DISCRETE-SIGNAL ANALYSIS AND DESIGN- P34

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P34:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P34 THE HILBERT TRANSFORM 151expensive. Receivers very often combine the phasing and Þlter methodsin the same or different signal frequency ranges to get greatly improvedperformance in difÞcult-signal environments. The comments for the SSB transmitter section also apply to the receiver,and no additional comments are needed for this chapter, which is intendedonly to show the Hilbert transform and its mathematical equivalent ina few speciÞc applications. Further and more complete information isavailable from a wide variety of sources [e.g., Sabin and Schoenike, 1998],that cannot be pursued adequately in this introductory book, which hasemphasized the analysis and design of discrete signals in the time andfrequency domains.REFERENCESBedrosian, S. D., 1963, Normalized design of 90◦ phase-difference networks, IRE Trans. Circuit Theory, vol. CT-7, June.Carlson, A. B., 1986, Communication Systems, 3rd ed., McGraw-Hill, New York.Cuthbert, T. R., 1987, Optimization Using Personal Computers with Applications to Electrical Networks, Wiley-Interscience, New York. See trcpep@aol.com or used-book stores.Dorf, R. C., 1990, Modern Control Systems, 5th ed., Addison-Wesley, Reading, MA, p. 282.Krauss H. L., C. W. Bostian, and F. H. Raab, 1980, Solid State Radio Engineer- ing, Wiley, New York.Mathworld, http://mathworld.wolfram.com/AnalyticFunction.html.Sabin, W. E., and E. O. Schoenike, 1998, HF Radio Systems and Circuits, SciTech, Mendham, NJ.Schwartz, M., 1980, Information Transmission, Modulation and Noise, 3rd ed., McGraw-Hill, New York.Van Valkenburg, M. E., 1982, Analog Filter Design, Oxford University Press, New York.Williams, A. B., and F. J. Taylor, 1995, Electronic Filter Design Handbook , 3rd ed., McGraw-Hill, New York.APPENDIXAdditional Discrete-SignalAnalysis and DesignInformation†This brief Appendix will provide a few additional examples of how Math-cad can be used in discrete math problem solving. The online sources andMathcad User Guide and Help (F1) are very valuable sources of infor-mation on speciÞc questions that the user might encounter in engineeringand other technical activities. The following material is guided by, and issimilar to, that of Dorf and Bishop [2004, Chap. 3].DISCRETE DERIVATIVEWe consider Þrst Fig. A-1, the discrete derivative, which can be a usefultool in solving discrete differential equations, both linear and nonlinear.We consider a speciÞc example, the exponential function exp(·) from† Permission has been granted by Pearson Education, Inc., Upper Saddle River, NJ, to usein this appendix, text and graphical material similar to that in Chapter 3 of [Dorf andBishop, 2004].Discrete-Signal Analysis and Design, By William E. SabinCopyright  2008 John Wiley & Sons, Inc. 153154 DISCRETE-SIGNAL ANALYSIS AND DESIGN n − N N := 256 n := 0,1.. N x(n) := e T := 1 1 x(n) 0.5 x(N) = 36.79% 0 0 50 100 150 200 250 n (a) y(N) − x(N) = 0.67% x(N) y(n):= x(0) if n = 0 Error for the x(n + T) − x(n) discrete derivative y(n−1) + if n > 0 T (b) 1 y(N) = 37.03% y(n) 0.5 0 0 50 100 150 200 250 n (c)Figure A-1 Discrete derivative: (a) exact exponential decay; (b) deÞ-nition of the discrete derivative; (c) exponential decay using the discretederivative.n = 0 to N − 1 that decays as −n x(n) = exp , 0 ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 155 Now consider the discrete approximation to this derivative, called y(n),and deÞne y(n)/ n as an approximation to the true derivative, as fol-lows: x(0) if n = 0 y(n) = (A-2) y(n − 1) + x(n+TT)−x(n) if n > 0T = 1 in this example. In this equation the second additive term is derived from an incre-ment of x (n). In other words, at each step in this process, y(n) hopefullydoes not change too much (in some situations with large sudden transi-tions, it might). The advantage that we get is an easy-to-calculate discreteapproximation to the exact derivative. Figure A-1c shows the decay of x (n) using the discrete derivative.In part (b) the accumulated error in the approximation is about 0.67%,which is pretty good. Smaller values of T can improve the accuracy; forexample, T = 0.1 gives an improvement to about 0.37%, but values ofT smaller than this are not helpful for this example. A ...