Discrete-Time Signals and Systems

Chapter 9 - Discrete-Time Signals and Systems

Programmable logic controller (PLC) is connected with analog and digital input and output modules to process physical variables which are to be maintained at the desired values. Each module has a processor to process analog or digital signals. This paper has designed analog input module (AIM) using field programmable gate array (FPGA). It has used a digital to analog converter (DAC), comparator and FPGA for a single channel analog to digital converter (ADC). It becomes an analog input module (AIM) when a DAC and comparator are added to FPGA for every addition of a channel. Conversion time of a processor based AIM is n*tc where ‘n’, ‘tc’ are the number of channels and conversion time of ADC respectively. Conversion time of ‘n’ channel FPGA based AIM is ‘tc’ as all the analog signals are processed concurrently. The design was experimented using Multisim software and the conversion time of eight channels ADC was identified 0.13 ms.KeywordsProgrammable logic controllerADCMicroprocessorFPGA

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.

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 process of converting an analog, or continuous time, signal into a digital signal is known as analog‐to‐digital conversion (ADC). Many signals are inherently analog. Examples of analog signals obtained from a transducer abound. Other analog signals are inherently electrical, such as those obtained from biomedical electromyography or electroencephalography electrodes. In many cases, it is deemed desirable to process a signal by a microprocessor or computer to extract information, or to store the signal in some digital storage medium such as a hard drive, a floppy disk, or a CD ROM. If the original signal is in analog format, in such cases, it is necessary to convert the analog signal into an equivalent digital signal. This article describes the process of ADC. It introduces the topic and describes the operation of two of the most common methods of accomplishing ADC: the successive approximation ADC and the flash ADC. The sampling theorem is stated and proved, aliasing is described, and quantization noise is discussed. Oversampling is described, and the resultant improvements in ADC performance are documented. Then delta‐sigma ADCs are introduced and described. Conventional first‐order and second‐order delta‐sigma ADCs are described and evaluated. Performance comparisons among simple oversampling, first‐order delta‐sigma, and second‐order delta‐sigma ADCs are documented. The article concludes with an example of a higher order delta‐sigma ADC performance evaluation.

With the change in advanced technology and the need for electronic equipments, the development of electronic instruments stimulate more efforts for more innovative ideas and better performance of design. In real world physical values, such as pressure, humidity, temperature and voice can be measured as analog (continuous) signals. However, to process these analog signals by electronic digital equipments, we need to convert these analog signals into digital or discreet signals or binary. In this paper, there are many types of Analog-to-Digital Converters (ADCs) which can be classified according to the applications and concept on which they were designed. ADCs including, Direct conversion ADC (Flash), successive approximation ADC, integrating ADC, Pipeline ADC. In some cases, many comparators are used to reduce the complexity, power and improve the linearity which are also explained in this paper.

Analog signals are converted into digital signals for digital signal processing. Analog-to-Digital Converter (ADC) is used for the conversion process. When required, Digital-to-Analog Converter (DAC) is used for reconverting the processed digital signals into analog signals. Sample-Hold amplifier, Anti-aliasing filter, Reconstruction filter and Operational amplifier (Op-amp) are the analog components that are generally used in signal conversion circuits. The operation of the analog components is presented. Four ADC architectures, namely, Flash ADC, Dual slope ADC, Successive approximation ADC and Sigma-Delta ADC, are explained. Two DAC architectures, namely, Binary-weighted input DAC and R-2R Ladder DAC, are explained.

The conversion of analog signals into digital signals and vice versa, performed by sampling and interpolation, respectively, is an essential operation in signal processing. When digital computers are used to compute the analog signals, it is important to effectively control the approximation error. In this paper we analyze the computability, i.e., the effective approximation of band limited signals in the Bernstein spaces B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> , 1 ≤ p <; ∞,andofthe corresponding discrete-time signals that are obtained by sampling. We show that for 1 <; p <; ∞, computability of the discrete-time signal implies computability of the continuous-time signal. For p = 1 this correspondence no longer holds. Further, we give a necessary and sufficient condition for computability and show that the Shannon sampling series provides a canonical approximation algorithm for p > 1. We discuss BIBO stable LTI systems and the time-domain concentration behavior of bandlimited signals as applications.

Synthetic instruments apply the flexibility and computational capabilities of a digital signal-processing platform to synthesize a wide variety of synthetic instruments. The emphasis presented in traditional descriptions of a synthetic instrument is the architecture and the capabilities of the digital signal processing (DSP) platform and the user interfaces to the same. In this paper we highlight and discuss considerations of the signal conditioning and signal collection tasks. The focal point of the signal collection process is the analog-to-digital converter (ADC), the element that defines the precision and the bandwidth of the sampled data representation of the signal processed by the DSP platform. Analog signal conditioning prior to the ADC performs the task of limiting the input signal bandwidth and possible translation of the spectral band center. Digital signal conditioning following the ADC continues to perform the same tasks by further limiting the bandwidth of the digitized input signal as well as performing spectral translation with appropriate sample rate changes. In addition, the post conversion process can correct gain, phase, and time delay imbalances between input signal paths as well as gain and phase distortion encountered in the analog signal path. We identify and address performance constraints of existing ADCs and present a number of signal processing-based options to enhance and extend the operating regimes of the analog signal conditioning and the ADC conversion process.

