[INDEX](#page-15-0)

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INDEX

Abscissa of convergence, 336 Accumulator systems, 259, 295, 519 discrete-time Fourier transform of, 875–76 Active circuits, 382–85 Adders, 388, 396, 399, 403 Addition of complex numbers, 11–12 of matrices, 38 of sinusoids, 18–20 Additivity, 97–98 Algebra of complex numbers, 5–15 matrix, 38–42 Aliasing, 536–38, 788–95, 805, 811–12, 817, 834 defined, 536 general condition for in sinusoids, 793–96 treachery of, 788–91 verification of in sinusoids, 792–93 Aliasing error, 659, 811 Amplitude, 16 Amplitude modulation, 711–13, 736–49, 762 Amplitude response, 413, 416–17, 421, 424, 435, 437–38, 440–42 Amplitude spectrum, 598, 607, 615–25, 667, 668, 707, 848, 870, 878 Analog filters, 261 Analog signals, 133 defined, 78 digital processing of, 547–54 properties of, 78 Analog systems, 109, 135, 261 Analog-to-digital (A/D) conversion, 799–802, 831 Analog-to-digital converters (ADC) bit number, 801–2 bit rate, 801–2 Angle modulation, 736, 763 Angles electronic calculators in computing, 8–11 principal value of, 9 Angular acceleration, 116 Angular position, 116 Angular velocity, 116

Anti-aliasing filters, 537, 791, 834 Anti-causal exponentials, 862–63 Anti-causal signals, 81 Aperiodic signals, 133 discrete-time Fourier integral and, 855–67 Fourier integral and, 680–89, 762 properties of, 78–82 Apparent frequency, 534–36, 792–96 Ars Magna (Cardano), 2–5 Associative property, 171, 283 Asymptotic stability. See Internal stability. Audio signals, 713–14, 725, 746 Automatic position control system, 406–8 Auxiliary conditions, 153 differential equation solution and, 161 Backward difference system, 258, 295, 519, 568–69 Bandlimited signals, 533, 788, 792, 802 Bandpass filters, 441–43, 542–44, 749, 882–83, 896 group delay and, 726–27 ideal, 730–31, 882–83 poles and zeros of H(s) and, 443 Bandstop filters, 441–42, 445, 545–46 Bandwidth, 628 continuous-time systems and, 208–10 data truncation and, 751–53 essential, 736, 758–59 Fourier transform and, 692, 706, 762–63 Bartlett window, 753–55 Baseband signals, 737–40, 746–47, 749 Basis signals, 651, 655, 668 Basis vectors, 648 Beat effect, 741 Bhaskar, 2 Bilateral Laplace transform, 330, 335–37, 445–55, 467 properties of, 451–55 Bilateral z-transform, 431, 490, 554–63 in discrete-time system analysis, 563 properties of, 559–60 Bilinear transformation, 569–70 Binary digital signals, 799 Black box, 95, 119, 120

Blackman window, 755, 761 Block diagrams, 386–88, 405, 407, 408, 519 Bôcher, M., 620–21 Bode plots, 419–35 constant of, 421 first-order pole and, 424–27 pole at the origin and, 422–23 second-order pole and, 426–35 Bombelli, Raphael, 3 Bonaparte, Napoleon, 347, 610–11 Bounded-input/bounded-output (BIBO) stability, 110, 135, 263 of continuous-time systems, 196–97, 199–203, 222–23 of discrete-time systems, 298–99, 301–4, 314, 526, 527 frequency response and, 412–13 internal stability relationship to, 199–203, 301–4 of the Laplace transform, 371–73 signal transmission and, 721 steady-state response and, 418 of the z-transform, 518 Butterfly signal flow graph, 825 Butterworth filters, 440, 551–54 cascaded second-order sections for Butterworth filter realization, 461–63 MATLAB on, 459–63 transformation of, 571–72

Canonic direct realization. See Direct form II realization Cardano, Gerolamo, 2–5 Cartesian form, 8–15 Cascade realization, 394, 526, 920, 923 Cascade systems, 190, 192, 372–73 Cascaded RC filters, 461–62 Causal exponentials, 861–62 Causal signals, 81, 83, 134 Causal sinusoidal input in continuous-time systems, 418–19 in discrete-time systems, 527 Causal systems, 104–6, 135, 263 properties of, 104–6 zero-state response and, 172, 283 Cayley–Hamilton theorem, 910–12 Characteristic equations of continuous-time systems, 153–55 of discrete-time systems, 271, 273, 309 of a matrix, 910–12, 933 Characteristic functions, 192 Characteristic modes of continuous-time systems, 153–55, 162–65, 167, 170, 196, 198–99, 203–6 of discrete-time systems, 271–74, 278–79, 297–301, 305, 313 Characteristic polynomials of continuous-time systems, 153–56, 164, 166, 202–3, 220 of discrete-time systems, 271, 274–75, 279, 303–4 of the Laplace transform, 371–72 of the z-transform, 518

Characteristic roots of continuous-time systems, 153–56, 162, 166, 198–203, 206, 209, 211–12, 214–17, 222–24 of discrete-time systems, 271, 273–75, 297, 299–301, 303–5, 309, 314 invariance of, 942–43 of a matrix, 911, 932–33 Characteristic values. See Characteristic roots Characteristic vectors, 910 Chebyshev filters, 440, 463–66 Circular convolution, 819–20, 821 Clearing fractions, 26–27, 32–33, 342–43 Closed Loop systems. See Feedback systems Coherent demodulation. See Synchronous demodulation Coefficients of Fourier series, computation, 595–98 Column vectors, 36 Commutative property of the convolution integral, 170, 173, 181, 191–92 of the convolution sum, 283 Compact disc (CD), 801 Compact form of Fourier series, 597–98, 599, 600, 604–7 Complex factors of Q(x), 29 Complex frequency, 89–91 Complex inputs, 177, 297 Complex numbers, 1–15, 54 algebra of, 5–15 arithmetical operations for, 12–15 conjugates of, 6–7 historical note, 1–5 logarithms of, 15 origins of, 2–5 standard forms of, 14–15 useful identities, 7–8 working with, 13–14 Complex poles, 395, 432, 497, 542 Complex roots, 154–56, 274–76 Complex signals, 94–95 Conjugate symmetry of the discrete Fourier transform, 818–19 of the discrete-time Fourier transform, 858–59, 867–68 of the Fourier transform, 684, 703 Conjugation, 684, 703 Constants, 54, 98, 100, 103, 130, 422 Constant-parameter systems. See Time-invariant systems Continuous functions, 858 Continuous-time filters, 455–63 Continuous-time Fourier transform (CTFT), 867, 884–85 Continuous-time signals, 107–8, 135 defined, 78 discrete-time systems and, 238 Fourier series and, 593–679 Fourier transform and, 678–769, 680–775 Continuous-time systems, 135, 150–236 analog systems compared with, 261 differential equations of, 161, 196, 213

