preface xi 1 signals and systems 1 1.1 signals 2 1.1.1 audio signals 3 probing further: household electrical power 7 1.1.2 images 9 1.1.3 video signals 11 probing further: color and light 12 1.1.4 signals representing physical attributes 15 1.1.5 sequences 16 1.1.6 discrete signals and sampling 18 1.2 systems 23 1.2.1 systems as functions 24 1.2.2 telecommunications systems 25 probing further: wireless communication 27 probing further:leo telephony 28 probing further: encrypted speech 32 .1.2.3 audio storage and retrieval 33 1.2.4 modem negotiation 34 1.2.5 feedback control systems 35 1.3 summary 40 2 defining signals and systems 45 2.1 defining functions 46 2.1.1 declarative assignment 47 2.1.2 graphs 48 probing further:relations 50 2.1.3 tables 51 2.1.4 procedures 52 2.1.5 composition 53 2.1.6 declarative versus imperative 56 probing further:declarative interpretation of imperative definitions 57 2.2 defining signals 59 2.2.1 declarative definitions 59 2.2.2 imperative definitions 60 2.2.3 physical modeling 61 probing further:physics of a tuningfork 61 2.3 defining systems 63 2.3.1 memoryless systems and systems with memory 63 2.3.2 differential equations 65 2.3.3 difference equations 66 2.3.4 composing systems by using block diagrams 68 basics:summations 69 probing further: composition of graphs 71 2.4 summary 74 interview: panos antsaklis 83 3 state machines 85 3.1 structure of state machines 86 3.1.1 updates 87 3.1.2 stuttering 88 3.2 finite-state machines 90 3.2.1 state transition diagrams 90 3.2.2 update table 96 3.3 nondeterministic state machines 100 3.3.1 state transition diagram 100 3.3.2 sets and functions model 103 3.4 simulation and bisimulation 106 3.4.1 relating behaviors 112 3.5 summary 115 4 composing state machines 123 4.1 synchrony 123 4.2 side-by-side composition 125 4.3 cascade composition 128 4.4 product-form inputs and outputs 132 4.5 general feed-forward composition 135 4.6 hierarchical composition 138 4.7 feedback 139 4.7.1 feedback composition with no inputs 140 4.7.2 state-determined output 145 4.7.3 feedback composition with inputs 149 4.7.4 constructive procedure for feedback composition 153 4.7.5 exhaustive search 156 probing further: constructive semantics 157 4.7.6 nondeterministic machines 158 4.8 summary 158 interview: gerard berry 166 5 linear systems 169 5.1 operation of an infinite-state machine 170 basics:functions yielding tuples 172 5.1.1 time 173 basics:matrices and vectors 174 basics:matrix arithmetic 175 5.2 linear functions 176 5.3 the [a,b,c,d] representation of a discrete linear system 179 5.3.1 impulse response 181 5.3.2 one-dimensional siso systems 183 5.3.3 zero-state and zero-input response 188 5.3.4 multidimensional siso systems 191 5.3.5 multidimensional mimo systems 199 probing further: impulse responses of mimo systems 200 5.3.6 linear input-output function 201 5.4 continuous-time state-space models 201 probing further: approximating continuous-time systems 202 5.5 summary 203 6 hybrid systems 209 6.1 mixed models 211 6.2 modal models 213 6.3 timed automata 216 probing forther: internet protocols 224 6.4 more interesting dynamics 226 6.5 supervisory control 231 6.6 formal model 237 6.7 summary 239 interview: pr. kumar 244 7 frequency domain 247 7.1 frequency decomposition 248 basics: frequencies in hertz and radians 248 basics: ranges of frequencies 249 probing further: circle of fifths 251 7.2 phase 253 7.3 spatial frequency 254 7.4 periodic and finite signals 255 7.5 fourier series 258 probing further: uniform convergence of the fourier series 262 probing further: mean square convergence of the fourier series 263 probing further: dirichlet conditions for validity of the fourier series 263 7.5.1 uniqueness of the fourier series 265 7.5.2 periodic, finite, and aperiodic signals 266 7.5.3 fourier series approximations to images 266 7.6 discrete-time signals 268 7.6.1 periodicity 268 basics: discrete-time frequencies 269 7.6.2 the discrete-time fourier series 270 7.7 summary 270 8 frequency response 277 8.1 ltl systems 278 8.1.1 time invariance 278 8.1.2 linearity 283 8.1.3 linearity and time invariance 286 8.2 finding and using the frequency response 289 8.2.1 linear difference and differential equations 292 basics: sinusoids in terms of complex exponentials 294 tips and tricks:phasors 294 8.2.2 the fourier series with complex exponentials 301 probing further: relating dfs coefficients 303 8.2.3 examples 304 8.3 determining the fourier series coefficients 305 probing further: formula for fourier series coefficients 306 probing further: exchanging integrals and summations 307 8.3.1 negative frequencies 307 8.4 frequency response and the fourier series 307 8.5 frequency response of composite systems 309 8.5.1 cascade connection 309 8.5.2 feedback connection 311 probing further: feedback