ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 161 VC Vc • IL • −R/L −1/C 1/L I L(0) I L VC(0) R (a) V O 1/s 1/s 1.0 1.0 X1(s) X1(s) • X2 X2 • −R/L (b) R V O 1/C 1/L U(s) 1/s −1/C 1/s 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 but with a sine-wave input signal u(t). The book by [Dorf and Bishop] explores this problem using several different methods that are very instructional but that we do not pursue in this book. The reader is encouraged to become more familiar with the network analysis methods described in this appendix. It is good practical engineering. Finally, Fig. A-4 illustrates the two varieties of ßow graph for the network discussed in this appendix. We can understand Fig. A-4a by referring to Eq. (A-5) with u set to zero (no external inputs) and with initial values of V C (0) and I L (0), as shown also in Fig. A-2. In Fig. A-4b, V C , I L , and their derivatives correspond to those in Eq. (A-5) with initial conditions V C and I L set to zero, as shown in Fig. A-3, and the input u drives the network from a zero start with a sine wave that starts at zero value. The output peak amplitude V O (t) ßuctuates for at least the 1000 time increments illustrated. It is also an interesting exercise for the reader to calculate and plot the inductor voltage and current and the capacitor voltage and current as functions of time n in Figs. A-2 and A-3. 162 DISCRETE-SIGNAL ANALYSIS AND DESIGN REFERENCES Dorf, 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. GLOSSARY Adjacent 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. Analytic signal (sequence). An X (k), its Hilbert transform ˆ 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. Sabin Copyright 2008 John Wiley & Sons, Inc. 163  164 GLOSSARY Boltzmann’s constant. 1.38 ×10 −23 joules per Kelvin. Used in noise calculations. Coherent. Two time signals x 1 (n)andx 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 −1to+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 =1tok =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 linear system analysis. GLOSSARY 165 Discrete frequency. Signals X (k ) in the frequency domain occur at dis- crete values of frequency (k) from 0 to N −1. Discrete time. Signals x(n) in the time domain occur at discrete values of time (n) from 0 to N −1. Digital signal processing (DSP). Signal processing in which signal amplitudes are also discrete (quantized). Even symmetry. The two sides, X (k )andX (N −k ), of a phasor spec- trum have the same phase. Expected value. The sum of products of a signal amplitude at time T and the probability of occurrence of the signal at time T [Eq. (6-1)]. Also known as the Þrst moment . Fast Fourier transform (FFT). A high-speed algorithm for the DFT. Flow graph. A graphical method of tracing the ßow of signals within a network. Fourier, Joseph. French mathematician who originated the trigonometric series method of analysis and design of mathematical and physical phenomena. Frequency domain. Signals are classiÞed according to their occurrence in frequency (f ) continuous or discrete X (k ). Frequency scaling. A sequence of frequency values have a certain sequential relationship from low end to high end. The maximum fre- quency minus the minimum frequency, divided by the number of frequencies, is the frequency scale factor. Gaussian noise. Random electrical noise, perhaps thermally generated noise, that has the Gaussian (normal) amplitude probability density function. Hermitian symmetry. A spectral property such that positive- and negative-frequency values are complex conjugates. The sine and cosine- wave phasors are Hermitian Hilbert transform. In RF work, an algorithm that modiÞes a two-sided phasor spectrum so that positive-frequency phasors are phase shifted −90 ◦ and negative-frequency phasors are phase shifted +90 ◦ . This idea is useful in many applications, especially in SSB. Integer. A collection of whole numbers: such as ±(1, 2, 3, ). . calculate and plot the inductor voltage and current and the capacitor voltage and current as functions of time n in Figs. A-2 and A-3. 162 DISCRETE-SIGNAL ANALYSIS AND DESIGN REFERENCES Dorf, R. C., and. understand Fig. A-4a by referring to Eq. (A-5) with u set to zero (no external inputs) and with initial values of V C (0) and I L (0), as shown also in Fig. A-2. In Fig. A-4b, V C , I L , and their. ADDITIONAL DISCRETE-SIGNAL ANALYSIS AND DESIGN INFORMATION 161 VC Vc • IL • −R/L −1/C 1/L I L(0) I L VC(0) R (a) V O 1/s