discrete-time systems compared with, 261 external input, response to, 168–96 frequency response of, 412–18, 732–33 internal conditions, response to, 151–63 intuitive insights into, 189–90, 203–5 Laplace transform, 330–487 Periodic inputs and, 637–641 properties of, 107–8 signal transmission through, 721–29 stability of, 196–203, 222–23 state equations for, 915–16 Control systems, 404–12 analysis of, 406–12 design specifications, 411 step input and, 407–9 Controllability/observability, 123–24 of continuous-time systems, 197, 200–2, 223 of discrete-time systems, 303, 965–68 in state-space analysis, 947–53, 961 Convergence abscissa of, 336 of Fourier series, 613–14 to the mean, 613, 614 region of. See region of convergence Convolution, 507–9 with an impulse, 283 of the bilateral z-transform, 560 circular, 819–20, 821 discrete-time, 311–12 fast, 821, 886 frequency. See Frequency convolution of the Fourier transform, 714–16 linear, 821 periodic, 886 time. See Time convolution Convolution integral, 170–93, 222, 282, 288, 313, 722 explanation for use, 189–90 graphical understanding of, 178–90, 217–20 properties of, 170–72 Convolution sum, 282–86, 313 graphical procedure for, 288–93 properties of, 282–83 from a table, 285–86 Convolution table, 175–76 Cooley, J. W., 824 Corner frequency, 424 Cramer’s rule, 23–25, 40, 51, 379, 385 Critically damped systems, 409, 410 Cubic equations, 2–3, 58 Custom filter function, 310–11 Cutoff frequency, 208, 209

Damping coefficient, 115–18 Dashpots linear, 115 torsional, 116

Data truncations, 749–55, 763 Decades, 422 Decibels, 421 Decimation-in-frequency algorithm, 824, 827 Decimation-in-time algorithm, 825–27 Decomposition, 99–100, 151 Delayed impulse, 168 Demodulation, 714 of amplitude modulation, 744–46 of DSB-SC signals, 739–41 synchronous, 743–44 Depressed cubic equation, 58 Derivative formulas, 56 Descartes, René, 2 Detection. See Demodulation Deterministic signals, 82, 134 Diagonal matrices, 37 Difference equations, 259–60, 265–70 causality condition in, 265–66 classical solution of, 298 differential equation kinship with, 260 frequency response, 532 order of, 260 recursive and non-recursive forms of, 259 recursive solution of, 266–70 sinusoidal response of difference equation systems, 528 z-transform solution of, 488, 510–19, 574 Differential equations, 161 classical solution of, 196 difference equation kinship with, 260 Laplace transform solution of, 346–48, 360–73 Differentiators digital, 256–58 ideal, 369–71, 373, 416–17 Digital differentiator example, 258–59 Digital filters, 108, 238, 261–62 Digital integrators, 258–59 Digital processing of analog signals, 547–53 Digital signals, 135, 797–99 advantages of, 261–62 binary, 799–801 defined, 78 L-ary, 799 properties of, 78 See also Analog-to-digital conversion Digital systems, 109, 135, 261 Dirac definition of an impulse, 88, 134 Dirac delta train, 696–97 Dirac, P.A.M., 86 Direct discrete Fourier transform (DFT), 808, 857 Direct form I (DFI) realization Laplace transform and, 390–91, 394 z-transform and, 521 See also Transposed direct form II realization

Direct form II (DFII) realization, 920–25, 954, 965, 967 Laplace transform and, 391, 398 z-transform and, 520–22, 525 Direct Fourier transform, 683, 702–3, 762 Direct z-transform, 488–592 Dirichlet conditions, 612, 614, 686 Discrete Fourier transform (DFT), 659, 805–23, 827–34, 835 aliasing and leakage and, 805–6 applications of, 820–23 computing Fourier transform, 812–18 derivation of, 807–10 determining filter output, 822–23 direct, 808, 857 discrete-time Fourier transform and, 885–86, 898 inverse, 808, 835, 857 MATLAB on, 827–34 picket fence effect and, 807 points of discontinuity, 807 properties of, 818–20 zero padding and, 810–11, 829–30 Discrete-time complex exponentials, 252 Discrete-time convolution, 311–12 Discrete-time exponentials, 247–49 Discrete-time Fourier integral, 855–67 Discrete-time Fourier series (DTFS), 845–55 computation of, 885–86 MATLAB on, 889–97 of periodic gate function, 853–55 periodic signals and, 846–47, 898 of sinusoids, 849–52 Discrete-time Fourier transform (DTFT), 857–88 of accumulator systems, 875–76 of anti-causal exponentials, 862–63 of causal exponentials, 861–62 continuous-time Fourier transform and, 883–86 existence of, 859, 886 inverse, 886 linear time-invariant discrete-time system analysis by, 879–80 MATLAB on, 889–97 physical appreciation of, 859 properties of, 867–78 of rectangular pulses, 863–65 table of, 860 z-transform connection with, 866–67, 886–88, 898 Discrete-time signals, 78, 79, 107–8, 133, 237–53 defined, 78 Fourier analysis of, 845–907 inherently bandlimited, 533 size of, 238–40 useful models, 245–53 useful operations, 240–45 Discrete-time systems, 135, 237–329 classification of, 262–64 controllability/observability of, 303, 965–68 difference equations of, 259–60, 265–70, 298

discrete-time Fourier transform analysis of, 878–83 examples of, 253–65 external input, response to, 280–98 frequency response of, 526–38 internal conditions, response to, 270–76 intuitive insights into, 305–6 properties of, 107–8, 264–65 stability of, 263, 298–305, 314 state-space analysis of, 953–64 z-transform analysis of, 488–592 Distinct factors of Q(x), 27 Distortionless transmission, 724–28, 730, 763, 880–82 bandpass systems and, 726–27, 881–82 measure of delay variation, 881 Distributive property, 171, 283 Division of complex numbers, 12–14 Double-sideband, suppressed-carrier (DSB-SC) modulation, 737–41, 742, 746–49 Downsampling, 243–44 Duality, 703–4 Dynamic systems, 103–4, 134–35, 263