Chapter 6 - Analogue to Digital and Digital to Analogue Conversion

In this article, the propagation of digital and analog signals through media which, in general, are both dissipative and dispersive is modeled using the one-dimensional telegraph equation. Input signals are represented using impulsive, Heaviside unit step, Gaussian, rectangular pulse, and both unmodulated and modulated sinusoidal pulse type boundary data. Applications to coaxial transmission lines and freshwater signal propagation, for both digital and analog signals, are included. The analysis presented here supports the finding that digital transmission in dispersive media is generally superior to that of analog. The boundary data (input signals) give rise to solutions of the telegraph equation which contain propagating discontinuities. It is shown that the magnitudes of these discontinuities, as a function of distance, can be found without the need of solving the governing equation. Thus, for digital signals in particular, signal strength at a given distance from the input source can be easily determined. Furthermore, the magnitudes of these discontinuities are found to be independent of both the dispersion coefficient k and the elastic coefficient b. In addition, it is shown that, depending on the algebraic sign of k, one of two distinct forms of dispersion is possible and that for small-time intervals, solutions are approximately independent of k.

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.

Widely used in telecom, television, music players, and controllers, a digital-to-analog converter (DAC) is a device that converts a digital signal into an analog signal. As integrated circuits (ICs) technologies develop, the modulation process has shifted from analog circuits to digital circuits, and the signal has been converted from a digital signal field to an analog signal field. Generally, the design of digital circuits is simpler than that of analog circuits, so the transferring can decline the requirement and difficulty of the system design. How the final DAC performs on the printed circuit board (PCB) directly affects the system's specifications. Therefore, verifying the final parameters of the DAC becomes an important task for the whole test. The DAC chip static parameters DNLity (DNL) and integral nonlinearity (INL) require measuring all points to get the calculated results. It will cost a long time if reading and recording are done by humans. This paper presents a fast and effective method to measure the static parameters of the DAC chip. The test time depends on the measurement accuracy and data. In this paper, the chip independently developed by Beijing Smart-Chip Microelectronics Technology Co. is used as a test case for testing. The results show that the DAC testing method mentioned in this paper can greatly shorten the testing time compared with manual testing.

Data converters bridge the gap between the physical world of analog signals and the abstract world of mathematics and representational data. Digital-toanalog converters (DACs) convert a sequence of digital numbers into an analog signal, and analog-to-digital converters (ADCs) convert an analog signal into a sequence of digital numbers. In this book, the analog signals we will be discussing most will be derived from imagery collected through detecting elements. Data converters have been around longer than you might expect. Walt Kester of Analog Devices published the Analog-Digital Conversion Handbook online in 2004 and included a wonderfully extensive history of data conversion that extended back to the 18th century. Kester outlines an extremely early DAC developed in the early 18th century in the Ottoman Empire. This early data converter was an 8-bit binary-weighted hydraulic DAC used to control water distribution. Nozzle size was weighted for water flow in a binary fashion, allowing controllers to change the flow with 28 or 256 individual settings using only eight nozzles. In this case, the input is coming directly from human operators, and the output is water current, making this also the first documented current DAC with flowing water molecules instead of electrons. Analog-to-digital converters were developed much later. In a 1921 patent, Rainey disclosed what might be the first electro-mechanical ADC for his facsimile system. This converter utilized a light beam, photocells, a galvanometer, and 32 relays to reproduce data printed on a transparency into an "m-hot out of 32" digital word. This is a 5-bit thermometric output similar to a typical 5-bit flash ADC, which will be discussed in Chapter 4. It might be fascinating to image sensor professionals that this ADC utilized photonic components almost 100 years ago. In 1937, Reeves, at the International Telephone and Telegraph Corporation in Paris, is credited with the first all-electronic DAC and ADC design with his pulse code modulation patents. Electronic converters quickly became necessary components in every electronic component interfacing to real-world signals. Jumping forward a little more than 50 years to the late 1980s brings us to the beginning of the era of the low-cost, high-volume image sensor. Eric Fossum, the father of the modern CMOS active pixel sensor, published a very early report of image processing that included ADCs integrated on a focal plane array.2 In Smart CMOS Image Sensors and Applications, Jun Ohta suggests that it is possible that Eklund, Svensson, and Astrom published the first column-level ADC on an image-sensing chip in 1995. So began the revolution in digital imaging systems-on-chip, which has led to the prevalence of high-quality, low-cost image sensors in modern electronics.

The ADCs characteristics that make up the MC don't always correspond to the specified requirements for accuracy and noise level. The thesis paper describes a new type of ADC with phase-locked loop (ADC with PLL). Currently, the development of a number of electronic devices is conducted jointly with the use of various microcontrollers (MK), which includes analog-digital converters (ADCs). A new type of ADC allows improve these characteristics, having a relatively simple circuit design implementation. Using the PLL loop, the input level voltage (analog signal) is converted into a digital frequency signal of a variable pulse ratio. The converted signal (analog voltage) is applied to one of the low-pass filter (LPF) inputs included in the PLL, thereby removing the circuit from equilibrium, changing the pulse ratio of the output signal.