Discretization As pointed out in Chapter 1, w ith the widespread use of computers in control loops it is inevitable that control engineers w ill face problems associated w ith sampled-data systems. Such systems by their very definition contain a mixture of continuous-time and discrete-time signals. A common problem that arises w ith sampled-data control systems is to find the equivalent effect o f continuous-time operations as seen by the computer in the loop. Typically, the modeling o f the signal converters assumes an ideal uniform sampler for the analog-to-digital converter and a simple (zero-order) hold device synchronized w ith the samples for a digital-to-analog converter. W ith these assumptions one may find in many references the standard zero-order h old model, also known as the step invariant (SI) model which w ill be discussed subsequently. 2.1 Introduction In addition to simple plant modeling with SI equivalents there are occasions, such as in digital redesign or system identification, that demand more accuracy between a given continuous-time (C-T) system and its discrete-time (D-T) equivalent model. In these instances higher-order discrete models are required. Two models which have been introduced for this purpose are the bilin ear transform ation (BT) (without prewarping) and a method which assumes a linearly interpolated input, also known as the trapezoidal rule. This latter method is referred to as a ram p-invariant (RI) model in contrast to the standard Z O H model being a step -in va ria n t (57) model. This model was introduced as a linearly interpolated (input) model in Chapter 1; see Eq.(1.29). There are many other useful models, but this chapter w ill focus on only these three methods o f discretiza tion as being the most useful in practice. The reverse problem, called continualization , is that o f reconstructing a C-T model from a given D-T model.
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