Eigenfunctions, 193 Eigenvalues. See Characteristic roots Eigenvectors, 910 Einstein, Albert, 348 Electrical systems, 95–96, 111–14 Laplace transform analysis of, 373–85, 467 state equations for, 916–19 Electromechanical systems, 118–19 Electronic calculators, 8–11 Energy signals, 67, 82, 134, 239–40 Energy spectral density, 734, 763 Envelope delay. See Group delay Envelope detector, 743–45 Equilibrium states, 196, 198 Error signals, 650–51 Error vectors, 642 Essential bandwidth, 736, 758–59 Euler, Leonhard, 2, 3 Euler’s formula, 5–6, 45, 252 Even component of a signal, 93–95 Even functions, 92–93, 134 Everlasting exponentials continuous-time systems and, 189, 193–95, 222 discrete-time systems and, 296–97, 313 Fourier series and, 637, 638, 641 Fourier transform and, 687 Laplace transform and, 367–68, 412, 419 Everlasting signals, 81, 134 Exponential Fourier series, 621–37, 661, 803 periodic inputs and, 637–41 reasons for using, 640 symmetry effect on, 630–32

Exponential Fourier spectra, 624–32, 664, 667, 668 Exponential functions, 89–91, 134 Exponential input, 193, 296 Exponentials computation of matrix, 922–913 discrete-time, 247–49 discrete-time complex, 252 everlasting. See Everlasting exponentials matrix, 968–69 monotonic, 20–22, 90, 91, 134 sinusoid varying, 22–23, 90, 134 sinusoids expressed in, 20 Exposition du système du monde (Laplace), 346 External description of a system, 119–20, 135 External input continuous-time system response to, 168–96 discrete-time system response to, 280–98 External stability. See Bounded-input/bounded-output stability Fast convolution, 821, 886 Fast Fourier transform (FFT), 659, 811, 821, 824–27, 835 computations reduced by, 824 discrete-time Fourier series and, 847 discrete-time Fourier transform and, 885–86, 898 Feedback systems Laplace transform and, 386–88, 392–95, 399, 404–12 z-transform and, 521 Feedforward connections, 392–94, 403 Filtering discrete Fourier transform and, 821–23 MATLAB on, 308–10 selective, 748–49 time constant and, 207–8 Filters analog, 261 anti-aliasing, 537, 791, 834 bandpass, 441–43 bandstop, 441–42, 445, 545–46 Butterworth. See Butterworth Filters cascaded RC, 461–62 Chebyshev, 440, 463–66 continuous-time, 455–63 custom function, 310–11 digital, 108, 238, 261–62 finite impulse response, 524, 892–97 first-order hold, 785 frequency response of, 412–18 highpass, 443, 445, 542, 730–31, 882–83 Ideal. See Ideal filters impulse invariance criterion of, 548 infinite impulse response, 524, 565–74 lowpass, 439–41 lowpass. See Lowpass filters notch, 441–43, 540, 545–46

poles and zeros of H(s) and, 436–45 practical, 444–45, 882–83 sharp cutoff, 748 windows in design of, 755 zero-order hold, 785 Final value theorem, 359–61, 508 Finite impulse response (FIR) filters, 524, 892–97 Finite-duration signals, 333 Finite-memory systems, 104 First-order factors, method of, 497 First-order hold filters, 785 Folding frequency, 789–91, 793, 795, 817 For-loops, 216–18 Forced response difference equations and, 298 differential equations and, 198 Forward amplifiers, 405–6 Fourier integral, 722 aperiodic signal and, 680–89, 762 discrete-time, 855–67 Fourier series, 593–679 compact form of, 597–98, 599, 600, 604–7 computing the coefficients of, 595–98 discrete time. See Discrete-time Fourier series existence of, 612–13 exponential. See Exponential Fourier series generalized, 641–59, 668 Legendre, 656–57 limitations of analysis method, 641 trigonometric. See Trigonometric Fourier series waveshaping in, 615–17 Fourier spectrum, 598–607, 777 exponential, 624–32, 664, 667, 668 nature of, 858–59 of a periodic signal, 848–55 Fourier transform, 680–755, 778, 802–3 continuous-time, 867, 883–86 discrete. See Discrete Fourier transform discrete-time. See Discrete-time Fourier transform direct, 683, 702–3, 762 existence of, 685–86 fast. See fast Fourier transform interpolation and, 785 inverse, 683, 693–95, 699, 762, 786–87 physical appreciation of, 687–89 properties of, 701–21 useful functions of, 689–701 Fourier transform pairs, 683, 700 Fourier, Baron Jean-Baptiste-Joseph, 610–12 Fractions, 1–2 clearing, 26–27, 32–34, 342–43 partial. See Partial fractions Frequency apparent, 534–36, 793–94 complex, 89–91

