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GRAPHICAL STUDY OF SIGNALS AND SYSTEMS Elisa Sayrol', Antoni Gasull', Josep Salavedra', Asunción Moreno', Francesc Vallverdú? 'Universitat Politècnica de Catalunya, Barcelona, Spain elisa.gasull.asuncion,[email protected] 2 Universitat Oberta de Catalunya, Barcelona, Spain [email protected] ABSTRACT We present a simulation software devoted to the practical study of analog signals and systems. The program exhaustively exploits the graphical tools provided by MATLAB to obtain a user-friendly environment. It can be used as a self-instruction teaching material. On the other hand it contains a complete and very large set of graphic exercises that makes it unique. Some examples of the possibilities and features of the developed software will be shown in this paper. 1. INTRODUCTION The learning material we present was made after several attempts to design an attractive set of practices for students undergoing a course in analog signals and systems. The course is offered in the second year, first semester, of an undergraduate engineering school of telecommunication. A practical tool that would fulfill the laboratory requirements (15 hours per semester) was not available in the market. On the other hand, there is a growing interest in elaborate multimedia learning material to complement or even to substitute existing courses. Thus, our second goal was to develop a self-instruction teaching material that would offer a graphical study of signals and systems concepts. There is a large number of references devoted to the study of signals and systems that provide practical exercises using software tools. Most of these tools concentrate on the digital case in detriment of the analog case, as for example in [1]. This is understandable since data used in computers is digital in nature. Still, for educational purposes, there is a need to develop complete software tools exclusively dedicated to the analog case. Analog signals and systems can be simulated through appropriate tools. MATLAB offers this possibility and, if some care is used in the design, digital processing is transparent to its use. Some MATLAB functions were created for this purpose. Although other programming tools are now being used for educational purposes, as Java-based tools [2], MATLAB is widely used in teaching signal processing concepts. In fact most new text books in the area include MATLAB code together with the text. In this work, the graphical tools provided by MATLAB are exploited. One of the advantages is that tools are easily programmed by the teacher, thus, students focus only on acquiring concepts (which is the main goal of this material), not on programming The graphical representation is the most important feature of this learning material. Signals, input and output data, etc, are presented in a user-friendly environment. Most options are selected by using the mouse button. The student has the possibility to visualize and experiment with the basic concepts of the course, as well as to complement them with some applications. We believe that its utilization can lead to higher comprehension of signals and systems concepts. This software includes a large collection of practical exercises exclusively dedicated to the study of analog signals and systems. Thus, signals and systems properties and, in particular linear time-invariance systems are covered in detail. The Fourier Transform, its properties and applications are extensively studied. The program contains a whole block of practical exercises devoted to the design of analog filters. Next, in section 2, we describe the most important features of the software. In section 3 we present the structure of the program carefully designed to achieve the educational goals. The contents of the program are also summarized in this section. Section 4 includes some examples that illustrate the graphical design and features of the program. Finally, section 5 is devoted to some conclusions. 2. PROGRAM FEATURES 2.1. Graphic configuration All practical exercises are solved through graphical windows, where input and output data are displayed. Some
specific tools were specially developed to build the window graphical representation. These were programmed in MATLAB (3). The location of graphics as well as buttons within the window were conveniently chosen to fit the requirements of most practical exercises in a very comprehensible fashion. 2.2. Flexibility The structure of the program is flexible enough to allow adding, suppressing or modifying sections of the program. It also allows adding new functions related either to the contents or the graphical utilities. 2.3. Robustness Input data through the keyboard or the mouse can produce numerical or executing errors. The program is protected against non correct button selection or wrong input data. 2.4. Self-Instruction Material This software can be we used as a laboratory tool but also as a self-instruction tool since it contains many interactive functions and a help menu for this purpose. The usage, the goals of every section and its importance and difficulty are explained. 2.5. Completeness It covers many topics and contains a large set of practical exercises devoted to a first study of analog signals and systems. In this sense, we believe it is a very complete tool. Utilize real signals (speech, audio and imaging) and simulated signals (analytic signals). Given the above points the program has been divided into thematic blocks and each of these in different sections. 