THE HILBERT TRANSFORM 151 expensive. Receivers very often combine the phasing and Þlter methods in the same or different signal frequency ranges to get greatly improved performance 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 intended only to show the Hilbert transform and its mathematical equivalent in a few speciÞc applications. Further and more complete information is available from a wide variety of sources [e.g., Sabin and Schoenike, 1998], that cannot be pursued adequately in this introductory book, which has emphasized the analysis and design of discrete signals in the time and frequency domains. REFERENCES Bedrosian, 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. APPENDIX Additional Discrete-Signal Analysis and Design Information † This brief Appendix will provide a few additional examples of how Math- cad can be used in discrete math problem solving. The online sources and Mathcad User Guide and Help (F1) are very valuable sources of infor- mation on speciÞc questions that the user might encounter in engineering and other technical activities. The following material is guided by, and is similar to, that of Dorf and Bishop [2004, Chap. 3]. DISCRETE DERIVATIVE We consider Þrst Fig. A-1, the discrete derivative, which can be a useful tool 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 use in this appendix, text and graphical material similar to that in Chapter 3 of [Dorf and Bishop, 2004]. Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 153 154 DISCRETE-SIGNAL ANALYSIS AND DESIGN N := 256 n := 0,1 N − x(n) := e n N T := 1 0 50 100 150 200 250 0 0.5 1 y(n) n (a) (b) + (c) y(n):= x(0) if n = 0 y(n−1) x(n + T) − x(n) T if n > 0 0 50 100 150 200 250 0 0.5 x(n) 1 n x(N) = 36.79% y(N) − x(N) x(N) = 0.67% y(N) = 37.03% Error for the discrete derivative Figure A-1 Discrete derivative: (a) exact exponential decay; (b) deÞ- nition of the discrete derivative; (c) exponential decay using the discrete derivative. n =0toN −1 that decays as x(n) = exp −n N ,0<n<N−1(A-1) The decay of this function from n =0toN is from 1.0 to 0.3679, corresponding to a time constant of 1.0. Figure A-1 shows the exact decay. 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: y(n) = x(0) if n = 0 y(n −1) + x(n+T)−x(n) T if n>0 (A-2) T =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) hopefully does not change too much (in some situations with large sudden transi- tions, it might). The advantage that we get is an easy-to-calculate discrete approximation 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; for example, T =0.1 gives an improvement to about 0.37%, but values of T smaller than this are not helpful for this example. A larger number of samples, such as 2 9 , is also helpful. The discrete derivative can be very useful in discrete signal analysis and design. STATE-VARIABLE SOLUTIONS We will use the discrete derivative and matrix algebra to solve the two- state differential equation for the LCR network in Fig. A-2. There are two energy storage elements, L and C , in the circuit. There is a voltage across and a displacement current through the capacitor C , and a voltage across and an electronic current through the inductor L. We want all of these as a function of time t. There are also possible initial conditions at t =0, which are a voltage V C0 on the capacitor and a current I L0 through the inductor, and a generator (u) (in this case, a current source) is con- nected as shown. The two basic differential equations are, in terms of v C and i L , . and graphical material similar to that in Chapter 3 of [Dorf and Bishop, 2004]. Discrete-Signal Analysis and Design, By William E. Sabin Copyright 2008 John Wiley & Sons, Inc. 153 154 DISCRETE-SIGNAL. Press, New York. Williams, A. B., and F. J. Taylor, 1995, Electronic Filter Design Handbook , 3rd ed., McGraw-Hill, New York. APPENDIX Additional Discrete-Signal Analysis and Design Information † This. sources [e.g., Sabin and Schoenike, 1998], that cannot be pursued adequately in this introductory book, which has emphasized the analysis and design of discrete signals in the time and frequency domains. REFERENCES Bedrosian,