Frequency (continued) corner, 424 cutoff, 208, 209 folding, 789–91, 793, 795, 817 fundamental, 594, 609–10, 846 negative, 626–28 neper, 91 radian, 16, 91, 594 reduction in range, 535 of sinusoids, 16 time delay variation with, 724–25 Frequency convolution of the bilateral Laplace transform, 452 of the discrete-time Fourier transform, 875–76 of the Fourier transform, 714–16 of the Laplace transform, 357 Frequency differentiation, 869 Frequency domain analysis, 368, 722–23, 848 of electrical networks, 374–78 of the Fourier series, 598, 601 two-dimensional view and, 732–33 See also Laplace transform Frequency inversion, 706 Frequency resolution, 807, 810–12, 815, 817 Frequency response, 724 Bode plots and, 419–22 of continuous-time systems, 412–18, 732–33 of discrete-time systems, 526–38 MATLAB on, 456–57, 531–32 periodic nature of, 532–36 from pole-zero location, 538–47 pole-zero plots and, 566–68 poles and zeros of H(s) and, 436–39 transfer function from, 435 Frequency reversal, 868–69 Frequency shifting of the bilateral Laplace transform, 451 of the discrete Fourier transform, 819 of the discrete-time Fourier transform, 871–74 of the Fourier transform, 711–13 of the Laplace transform, 353–54 Frequency spectra, 598, 601 Frequency-division multiplexing (FDM), 714, 749–50 Function M-files, 214–15 Functions characteristic, 193 continuous, 858 even, 92–93, 134 exponential, 89–91, 134 improper, 25–26, 34 interpolation, 690 MATLAB on, 126–33 odd, 92–95, 134 proper, 25–27

rational, 25–29, 338 singularity, 89 Fundamental band, 533, 534, 537, 793 Fundamental frequency, 594, 609–10, 846 Fundamental period, 79, 133, 239–40, 593, 595, 846

Gain enhancement by poles, 437–38 Gauss, Karl Friedrich, 3–4 Generalized Fourier series, 641–59, 668 Generalized linear phase (GLP), 726–27 Gibbs phenomenon, 619–21, 661–63 Gibbs, Josiah Willard, 620–21 Graphical interpretation of convolution integral, 178–90, 217–20 of convolution sum, 288–93 Greatest common factor of frequencies, 609–10 Group delay, 725–28, 881

H(s)

filter design and, 436–45 realization of, 548–49 See also Transfer functions Half-wave symmetry, 608 Hamming window, 754–55, 761 Hanning window, 754–55, 761 Hardware realization, 64, 95, 133 Harmonic distortion, 634 Harmonically related frequencies, 609 Heaviside “cover-up” method, 27–30, 33–35, 341, 342–43, 497 Heaviside, Oliver, 347–48, 612 Highpass filters, 443, 445, 542, 745, 747, 882–83 Homogeneity, 97–98

Ideal delay, 369, 416 Ideal differentiators, 369–71, 373, 416–17 Ideal filters, 730–33, 763, 785, 791, 834, 882–83 Ideal integrators, 369, 370, 373, 400, 416–18 Ideal interpolation, 786–87 Ideal linear phase (ILP), 725, 727 Ideal masses, 114 Identity matrices, 37 Identity systems, 109, 192, 263 Imaginary numbers, 1–5 Impedance, 374–77, 379, 380, 382, 384, 387, 399 Improper functions, 25–26, 34 Impulse invariance criterion of filter design, 548 Impulse matching, 164–66 Impulse response matrix, 938 Indefinite integrals, 57 Indicator function. See Relational operators Inertia, moment of, 116–18 Infinite impulse response (IIR) filters, 524, 565–74 Information transmission rate, 209–10

Initial conditions, 97–100, 102, 122, 134, 335 at 0− and 0+, 363–64 continuous-time systems and, 158–61 generators of, 376–83 Initial value theorem, 359–61, 508 Input, 64 complex, 177, 297 exponential, 193, 296 external. See External input in linear systems, 97 multiple, 178, 287–88 ramp, 410–11 sinusoidal. See Sinusoidal input step, 407–10 Input–output description, 111–19 Instantaneous systems, 103–4, 134, 263 Integrals convolution. See Convolutional integral discrete-time Fourier, 855–67 Fourier. See Fourier integral indefinite, 57 of matrices, 909–10 Integrators digital, 258–59 ideal, 369, 370, 373, 400, 416–18 system realization and, 400 Integro-differential equations, 360–73, 466, 488 Interconnected systems continuous-time, 190–93 discrete-time, 294–97 Internal conditions continuous-time system response to, 151–63 discrete-time system response to, 270–76 Internal description of a system, 119–21, 135, 908 See also State-space description of a system Internal stability, 110, 135, 263 BIBO relationship to, 199–203, 301–4 of continuous-time systems, 196–203, 222–23 of discrete-time systems, 298–302, 305, 314, 526, 527 of the Laplace transform, 372 of the z-transform, 518 Interpolation, 785–88 of discrete-time signals, 243–44 ideal, 786–87 simple, 785–86 spectral, 804 Interpolation formula, 779, 787 Interpolation function, 690 Intuitive insights into continuous-time systems, 189–90, 203–12 into discrete-time systems, 305–6 into the Laplace transform, 367–68 Inverse continuous-time systems, 192–93 Inverse discrete Fourier transform (IDFT), 808, 827, 857 Inverse discrete-time Fourier transform (IDTFT), 886 of rectangular spectrum, 865–66 Inverse discrete-time systems, 294–95 Inverse Fourier transform, 683, 693–95, 699, 762, 786–87 Inverse Laplace transform, 333, 335, 445, 549 finding, 338–46 Inverse z-transform, 488–89, 491, 499, 500, 501, 510, 554, 555, 559 finding, 495 Inversion frequency, 706 matrix, 40–42 Invertible systems, 109–10, 135, 263 Irrational numbers, 1–2 Kaiser window, 755, 760–62 Kelvin, Lord, 348 Kennelly-Heaviside atmosphere layer, 348 Kirchhoff’s laws, 95 current (KCL), 111, 213, 374 voltage (KVL), 111, 374 Kronecker delta functions, 245 bandlimited interpolation of, 787–88 L-ary digital signals, 799 L’Hôpital’s rule, 58, 211, 690 Lagrange, Louis de, 347, 612, 613 Laplace transform, 167, 330–487, 721 bilateral. See Bilateral Laplace transform differential equation solutions and, 346–48, 360–73 electrical network analysis and, 373–85, 467 existence of, 336–37 Fourier transform connection with, 699–701, 866 intuitive interpretation of, 367–69 inverse, 549, 938–39 properties of, 349–62 stability of, 371–74 state equation solutions by, 927–33 system realization and, 388–404 unilateral, 333–36, 337, 338, 345, 360, 445, 467 z-transform connection with, 488, 489, 491, 563–65 Laplace transform pairs, 333 Laplace, Marquis Pierre-Simon de, 346–47, 611, 612, 613 Leakage, 751, 753–55, 763, 805–6 Left half plane (LHP), 91, 198–99, 202, 211, 223, 435 Left shift, 71, 73, 130, 134, 503, 509, 510, 512 Left-sided sequences, 555–56 Legendre Fourier series, 656–57 Leibniz, Gottfried Wilhelm, 801 Linear convolution, 821 Linear dashpots, 115 Linear phase distortionless transmission and, 725, 881 generalized, 726–27 ideal, 725, 727