3.2. Program Structure The program includes 5 thematic blocks. The first one is devoted to the analysis of signals and systems in the temporal domain. The second one examines the particular case of linear time-invariant systems, introducing the convolution operation. The third block contains practical exercises related to the definition and properties of the Fourier Transform. Periodic signals and sampling are studied in the fourth block. Finally the last block is devoted to the design of analog filters. The content of every block is divided into several sections. Sections are ordered from graphical solving of theoretical exercises to applications that illustrate the usefulness of the concepts. Figure 1 shows the initial window of the program with the five blocks and the buttons to select each section. The design of the main window is different from the one of the sections as it will be seen. Next we summarize the contents of each section. 3.3. Contents 3.3.1. Block 1: Signals and Systems. • Transformation of the independent variable. Interpretation of the transformation of the independent variable as a system and the importance of the order when applying different transformations. Analyzing system properties for unknown systems. Application of the linear and time-invariant properties of a system. Interpretation of the delta function and determination of the impulse response of a system. 3.3.2. Block 2: Linear time-invariant systems. • Examples of the convolution operation in ID. • Deconvolution by finding the impulse response of a system from a set of possible functions given several pairs of input-output signals. • Convolution in 2D applied to noise reduction and contour detection using 2-D linear space-invariant filters • Smoothing filter application to reduce noise in the transmission of data through electric networks. Digital transmission simulation when different waveforms are considered to transmit digital information. Detection is analyzed in the presence of noise using smoothing filters and adaptive filters. 3. PROGRAM STRUCTURE 3.1. Objectives The objective of this learning material is to provide a strong support in apprehending the basic concepts of a course devoted to the study of analog signals and systems. Graphic simulations are a very helpful way to acquire the knowledge to easily understand and handle signals and systems in the temporal and frequency domains. The following points where considered to define the structure of the program: • Develop graphical tools that help understand the basic theoretical concepts. Develop graphical tools that help understand the utilization of such basic concepts in different situations. Include simple applications directly related to the theoretical concepts. Include more complex applications close to existing signal processing applications,
3.3.3. Block 3: Fourier Transform. Basic properties of the Fourier Transform. Filters in the time and frequency domain studying different functions and representations (impulse response, linear and log magnitude of the FT, phase of the FT, etc...) The frequency shift property as the base for amplitude modulations. Signal multiplexing in the frequency domain and the effect of filtering with non-ideal filters. Encryption of speech signals by dividing and shuffling the frequency band. Shifting and filtering operations are used. 3.3.4. Block 4: Fourier Transform of periodic signals and sampling of signals. Approximation of periodic signals using Fourier series. Generation of the square wave using a non-linear system. Effects of non-linear amplifiers on multiplexed speech signals causing crosstalk. Display of new harmonics. Modulation by sampling. Multiplexing the right and left channel of FM stereo signals by sampling and filtering. • Finding periodicities in electrocardiogram signals retrieved from fetuses in pregnant women. 3.3.5. Block 5: Design of analog systems. Design of low-pass filters (Butterworth, Chebychew, Inverse Chebychew and elliptic filters) and its study using different functions and representations. Influence of the location of zeros and poles in the impulse response of a filter by varying its position. Filtering the sum of sine waves at different frequencies. Study of a multiplex/demultiplex system using real filters and different frequency carriers. Frequency Transformation to obtain high pass, band pass and eliminated band filters. 4. EXAMPLES Each of the exercises summarized above was carefully designed. Here we are obliged to select only some examples to illustrate the features of the program. Some other examples will be available at the website: http://gps- tsc.upc.es/imatge/ Elisa/Elisa.html As it can be seen in figures from 2 to 4 graphical windows are divided into two. The right column corresponds to the menu and is mainly devoted to the following operations: selection of signals, system schemes or other functions; selection of parameters using slide buttons or introduced by the user; binary selection or checking buttons; a message window; and the quit button to exit the exercise. The left-hand side displays the signals and system functions, and in some cases the block diagram of a system configuration. Zoom operations, display of specific values, shifting, scaling and magnitude transformations can be carried out using the mouse. 