DISCRETE-SIGNAL ANALYSIS AND DESIGN- P9:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | 26 DISCRETE-SIGNAL ANALYSIS AND DESIGN be shown if we like on a separate phase-angle graph Fig. 1-6 . Finally to reconstruct the time plot in Fig. 1-2c the two rotating X k phasors in Fig. 1-2b re-create the sinusoidal time sequence x n using the IDFT of Eq. 1-8 . Figure 1-6 should be studied as an example of converting exp -n 20 from time to frequency and phase and back to time. Note that parts a and d show only the positive-time part of the x n waveform. The negative-time part is a mirror image and is occasionally not shown but it is never ignored. There is one other thing about sequences. Because in this book they are steady-state signals in which all transients have disappeared it does not matter where they came from. They can be solutions to differential equations or signal generator output at the end of a long nonlinear transmission line etc. etc. The DFT and IDFT do not identify the source of the sequences only tell the relationship between the steady-state time domain and the steady-state frequency domain. We should avoid trying to make anything more than that out of them. Other methods do a much better job of tracing the origins of sequences in time and frequency. The Appendix shows a simple example of this interesting and very important activity. REFERENCES Bracewell R. 1986 The Fourier Transform and its Applications McGraw-Hill New York. Carlson A. B. 1986 Communication Systems 3rd ed. McGraw-Hill New York. Oppenheim . A. Willsky and I. Young 1983 Signals and Systems Prentice-Hall Englewood Cliffs NJ. Stanley W. D. et al. 1984 Digital Signal Processing 2nd ed. Reston Publishing Reston VA. 2 Sine Cosine and 0 In the spectrum X k where k in this chapter is conhned to integer values the hrst N 2 1 are a collection of positive-frequency phasors. It is sometimes sufficient to work with just this information. Another approach is usually more desirable and is easy to accomplish. Figure 2-1 illustrates a problem that occurs frequently when we use only .
