DISCRETE-SIGNAL ANALYSIS AND DESIGN- P36

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P36: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- P36 ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 161 VC(0) IL(0) • 1.0 • 1.0 VC IL VO 1/s 1/L 1/s R Vc −R/L IL −1/C (a) • • 1/C X1(s) 1/s 1/L X2 1/s X2 R VO U(s) X1(s) −R/L −1/C (b)Figure A-4 Flow-chart for the network of Fig. A-2: (a) with no input(u) but with initial values of V C and I L ; (b) with no initial conditions butwith a sine-wave input signal u(t). The book by [Dorf and Bishop] explores this problem using severaldifferent methods that are very instructional but that we do not pursuein this book. The reader is encouraged to become more familiar with thenetwork analysis methods described in this appendix. It is good practicalengineering. Finally, Fig. A-4 illustrates the two varieties of ßow graph for thenetwork discussed in this appendix. We can understand Fig. A-4a byreferring to Eq. (A-5) with u set to zero (no external inputs) and withinitial values of VC (0) and IL (0), as shown also in Fig. A-2. In Fig. A-4b,VC , IL , and their derivatives correspond to those in Eq. (A-5) with initialconditions VC and IL set to zero, as shown in Fig. A-3, and the input udrives the network from a zero start with a sine wave that starts at zerovalue. The output peak amplitude VO (t) ßuctuates for at least the 1000time increments illustrated. It is also an interesting exercise for the reader to calculate and plotthe inductor voltage and current and the capacitor voltage and current asfunctions of time n in Figs. A-2 and A-3.162 DISCRETE-SIGNAL ANALYSIS AND DESIGNREFERENCESDorf, R. C., and R. H. Bishop, 2004, Modern Control Systems, 10th ed., Prentice Hall, Upper Saddle River, NJ, Chap. 3.Zwillinger, D., Ed., 1996, CRC Standard Mathematical Tables and Formulae, 30th ed., CRC Press, Boca Raton, FL.GLOSSARYAdjacent channel interference. One or more adjacent channel signals create interference in a desired channel by aliasing or wideband emissions.Aliasing (classical). In positive-only frequency systems, a signal in part of the positive-frequency region is invaded by a second signal that is in an adjacent part of the positive-frequency region.Aliasing. The overlapping (invasion) from one 0 to N − 1 time or fre- quency sequence to an adjacent 0 to N − 1 time or frequency sequence.Amplitude noise. Noise created by variations in the amplitude of a signal. ˆAnalytic signal (sequence). An X (k ), its Hilbert transform X(k) and the ±j operator combine to create a phasor sequence that is one- sided in the positive- or negative-frequency domain. The phasor A exp(±j θ) is an analytic signal. The analytic phasor sequence is used to construct SSB signals digitally or discretely. It is synthesized to design analog SSB systems.Auto-covariance. The ac component of an autocorrelation.Average value. The time average of a signal between two time limits, often minus inÞnity to plus inÞnity.Discrete-Signal Analysis and Design, By William E. SabinCopyright  2008 John Wiley & Sons, Inc. 163164 GLOSSARYBoltzmann’s constant. 1.38 × 10−23 joules per Kelvin. Used in noise calculations.Coherent. Two time signals x 1 (n) and x 2 (n) are coherent if their x (n) values add together algebraically at each (n). In the frequency domain the X (k )s add in a similar manner.Complex frequency domain. Values of X (k ) phasors contain a real part, an imaginary part, an amplitude value, a frequency value, and a phase value relative to some reference phase value. The domain has a positive- frequency side and an equal-length negative-frequency side.Complex plane. The two-dimensional rectangular plane of the real axis (x ) and the imaginary axis (jy) (see Fig. 1-5).Complex signal. A signal that is deÞned as part real and part imaginary on the complex plane. In the time domain, sequences can be complex. In the frequency domain, a single phasor can be complex.Convolution. A fold, slide, and multiply operation to obtain an overlap area between two geometric or mathematical regions.Correlation. A measure of the similarity of a function and a time- or frequency-shifted copy of the function (auto correlation) or the similar- ity of two different functions, one of which is shifted (cross-correlation).Correlation coefÞcient. A measure of the “relatedness” in some sense, from −1 to +1, of two nondeterministic or deterministic processes.Cross-covariance. The ac component of a cross-correlation.Cross power spectrum. The commonality of power spectrum in two associated signals.Discrete derivative. An approximate implementation of a time-derivative that uses the discrete sequence x (n).Discrete Fourier series. In discrete-signal length-N analysis, a periodic repeating waveform can be deÞned as a useful set of positive-frequency harmonics from k = 1 to k = N /2 − 1.Discrete Fourier transform (DFT). Converts the time domain x (n) to the frequency domain X (k ).Discrete Fourier transform of convolution. Converts a convolution of two time sequences to the product of two frequency sequences: the system function. Used in line ...