Linear phase (continued) physical description of, 707–9 physical explanation of, 870–71 Linear springs, 114 Linear systems, 97–101, 134 heuristic understanding of, 722–23 response of, 98–100 Linear time-invariant continuous-time (LTIC) systems. See Continuous-time systems Linear time-invariant discrete-time (LTID) systems. See Discrete-time systems Linear time-invariant (LTI) systems, 103, 194–95 Linear time-invariant discrete-time (LTID) systems, 879–80 Linear time-varying systems, 103 Linear transformation of vectors, 36, 939–47, 961 Linearity of the bilateral Laplace transform, 451 of the bilateral z-transform, 559 concept of, 97–98 of the discrete Fourier transform, 818, 824 of the discrete-time Fourier transform, 867 of discrete-time systems, 262 of the Fourier transform, 686–87, 824 of the Laplace transform, 331–32 of the z-transform, 489 Log magnitude, 27, 422–24 Loop currents continuous-time systems and, 159–63, 175 Laplace transform and, 375 Lower sideband (LSB), 738–39, 747 Lowpass filters, 439–41, 540–42 ideal, 730, 784–85, 788–89, 882–83 poles and zeros of H(s) and, 436–45 M-files, 212–20 function, 214–15 script, 213–14, 218 Maclaurin series, 6, 55 Magnitude response. See Amplitude response Marginally stable systems continuous-time, 198–200, 203, 211, 222–24 discrete-time, 301–2, 304, 314 Laplace transform, 373 signal transmission and, 721 z-transform, 519 Mathematical models of systems, 95–96, 125 MATLAB on Butterworth filters, 459–63 calculator operations in, 43–45 on continuous-time filters, 455–63 on discrete Fourier transform, 827–34 on discrete-time Fourier series and transform,

889–97

on discrete-time systems/signals, 306–12

elementary operations in, 42–53 on filtering, 308–10 Fourier series applications in, 661–67 Fourier transform topics in, 755–62 frequency response plots, 531–32 on functions, 126–33 impulse invariance, 553 impulse response and, 167 on infinite-impulse response filters, 565–74 M-files in, 212–20 matrix operations in, 49–53 multiple magnitude response curves, 544 partial fraction expansion in, 53 periodic functions, 661–63 phase spectrum, 664–67 polynomial roots and, 157 simple plotting in, 46–48 state-space analysis in, 961–69 vector operations in, 45–46 zero-input response and, 157–58 Matrices, 36–42 algebra of, 38–42 characteristic equation of, 909–10, 933 characteristic roots of, 932–33 computing exponential of, 912–13 definitions and properties of, 37–38 derivatives of, 909–10 diagonal, 37 diagonalization of, 943–44 equal, 37 functions of, 911–12 identity, 37 impulse response, 938 integrals of, 909–10 inversion of, 40–42 MATLAB operations, 49–53 nonsingular, 41 square, 36, 37, 41 state transition, 936 symmetric, 37 transpose of, 37–38 zero, 37 Matrix exponentials, 968–69 Matrix exponentiation, 968–69 Mechanical systems, 114–18 Memory, systems and, 104, 263 Memoryless systems. See Instantaneous systems Method of residues, 27 Michelson, Albert, 620–21 Minimum phase systems, 435, 436 Modified partial fractions, 35, 496 Modulation, 713–14, 736–49 amplitude, 711–13, 736, 742–46, 762 angle, 736, 763 of the discrete-time Fourier transform, 872

double-sideband, suppressed-carrier, 737–41, 742, 746–49 pulse-amplitude, 796 pulse-code, 796, 799 pulse-position, 796 pulse-width, 796 single-sideband, 746–49 Moment of inertia, 116–18 Monotonic exponentials, 20–22, 90, 91, 134 Multiple inputs, 178, 287–88 Multiple-input, multiple-output (MIMO) systems, 98, 125, 908 Multiplication bilateral z-transform and, 560 of complex numbers, 12–14 discrete-time Fourier transform and, 869 of a function by an impulse, 87 matrix, 38–40 scalar, 38, 400–1, 505 z-transform and, 506–7 Natural binary code (NBC), 799 Natural modes. See Characteristic modes Natural numbers, 1 Natural response difference equations and, 298 differential equations and, 196 Negative feedback, 406 Negative frequency, 626–28 Negative numbers, 1–3, 45 Neper frequency, 91 Neutral equilibrium, 197, 198 Newton, Sir Isaac, 2, 346–47 Noise, 66, 151, 371, 417, 791, 797–99 Nonanticipative systems. See Causal systems Non-bandlimited signals, 792 Noncausal signals, 81 Noncausal systems, 104–7, 135, 263 properties of, 104–6 reasons for studying, 106–7 Non-invertible systems, 109–10, 135, 263 Non-inverting amplifiers, 382 Nonlinear systems, 97–101, 134 Nonsingular matrices, 41 Non-uniqueness, 533 Normal-form equations, 915 Norton theorem, 375 Notch filters, 441–43, 540, 545–46. See also Bandstop filters Numerical integration, 131–33 Nyquist interval, 778, 779 Nyquist rate, 778–81, 788–89, 792, 795, 821 Nyquist samples, 778, 781, 782, 788, 792