4.1. Example 1: System Properties In this practical exercise linearity, time invariance, causality and stability, of an unknown system are studied. On the menu part, the first button is used to choose the property under study. Figure 2 shows an example where the time-invariance property has been chosen. The menu as well as the signal display part for the other three properties change appropriately. The second button of this example selects the unknown system to analyze its properties. Input signals are chosen from a large set of functions by means of the third button. Once all selection buttons have been activated, the input signal (the one selected) and the output signal of the unknown signal appear in the upper part of the window. In the lower part of the window a shifted version of the input signal and the output to this signal are represented. This input signal can be shifted easily by clicking and dragging the mouse on the function. The output to the shifted signal is displayed in real time. Thus, the invariance property is studied by visually comparing the two output signals. By testing several signals from the menu, the time-invariant property can be verified. The user fills in the checking button for this property. When all properties have been checked, a message is prompted informing the user if the answer was right or wrong 4.2. Example 2 : The Convolution Operation In this exercise a signal is convolved with an impulse response function. Both waveforms are selected in the menu part of the window. They can be transformed by checking the right buttons on the menu part and modifying the waveform with the mouse situated over their representations. To start the convolution operation the pause button must be unselected. The result of the convolution is represented in the lower part of the window in an animated way. The process can be stopped at a certain time instance by checking the pause button again. Once done, the result of the convolution at a given point can also be visualized by clicking directly over the function. Figure 3 shows this operation when the result of convolution is stopped at 1 second approximately. Students recognized that animation was very helpful in understanding convolution.
ANALOG SIGNALS & SYSTEMS LAB PRACTICE 1 PRACTICE PRACTICE PRACTICE 2 DOWOLUTION PRACTICEJ FODOS PO Va Figure 1 Program main window Figure 3 Convolution operation window Figure 2 Time-invariance property window 4.3. Example 3 : Influence of the location of zeros and poles When this section is selected two different windows will be activated. In the first window an analog filter has to be designed. The filter can be built from the Butterworth, Chebychew, Inverse Chebychew or elliptic approximations. Then, when this operation is finished the zero-pole diagram appears in a second window. An example is shown in figure 4. It also contains the impulse response of such system and another function that is selected in the menu. In this particular example, the function corresponds to the frequency magnitude. The important feature of this practical exercise lies in the possibility to shift zeros and poles by clicking and dragging the mouse within the diagram. The influence of these modifications are visualized in real time in the functions represented below. Thus, zeros and poles can be graphically modified to obtain a certain impulse response. The filter can also be redefined by returning to the first window. Figure 4 Influence of zeros and poles window 5. CONCLUSIONS In this paper we have presented a program devoted to the digital simulation of analog signals and systems. The program is characterized by its graphical presentation, its completeness, flexibility and robustness and it can be used as a self-instruction material. The structure of the program has been defined with educational objectives in mind. The design of the different windows has been carefully chosen as it has been presented in the examples, exploiting the graphical tools provided by MATLAB. References [1] Denbigh, P.N., (1998), System Analysis and Signal Processing, With Emphasis on the use of MATLAB, Addison Wesley [2] J. Shaffer, J. Hamaker, J. Picone, Visualization of Signal Processing Concepts, ICASSP-98,vol. 3, pp. 1853-1856, Seattle 1998 [3] Gasull A., Sayrol E., Moreno A., Vallverdu F., Salavedra J., Oliveras A., (1999), Editor Gráfico de figuras MATLAB, usuarios de MATLAB conference, Madrid, 1999
Write a two-page (single-sided) summary report on the following research article: E. Sayrol, A. Gasull, J. Salavedra, A. Moreno and F. Vallverdu, "Graphical study of signals and systems," in Proc. the 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 2001, pp. 2705-2708 vol.5, doi: 10.1109/ICASSP.2001.940204. IEEE. Your report is expected to address the following. 1) What is the motivation that the authors mentioned in writing the paper? 2) What problems do the authors state in the paper? 3) What is the main objective that the authors addressed in this paper? 4) Briefly describe the methodology and the proposed model mentioned in this paper. 5) Point out major findings that the authors highlighted in this paper. 6) Finally, conclude your understanding of the paper as a whole and point out one to two further research directions. Note: This assignment is intended to let you understand the standard structure of a research paper authored by others and eventually help write scholarly research papers by yourself in your future endeavor.