systems are lti 312 8.6 summary 315 interview:dawn tilbury 323 9 filtering 325 9.1 convolution 328 9.1.1 convolution sum and integral 328 9.1.2 impulses 332 9.1.3 signals as sums of weighted delta functions 333 9.1.4 impulse response and convolution 335 9.2 frequency response and impulse response 338 9.3 causality 342 probing further: causality 342 9.4 finite impulse response filters 343 9.4.1 design of fir filters 346 9.4.2 decibels 349 probing further:decibels 350 9.5 infinite impulse response (iir) filters 351 9.5.1 designing iir filters 352 9.6 implementation of filters 355 9.6.1 matlab implementation 355 probing further:joua implementation of an htr filter 356 probing further: programmable dsp implementation of an fir filter 357 9.6.2 signal flow graphs 358 9.7 summary 361 lo the four fourier transforms 369 10.1 notation 370 10.2 the fourier series 370 probing further: showing inverse relations 372 10.3 the discrete fourier transform 376 10.4 the discrete-time fourier transform 380 10.5 the continuous-time fourier transform 383 10.6 fourier transforms versus fourier series 385 10.6.1 fourier transforms of finite signals 385 10.6.2 fourier analysis of a speech signal 387 10.6.3 fourier transforms of periodic signals 390 10.7 properties of fourier transforms 393 10.7.1 convolution 393 probing further:multiplying signals 398 10.7.2 conjugate symmetry 399 10.7.3 time shifting 401 10.7.4 linearity 404 10.7.5 constant signals 405 10.7.6 frequency shifting and modulation 407 10.8 summary 408 interview:jeff bier 422 il sampling and reconstruction 425 11.1 sampling 425 11.1.1 sampling a sinusoid 426 basics: units 426 11.1.2 aliasing 426 11.1.3 perceived pitch experiment 428 11.1.4 avoiding aliasing ambiguities 431 probing further: antialiasing for fonts 432 11.2 reconstruction 433 11.2.1 a model for reconstruction 434 probing further:sampling 437 probing further: impulse trains 438 11.3 the nyquist-shannon sampling theorem 438 11.4 summary 442 12 stability 447 12.1 boundedness and stability 450 12.1.1 absolutely summable and absolutely integrable 450 12.1.2 stability 452 probing further:stable systems and their impulse response 453 12.2 the z transform 456 12.2.1 structure of the region of convergence 458 12.2.2 stability and the z transform 463 12.2.3 rational z tranforms and poles and zeros 463 12.3 the laplace transform 467 12.3.1 structure of the region of convergence 469 12.3.2 stability and the laplace transform 472 12.3.3 rational laplace tranforms and poles and zeros 474 12.4 summary 475 intervlew: xavier rodet 481 13 laplace and z transforms 483 13.1 properties of the z tranform 485 13.1.1 linearity 485 13.1.2 delay 488 13.1.3 convolution 489 13.1.4 conjugation 490 13.1.5 time reversal 491 probing further: derivatives of z transforms 491 13.1.6 multiplication by an exponential 492 13.1.7 causal signals and the initial value theorem 493 13.2 frequency response and pole-zero plots 494 13.3 properties of the laplace transform 497 13.3.1 integration 497 13.3.2 sinusoidal signals 499 13.3.3 differential equations 500 13.4 frequency response and pole-zero plots, continuous time 501 13.5 the inverse transforms 503 13.5.1 inverse z transform 503 13.5.2 inverse laplace transform 512 probing further: inverse transform as on integral 514 probing further:differentiation property of the laplace transform 515 13.6 steady-state response 515 13.7 linear difference and differential equations 519 13.7.1 lti differential equations 525 13.8 state-space models 530 13.8.1 continuous-time state-space models 535 13.9 summary 541 14 composition and feedback control 549 14.1 cascade composition 550 14.1.1 stabilization 550 14.1.2 equalization 551 14.2 parallel composition 557 14.2.1 stabilization 558 14.2.2 noise cancelation 559 14.3 feedback composition 562 14.3.1 proportional controllers 564 14.4 pid controllers 574 14.5 summary 580 a sets and functions 589 a. 1 sets 589 a.1.1 assignment and assertion 591 a.1.2 sets of sets 592 a.1.3 variables and predicates 592 probing further:predicates in matlab 593 a.1.4 quantification over sets 594 a.1.5 some useful sets 596 a.1.6 set operations: union, intersection, complement 597 a.1.7 predicate operations 597 a.1.8 permutations and combinations 599 basics: tuples, strings, and sequences 600 a.1.9 product sets 601 a.1.10 evaluating an expression 605 a.2 functions 608 a.2.1 defining functions 610 a.2.2 tuples and sequences as functions 610 a.2.3 function properties 611 probing further:infinite sets 612 probing further:even bigger sets 613 a.3 summary 614 b complex numbers 619 b.1 imaginary numbers 619 b.2 arithmetic of imaginary numbers 621 b.3 complex numbers 622 b.4 arithmetic of complex numbers 622 b.5 exponentials 624 b.6 polar coordinates 626 basics: from cartesian to polar coordinates 627 symbols 635 index 637