Observability. See controllability/observability Octave, 422

Odd component of a signals, 93–95 Odd functions, 92–95, 134 Operational amplifiers, 382–83, 399, 467 Ordinary numbers, 1–5 Orthogonal signal space, 649–50 Orthogonal signals, 668 energy of the sum of, 647 signal representation by set, 647–59 Orthogonal vector space, 647–48 Orthogonality, 622 Orthonormal sets, 649 Oscillators, 203 Output, 64, 97 Output equations, 122, 124, 908, 930, 941 Overdamped systems, 409–10 Paley–Wiener criterion, 444, 731–32, 788 Parallel realization, 393–94, 525–26, 921, 924–25 Parallel systems, 190, 387 Parseval’s theorem, 632, 651–52, 734–35, 755, 758–59, 876–78 Partial fractions expansion of, 25–35, 53 inverse transform by partial fraction expansion and tables, 495–98 Laplace transform and, 338–39, 341, 344, 362, 394, 395, 419, 454 modified, 35 z-transform, 499 Passbands, 441, 444–45, 748, 755 Peak time, 409–10 Percent overshoot (PO), 409–10 Periodic (circular) convolution, 819–20 of the discrete-time Fourier transform, 875 Periodic extension of the Fourier spectrum, 848–55 properties of, 80–81 Periodic functions Fourier spectra as, 858 MATLAB on, 661–63 Periodic gate function, 853–55 Periodic signals, 133, 637–40 discrete-time Fourier series and, 846–47 Fourier spectra of, 848–55 Fourier transform of, 695–96 properties of, 78–82 and trigonometric Fourier series, 593–612, 661 Periods fundamental, 79, 133, 239–40, 593, 595, 846 sinusoid, 16 Phase response, 413–25, 427–35, 439, 467 Phase spectrum, 598, 607, 617–18, 707, 848 MATLAB on, 664–67 using principal values, 709–10

Phase-plane analysis, 125, 909 Phasors, 18–20 Physical systems. See Causal systems Picket fence effect, 807 Pickoff nodes, 190, 254–55, 396 Pingala, 801 Pointwise convergent series, 613 Polar coordinates, 5–6 Polar form, 8–15 arithmetical operations in, 12–15 sinusoids and, 18 Pole-zero location, 538–47 Pole-zero plots, 566–68 Poles complex, 395, 432, 497, 542 controlling gain by, 540 first-order, 424–27 gain enhancement by, 437–38 H(s), filter design and, 436–45 at the origin, 422–23 repeated, 395, 525, 926 in the right half plane, 371, 435–36 second-order, 426–35 wall of, 439–41, 542 Polynomial expansion, 458–59 Polynomial roots, 157, 572 Positive feedback, 406 Power series, 55 Power signals, 67, 82, 134, 239–40. See also Signal power Power, determining, 68–69 matrix, 912–13 Powers, of complex numbers, 13–16 Practical filters, 730–33, 882–83 Preece, Sir William, 349 Prewarping, 570–71 Principal values of the angle, 9 phase spectrum using, 709–10 Proper functions, 25–27 Pulse-amplitude modulation (PAM), 796 Pulse-code modulation (PCM), 796, 799 Pulse dispersion, 209 Pulse-position modulation (PPM), 796 Pulse-width modulation (PWM), 796 Pupin, M., 348 Pythagoras, 2

Quadratic equations, 58 Quadratic factors, 29–30 for the Laplace transform, 341–42 for the z-transform, 497 Quantization, 799, 831–34 Quantized levels, 799

Radian frequency, 16, 91, 594 Random signals, 82, 134 Rational functions, 25–29, 338 Real numbers, 2–7, 43 Real time, 105–6 Rectangular pulses, 863–65 Rectangular spectrum, 865–66 Rectangular windows, 751, 753–55, 763 Reflection property, 868–69 Region of convergence (ROC) for continuous-time systems, 193 for finite-duration signals, 333 for the Laplace transform, 331–33, 337, 347, 448, 449, 454–55, 467 for the z-transform, 489–91, 555–58, 561 Relational operators, 128–29 Repeated factors of Q(x), 31–32 Repeated poles, 395, 525, 926 Repeated roots of continuous-time systems, 154–56, 195, 198, 202, 223 of discrete-time systems, 270, 273–74, 297, 301, 313–14 Resonance phenomenon, 163, 204, 205, 210–12, 305 Right half plane (RHP), 91, 198, 200–3, 223, 371, 435–36 Right shift, 71–72, 131, 134, 501–4, 509, 510 Right-sided sequences, 555–56 Rise time, 206–7, 405, 409–10, 411 RLC networks, 914, 916–18 RMS value, 68–69, 70 Rolloff rate, 753, 754 Roots complex, 154–56, 274–76 of complex numbers, 11–15 polynomial, 157, 572 repeated. See Repeated roots unrepeated, 198, 202, 223, 301, 314 Rotational systems, 116–19 Rotational mass. See Moment of inertia Row vectors, 36, 45, 48–50

Sales estimate example, 255–56 Sallen–Key circuit, 383, 384, 461–62, 463, 466 Sampled continuous-time sinusoids, 527–31 Sampling, 776–844 practical, 781–84 properties of, 87–88, 134 signal reconstruction and, 785–99 spectral, 759–60, 802–4 See also Discrete Fourier transform; Fast-Fourier transform Sampling interval, 550–54 Sampling rate, 243–44, 536–37 Sampling theorem, 537, 776–84, 834–35 applications of, 796–99 spectral, 802 Savings account example, 253–55

