6 DISCRETE-SIGNAL ANALYSIS AND DESIGN capabilities of Mathcad are used. MATLAB users will have no trouble trans- lating everything in this book directly to their system. Keep printouts and notes for future reference. Mathcad also has an excellent relationship with an EXCEL program that has been conÞgured for complex algebra. EXCEL is an excellent partner to Mathcad for many purposes. An excellent, high-quality linear and nonlinear analog and digital cir- cuit simulator such as Multisim (Electronics Work Bench, a division of world-famous National Instruments Co., www.ni.com), which uses accu- rate models for a wide range of electronic components, linear and nonlin- ear, is another long-term investment for the serious electronics engineer and experimenter. And similar to Mathcad, your circuit diagram, with component values and many kinds of virtual test instruments, appears on the screen. A sophisticated embedded graphing capability is included. Less expensive (or even free) but fairly elementary alternatives are avail- able from many other sources. For example, the beginner may want to start with the various forms of SPICE. However, Multisim, although the up-front cost is signiÞcant, is a valuable long-term investment that should be considered. Multisim offers various learning editions at reduced cost. I recommend this software, especially the complete versions, very highly as a long-term tool for linear and nonlinear analysis and simulation. An added RF Design package is available for more sophisticated RF modelling. Mathcad is also interactive with LabVIEW, another product of National Instruments Co., which is widely used for laboratory data gathering and analysis. See http://www.ni.com/analysis/mathcad.htm for more informa- tion on this interesting topic. Another approach that is much less expensive, but also much less pow- erful, involves structured programming languages such as BASIC, Fortran, C ++ , Pascal, EXCEL, and others with which many readers have previous experience. However, my suggestion is to get involved early with a more sophisticated and long-enduring approach, especially with an excellent program such as Mathcad. For the website-friendly personal computer, the online search engines put us in touch very quickly with a vast world of speciÞc technical refer- ence and cross-referenced material that would often be laborious to Þnd using traditional library retrieval methods. INTRODUCTION 7 MathType, an Equation Editor for the word processor (http://www. dessci.com/en/), is another valuable tool that is ideal for document and report preparation. This book was written using that program. And of course these programs are all available for many other uses for many years to come. The time devoted to learning these programs, even at the introductory level, is well spent. These materials are not free, but in my opinion, a personal at-home modest long-term investment in productivity software should be a part of every electronics engineer’s and experimenter’s career (just like his education), as a supplement to that which is at a school or company location (which, as we know, can change occasionally). Keep in mind that although the computer is a valuable tool, it does not relieve the operator of the responsibility for understanding the core technology and math that are being utilized. Nevertheless, some pleasant and unexpected insights will occur very often. Remember also that the introductory treatment in this book is not meant to compete with the more scholarly literature that provides much more advanced coverage, but hopefully, it will be a good and quite useful initial contact with the more advanced topics. REFERENCES Oppenheim, A. V., and R. W., Schafer, 1999, Discrete-Time Signal Processing, 2nd ed., Prentice Hall, Upper Saddle River, NJ. Simon, B., Various Mathcad reviews, Department of Mathematics, California Institute of Technology. 1 First Principles This Þrst chapter presents an overview of some basic ideas. Later chapters will expand on these ideas and clarify the subtleties that are frequently encountered. Practical examples will be emphasized. The data to be pro- cessed is presented in a sampled-time or sampled-frequency format, using a number of samples that is usually not more than 2 11 =2048. The fol- lowing “shopping list” of operations is summarized as follows: 1. The user inputs, from a tabulated or calculated sequence, a set of numerical values, or possibly two sets, each with N =2 M (M =3, 4, 5, ,11) values. The sets can be real or complex in the “time” or “frequency” domains, which are related by the Discrete Fourier Transform (DFT) and its companion, the Inverse Discrete Fourier Transform (IDFT). This book will emphasize time and frequency domains as used in electronic engineering, especially communica- tions. The reader will become more comfortable and proÞcient in both domains and learn to think simultaneously in both. 2. The sequences selected are assumed to span one period of an eternal steady-state repetitive sequence and to be highly separated from Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 9  10 DISCRETE-SIGNAL ANALYSIS AND DESIGN adjacent sequences. The DFT (discrete Fourier transform), and DFS (discrete Fourier series) are interchangeable in these situations. 3. The following topics are emphasized: a. Forward transformation and inverse transformation to convert between “frequency” and “time”. b. Spectral leakage and aliasing. c. Smoothing and windowing operations in time and frequency. d. Time and frequency scaling operations. e. Power spectrum and cross-spectrum. f. Multiplication and convolution using the DFT and IDFT. g. Relationship between convolution and multiplication. h. Autocorrelation and cross-correlation. i. Relations between correlation and power spectrum using the Wiener-Khintchine theorem. j. Filtering or other signal-processing operations in the time domain or frequency domain. k. Hilbert transform and its applications in communications. l. Gaussian (normal) random noise. m. The discrete differential (difference) equation. The sequences to be analyzed can be created by internal algorithms or imported from data Þles that are generated by the user. A library of such Þles, and also their computed results, can be named and stored in a special hard disk folder. The DFT and IDFT, and especially the FFT and IFFT, are not only very fast but also very easy to learn and use. Discrete Signal Processing using the computer, especially the personal computer, is advancing steadily into the mainstream of modern electrical engineering, and that is the main focus of this book. SEQUENCE STRUCTURE IN THE TIME AND FREQUENCY DOMAINS A time-domain sequence x (n)ofinÞnite duration −∞ ≤ n ≤+∞that repeats at multiples of N is shown in Fig. 1-1a, where each x(n) is uniquely . and to be highly separated from Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 9  10 DISCRETE-SIGNAL ANALYSIS AND DESIGN adjacent sequences “frequency” and “time”. b. Spectral leakage and aliasing. c. Smoothing and windowing operations in time and frequency. d. Time and frequency scaling operations. e. Power spectrum and cross-spectrum. f and also their computed results, can be named and stored in a special hard disk folder. The DFT and IDFT, and especially the FFT and IFFT, are not only very fast but also very easy to learn and