Snals, *Systems*, & *Transforms*, Preface xvii1 Introduction 11.1 Modeling 11.2 Continuous-Time Physical *Systems* 4Electric Circuits, 4Operational Amplifier Circuits, 6Simple Pendulum, 9DC Power Supplies, 10Analogous *Systems*, 121.3 Samplers *and* Discrete-Time Physical *Systems* 14Analog-to-Dital Converter, 14Numerical Integration, 16Picture in a Picture, 17Compact Disks, 18Sampling in Telephone *Systems*, 19Data-Acquisition *System*, 211.4 MATLAB *and* Simulink 222 Continuous-Time Snals *and* *Systems* 232.1 Transformations of Continuous-Time Snals 24Time Transformations, 24Amplitude Transformations, 302.2 Snal Characteristics 32Even *and* Odd Snals, 32Periodic Snals, 342.3 Common Snals in Engineering 392.4 Singularity Functions 45Unit Step Function, 45Unit Impulse Function, 492.5 Mathematical Functions for Snals 542.6 Continuous-Time *Systems* 59Interconnecting *Systems*, 61Feedback *System*, 642.7 Properties of Continuous-Time *Systems* 65Stability, 69Linearity, 74Summary 76Problems 783 Continuous-Time Linear Time-Invariant *Systems* 903.1 Impulse Representation of Continuous-Time Snals 913.2 Convolution for Continuous-Time LTI *Systems* 923.3 Properties of Convolution 1053.4 Properties of Continuous-Time LTI *Systems* 108Memoryless *Systems*, 109Invertibility, 109Causality, 110Stability, 111Unit Step Response, 1123.5 Differential-Equation Models 113Solution of Differential Equations, 115General Case, 117Relation to Physical *Systems*, 1193.6 Terms in the Natural Response 120Stability, 1213.7 *System* Response for Complex-Exponential Inputs 124Linearity, 124Complex Inputs for LTI *Systems*, 125Impulse Response, 1293.8 Block Diagrams 130Direct Form I, 134Direct Form II, 134nth-Order Realizations, 134Practical Considerations, 136Summary 139Problems 1494 Fourier Series 1544.1 Approximating Periodic Functions 155Periodic Functions, 155Approximating Periodic Functions, 1564.2 Fourier Series 160Fourier Series, 161Fourier Coefficients, 1624.3 Fourier Series *and* Frequency Spectra 165Frequency Spectra, 1664.4 Properties of Fourier Series 1754.5 *System* Analysis 1784.6 Fourier Series Transformations 185Amplitude Transformations, 186Time Transformations, 188Summary 190Problems 1915 The Fourier Transform 2015.1 Definition of the Fourier Transform 2015.2 Properties of the Fourier Transform 210Linearity, 211Time Scaling, 212Time Shifting, 214Time Reversal, 215Time Transformation, 216Duality, 218Convolution, 220Frequency Shifting, 221Time Integration, 224Time Differentiation, 226Frequency Differentiation, 231Symmetry, 232Summary, 2335.3 Fourier *Transforms* of Time Functions 233DC Level, 233Unit Step Function, 233Switched Cosine, 234Pulsed Cosine, 234Exponential Pulse, 236Fourier *Transforms* of Periodic Functions, 236Summary, 2415.4 Application of the Fourier Transform 241Frequency Response of Linear *Systems*, 241Frequency Spectra of Snals, 250Summary, 2525.5 Energy *and* Power Density Spectra 253Energy Density Spectrum, 253Power Density Spectrum, 256Power *and* Energy Transmission, 258Summary, 260Summary 262Problems 2636 Applications of the Fourier Transform 2726.1 I deal Filters 2726.2 Real Filters 279RC Low-Pass Filter, 280Butterworth Filter, 282Bandpass Filters, 288Active Filters, 289Summary, 2916.3 Bandwidth Relationships 2916.4 Sampling Continuous-Time Snals 295Impulse Sampling, 296Shannon’s Sampling Theorem, 299Practical Sampling, 2996.5 Reconstruction of Snals from Sample Data 300Interpolating Function, 302Dital-to-Analog Conversion, 304Quantization Error, 3066.6 Sinusoidal Amplitude Modulation 308Frequency-Division Multiplexing, 3176.7 Pulse-Amplitude Modulation 319Time-Division Multiplexing, 321Flat-Top PAM, 323Summary 326Problems 3267 The Laplace Transform 3367.1 Definitions of Laplace *Transforms* 3377.2 Examples 3407.3 Laplace *Transforms* of Functions 3457.4 Laplace Transform Properties 349Real Shifting, 350Differentiation, 354Integration, 3567.5 Additional Properties 357Multiplication by t, 357Initial Value, 358Final Value, 359Time Transformation, 3607.6 Response of LTI *Systems* 363Initial Conditions, 363Transfer Functions, 364Convolution, 369Transforms with Complex Poles, 371Functions with Repeated Poles, 3747.7 LTI *Systems* Characteristics 375Causality, 375Stability, 376Invertibility, 