Scalar multiplication, 38, 400–1, 505, 509, 520 Scaling, 97–98, 130 of the Fourier transform, 705–6, 755, 757, 762 of the Laplace transform, 357 See also Time scaling Script M-files, 213–14, 216, 218 Selective-filtering method, 748–49 Sharp cutoff filters, 748 Shifting of the bilateral z-transform, 559 of the convolution integral, 171–72 of the convolution sum, 283 of discrete-time signals, 240 See also Frequency shifting; Time shifting Sideband, 746–49 sifting. See Sampling Signal distortion, 723–25 Signal energy, 65–66, 70, 131–33, 733–36, 757, 877–78. See also Energy signals Signal power, 65–67, 133. See also Power signals Signal reconstruction, 785–99. See also Interpolation Signal-to-noise power ratio, 66 Signal transmission, 721–29 Signals, 64–91, 133–34 analog. See Analog signals anti-causal, 81 aperiodic. See Aperiodic signals audio, 713–14, 725, 746 bandlimited, 533, 788, 792, 802 baseband, 737–40, 746–47, 749 basis, 651, 655, 668 causal, 81, 83, 134 classification of, 78–82, 133–34 comparison and components of, 643–45 complex, 94–95 continuous time. See continuous-time signals defined, 65 deterministic, 83, 134 digital. See Digital signals discrete time. See Discrete-time signals energy, 82, 134, 239–40 error, 650–51 even components of, 93–95 everlasting, 81, 134 finite-duration, 333 modulating, 711, 737–39 non-bandlimited, 792 noncausal, 81 odd components of, 93–95 orthogonal. See Orthogonal signals periodic. See Periodic signals phantoms of, 189 power, 82, 134, 239–40 random, 82, 134 size of, 64–70, 133

sketching, 20–23 time reversal of, 77 time limited, 802, 805, 807 two-dimensional view of, 732–33 useful models, 82–91 useful operations, 71–78 as vectors, 641–59 video, 725, 749 Sinc function, 757 Single-input, single-output (SISO) systems, 98, 125, 908 Single-sideband (SSB) modulation, 746–49 Singularity functions, 89 Sinusoidal input causal. See Causal sinusoidal input continuous-time systems and, 208 discrete-time systems and, 309 frequency response and, 413–17 steady-state response to causal sinusoidal input, 418–19 Sinusoids, 16–20, 89–91, 134 addition of, 18–20 apparent frequency of sampled, 795–96 compression and expansion, 76 continuous-time, 251–52, 533–37 discrete-time, 251, 527, 528, 533–37 discrete-time Fourier series of, 849–52 in exponential terms, 20 exponentially varying, 22–23, 80, 134 general condition for aliasing in, 793–96 power of a sum of two equal-frequency, 70 sampled continuous-time, 527–31 verification of aliasing in, 792–93 Sketching signals, 20–23 Sliding-tape method, 290–93 Software realization, 64, 95, 133 Spectral density, 688 Spectral folding. See Aliasing Spectral interpolation, 804 Spectral resolution, 807 Spectral sampling, 759–60, 802 Spectral sampling theorem, 802 Spectral spreading, 751–53, 755, 763, 807 Springs linear, 114 torsional, 116–17 Square matrices, 36, 37, 41 Square roots of negative numbers, 2–4 Stability BIBO. See Bounded-input/bounded-output stability of continuous-time systems, 196–203, 222–23 of discrete-time systems, 263, 298–305, 314 of the Laplace transform, 371–74 Internal. See Internal stability of the z-transform, 518–19 marginal. See marginally stable systems

Stable equilibrium, 196–97 Stable systems, 110, 263 State equations, 122–25, 135, 908–9, 969 alternative procedure to determine, 918–19 diagonal form of, 944–47 solution of, 926–39 for the state vector, 941–42 systematic procedure for determining, 913–26 time-domain method to solve, 936–37 State transition matrix (STM), 936 State variables, 121–25, 135, 908, 969 State vectors, 927–30, 961 linear transformation of, 941–42 State-space analysis, 908–73 controllability/observability in, 947–53, 961 of discrete-time systems, 953–64 in MATLAB, 961–69 transfer function and, 920–24 transfer function matrix, 931–32 State-space description of a system, 121–25 Steady-state error, 409–11 Steady-state response in continuous-time systems, 418–19 in discrete-time systems, 527 Stem plots, 306–8 Step input, 407–10 Stiffness of linear springs, 114 of torsional springs, 116–17 Stopbands, 441, 444, 445, 456, 457, 459, 460, 463, 755 Subcarriers, 749 Subtraction of complex numbers, 11–12 Superposition, 98, 99, 100, 123, 134 continuous-time systems and, 168, 170, 178 discrete-time systems and, 287 Symmetric matrices, 37 Symmetry conjugate. See Conjugate symmetry exponential Fourier series and, 630–32 trigonometric Fourier series and, 607–8 Synchronous demodulation, 743–44, 747 System realization, 388–404, 519–25, 567 cascade, 394, 525–26, 919–20, 923 of complex conjugate poles, 395 direct. See Direct form I realization; Direct form II realization differences in performance, 525–26 hardware, 64, 95, 133 parallel. See Parallel realization software, 64, 95, 129 Systems, 95–133, 134–35 accumulator, 259, 295, 519 analog, 109, 135, 261 backward difference, 258, 295, 519, 568–69 BIBO stability, assessing, 110 cascade, 190, 192, 372, 373

causal. See causal systems causality, assessing, 105 classification of, 97–110, 134–35 continuous time. See Continuous-time systems control. See control systems critically damped, 409, 410 data for computing response, 96–97 defined, 64 digital, 78, 135, 261 discrete time. See discrete time systems dynamic, 103–4, 134–35, 263 electrical, 95–96, 111–14 electrical. See Electrical systems electromechanical, 118–19 feedback. See feedback systems finite-memory, 104 identity, 109, 192, 263 input–output description, 111–19 instantaneous, 103–4, 263 interconnected. See interconnected systems invertible, 109–10, 135, 263 linear. See Linear systems mathematical models of, 95–96, 125 mechanical, 114–18 memory and, 104, 263 minimum phase, 435, 436 multiple-input, multiple-output, 98, 125, 908 noncausal, 104–7, 263 non-invertible, 109–10, 135 nonlinear, 97–101, 134 overdamped, 409–10 parallel, 190, 387 phantoms of, 189 properties of, 264–65 rotational, 116–19 single-input, single-output, 98, 125, 908 stable, 110, 263 translational, 114–16 time invariant. See Time-invariant systems time varying. See Time-varying systems two-dimensional view of, 732–33 underdamped, 409 unstable, 110, 263

Tacoma Narrows Bridge failure, 212 Tapered windows, 753–54, 763, 807 Taylor series, 55 Théorie analytique de la chaleur (Fourier), 612 Thévenin’s theorem, 375, 378, 379 Time constant of continuous-time systems, 205–10, 223 of the exponential, 21–22 filtering and, 207–9 information transmission rate and, 209–10

pulse dispersion and, 209 rise time and, 206–7 Time convolution of the bilateral Laplace transform, 452 of the discrete-time Fourier transform, 875–76 of the Fourier transform, 714–16 of the Laplace transform, 357 of the z-transform, 507–8 Time delay, variation with frequency, 724–25 Time differentiation of the bilateral Laplace transform, 451 of the Fourier transform, 716–18 of the Laplace transform, 354–56 Time integration of the bilateral Laplace transform, 451 of the Fourier transform, 716–18 of the Laplace transform, 356–57 Time inversion, 706 Time reversal, 134 of the bilateral Laplace transform, 452 of the bilateral z-transform, 560 of the convolution integral, 178, 181 described, 76–77 of the discrete-time Fourier transform, 868–69 of discrete-time signals, 242 of the z-transform, 506–7 Time scaling, 77 of the bilateral Laplace transform, 452 described, 73–74 Time shifting, 77, 79 of the bilateral Laplace transform, 451 of the convolution integral, 178 described, 71–73 of the discrete Fourier transform, 819 of the discrete-time Fourier transform, 870 of the Fourier transform, 707 of the Laplace transform, 349–51 of the z-transform, 501–5, 510 Time-division multiplexing (TDM), 749, 797 Time-domain analysis, 723 of continuous-time systems, 150–236 of discrete-time systems, 237–329 of the Fourier series, 598, 601 of interpolation, 785–88 state equation solution in, 933–39 two-dimensional view and, 732–33 Time-frequency duality, 702–3, 723, 753 Time invariant systems, 134 discrete-time, 262 linear. See Linear time-invariant systems properties of, 102–3 Time-varying systems, 134 discrete-time, 262 linear, 103 properties of, 102–3

Time-limited signals, 802, 805, 807 Torque, 116–18 Torsional dashpots, 116 Torsional springs, 116, 117 Total response of continuous-time systems, 195–96 of discrete-time systems, 297–98 Traité de mécanique céleste (Laplace), 346 Transfer functions, 522 analog filter realization with, 548–49 block diagrams and, 386–88 of continuous-time systems, 193–94, 222 of discrete-time systems, 296–97, 314, 514–15, 567–68 from the frequency response, 435 inadequacy for system description, 953 realization of, 389–99, 401, 524–25 state equations from, 916, 919–26 from state-space representations, 964–65 Translational systems, 114–16 Transpose of a matrix, 37–38 Transposed direct form II (TDFII) realization, 398, 967–69 state equations and, 920–24 z-transform and, 520–22, 525–26 Triangular windows, 751 Trigonometric Fourier series, 640, 652, 657–58, 667, 668 exponential, 621–37, 661 periodic signals and, 593–612, 661 sampling and, 777, 782 symmetry effect on, 607–8 Trigonometric identities, 55–56 Tukey, J. W., 824

Underdamped systems, 409 Uniformly convergent series, 613 Unilateral Laplace transform, 333–36, 337, 338, 345, 360, 445, 467 Unilateral z-transform, 489, 491, 492, 495, 554–55, 559 Uniqueness, 335 Unit delay, 517, 520, 521 Unit-gate function, 689 Unit-impulse function, 133 of discrete-time systems, 246–47, 280, 313 as a generalized function, 88–89 properties of, 86–89 Unit-impulse response of continuous-time systems, 163–68, 170, 189–93, 220–21, 222, 731 convolution with, 171 determining, 221 of discrete-time systems, 277–80, 286, 295, 313 Unit matrices, 37 Unit-step function, 84–86, 88–89 of discrete-time systems, 246–47 relational operators and, 128–30

988 Index

Unit-triangle function, 689–90 Unrepeated roots, 198, 202, 223, 301, 314 Unstable equilibrium, 196–97 Unstable systems, 110, 263 Upper sideband (USB), 737–39, 746–48 Upsampling, 243–44 Vectors, 36–37, 641–59 basis, 648 characteristic, 910 column, 36 components of, 642–43 error, 642 MATLAB operations, 45–46 matrix multiplication by, 40 orthogonal space, 647–48 row, 36, 45, 48–50 signals as, 641–59 state, 927–30, 961 Vestigial sideband (VSB), 749 Video signals, 725, 749 Waveshaping, 615–17 Weber–Fechner law, 421 Width of the convolution integral, 172, 187 of the convolution sum, 283 Window functions, 749–55, 760–62 z-transform, 488–592 bilateral. See Bilateral z-transform difference equation solutions of, 488, 510–19, 574 direct, 488–592 discrete-time Fourier transform and, 866–67, 886–88, 898 existence of, 491–95

inverse. See inverse z-transform

properties of, 501–9 stability of, 518–19 state-space analysis and, 956, 959–65 system realization and, 519–25, 567 time-reversal property, 506–7 time-shifting properties, 501–5 unilateral, 489, 491, 492, 495, 554–55, 559 z-domain differentiation property, 506 z-domain scaling property, 505 Zero matrices, 37 Zero padding, 810–11, 829–30 Zero-input response, 119, 123 of continuous-time systems, 151–63, 195–96, 203, 220–22 described, 98–100 of discrete-time systems, 270–76, 297–301, 309–11 insights into behavior of, 161–63 of the Laplace transform, 363, 368 in oscillators, 203 of the z-transform, 512–13 zero-state response independence from, 161 Zero-order hold (ZOH) filters, 785 Zero-state response, 119, 123 alternate interpretation, 515–18 causality and, 172–73 of continuous-time systems, 151, 161, 168–96, 221–22, 512–16 described, 98–101 of discrete-time systems, 280–98, 308–9, 311, 312, 313 of the Laplace transform, 358, 363, 366–67, 369, 370 zero-input response independence from, 161 Zeros controlling gain by, 540 filter design, 436–45 first-order, 424–27 gain suppression by, 439–40 at the origin, 422–23

second-order, 426–35