378Frequency Response, 379Step Response, 3807.8 Bilateral Laplace Transform 382Region of Convergence, 384Bilateral Transform from Unilateral Tables, 386Inverse Bilateral Laplace Transform, 3897.9 Relationship of the Laplace Transform to the Fourier Transform 391Summary 392Problems 3938 State Variables for Continuous-Time *Systems* 4018.1 State-Variable Modeling 4028.2 Simulation Diagrams 4068.3 *Solution* of State Equations 412Laplace-Transform *Solution*, 412Convolution *Solution*, 417Infinite Series *Solution*, 4188.4 Properties of the State-Transition Matrix 4218.5 Transfer Functions 423Stability, 4258.6 Similarity Transformations 427Transformations, 427Properties, 433Summary 435Problems 4379 Discrete-Time Snals *and* *Systems* 4469.1 Discrete-Time Snals *and* *Systems* 448Unit Step *and* Unit Impulse Functions, 450Equivalent Operations, 4529.2 Transformations of Discrete-Time Snals 453Time Transformations, 454Amplitude Transformations, 4599.3 Characteristics of Discrete-Time Snals 462Even *and* Odd Snals, 462Snals Periodic in n, 465Snals Periodic in , 4689.4 Common Discrete-Time Snals 4699.5 Discrete-Time *Systems* 475Interconnecting *Systems*, 4769.6 Properties of Discrete-Time *Systems* 478Systems with Memory, 478Invertibility, 479Inverse of a *System*, 480Causality, 480Stability, 481Time Invariance, 481Linearity, 482Summary 484Problems 48610 Discrete-Time Linear Time-Invariant *Systems* 49510.1 Impulse Representation of Discrete-Time Snals 49610.2 Convolution for Discrete-Time *Systems* 497Properties of Convolution, 50610.3 Properties of Discrete-Time LTI *Systems* 509Memory, 510Invertibility, 510Causality, 510Stability, 511Unit Step Response, 51310.4 Difference-Equation Models 514Difference-Equation Models, 514Classical Method, 516Solution by Iteration, 52110.5 Terms in the Natural Response 522Stability, 52310.6 Block Diagrams 525Two Standard Forms, 52710.7 *System* Response for Complex-Exponential Inputs 531Linearity, 532Complex Inputs for LTI *Systems*, 532Stability, 537Sampled Snals, 537Impulse Response, 537Summary 539Problems 54011 The z-Transform 55211.1 Definitions of z-*Transforms* 55211.2 Examples 555Two z-*Transforms*, 555Dital-Filter Example, 55811.3 z-*Transforms* of Functions 560Sinusoids, 56111.4 z-Transform Properties 565Real Shifting, 565Initial *and* Final Values, 56811.5 Additional Properties 570Time Scaling, 570Convolution in Time, 57211.6 L TI *System* Applications 573Transfer Functions, 573Inverse z-Transform, 575Complex Poles, 578Causality, 580Stability, 581Invertibility, 584Frequency Response, 58511.7 Bilateral z-Transform 588Bilateral *Transforms*, 592Regions of Convergence, 594Inverse Bilateral *Transforms*, 595Summary 598Problems 59912 Fourier *Transforms* of Discrete-Time Snals 60912.1 Discrete-Time Fourier Transform 610z-Transform, 61212.2 Properties of the Discrete-Time Fourier Transform 617Periodicity, 618Linearity, 619Time Shift, 619Frequency Shift, 620Symmetry, 620Time Reversal, 621Convolution in Time, 621Convolution in Frequency, 622Multiplication by n, 623Parseval’s Theorem, 62312.3 Discrete-Time Fourier Transform of Periodic Sequences 62412.4 Discrete Fourier Transform 630Shorthand Notation for the DFT, 632Frequency Resolution of the DFT, 632Validity of the DFT, 634Summary, 63812.5 Fast Fourier Transform 638Decomposition-in-Time Fast Fourier Transform Algorithm, 638Decomposition-in-Frequency Fast Fourier Transform, 643Summary, 64612.6 Applications of the Discrete Fourier Transform 646Calculation of Fourier *Transforms*, 646Convolution, 654Filtering, 663Correlation, 671Energy Spectral Density Estimation, 677Summary, 67812.7 The Discrete Cosine Transform, 678Summary 683Problems 68413 State Variables for Discrete-Time *Systems* 69213.1 State-Variable Modeling 69313.2 Simulation Diagrams 69713.3 *Solution* of State Equations 703Recursive *Solution*, 703z-Transform *Solution*, 70513.4 Properties of the State Transition Matrix 71013.5 Transfer Functions 712Stability, 71413.6 Similarity Transformations 715Properties, 719Summary 720Problems 721Appendices 718A. This text provides a clear, comprehensive presentation of both the theory *and* applications in snals, *systems*, *and* *transforms*. It presents the.

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