Signals and Systems Using MATLAB – Third Edition
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Luis F. Chaparro
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Signals and Systems Using MATLAB
Third Edition
Luis F. Chaparro
Department of Electrical and Computer Engineering
University of Pittsburgh
Pittsburgh, PA, USA
Aydin Akan
Department of Biomedical Engineering
Izmir Katip Celebi University
Izmir, Turkey
Table of contents
Part 1 Introduction

  1. From the Ground Up!
    Part 2 Theory and Application of Continuous-Time Signals and Systems
  2. Continuous-time Signals
  3. Continuous-time Systems
  4. The Laplace Transform
  5. Frequency Analysis: The Fourier Series
  6. Frequency Analysis: The Fourier Transform
    Part 3 Theory and Application of Discrete-Time Signals and Systems
  7. Sampling Theory
  8. Discrete-time Signals and Systems
  9. The Z-transform
    9: Fourier Analysis of Discrete-time Signals and Systems
  10. Introduction to the Design of Discrete Filters
    Appendix A: Useful Formulas
    Bibliography
    Index
    A
    Acoustic signal, 5, 61, 93, 124, 553, 751
    Adder circuit, 148
    Adders, 65, 123, 147, 367, 384, 403, 607, 771
    Aliasing, 466, 650
    All-pass filter, 237, 344, 374, 650, 672, 714, 724, 753,
    791
    ideal, 337
    Alternating current (AC), 124
    Amplitude
    analog, 60
    maximum, 293
    Amplitude modulation (AM), 94
    demodulation, 411
    envelope detector, 413
    suppressed carrier, 411
    tunable bandpass filter, 411
    Analog communication systems, 410
    AM, 410
    angle modulation, 418
    channel noise, 412
    FM, 418
    frequency division multiplexing (FDM), 416
    quadrature amplitude modulation (QAM), 416
    single sideband AM, 415
    Analog control systems, 368
    actuator, 370
    cruise control, 371
    feedback, 368
    open- and closed-loop, 368
    PI controller, 371
    positive- and negative-feedback, 367
    proportional (P) controller, 375
    stability and stabilization, 373
    transducer, 370
    Analog filtering, 421
    basics, 423
    Butterworth low-pass filter, 427
    Chebyshev low-pass filter, 429
    Chebyshev polynomials, 430
    eigenfunction property, 423
    factorization, 424
    frequency transformations, 434
    loss function, 424
    low-pass specifications, 424
    magnitude and frequency normalization, 426
    magnitude squared function, 423
    Analog LTI systems, 117
    BIBO stability, 148, 154, 531, 641, 727
    causality, 148, 149
    convolution integral, 137
    eigenfunction property, 242
    frequency response, 242, 333
    impulse response, 137
    impulse response, transfer function and frequency
    response, 333
    passivity, 155
    transient analysis, 375
    zero-input response, 128
    zero-state response, 128
    Analog systems
    dc source, 335
    windowing, 335
    Analog-to-digital converter
    A/D converter, 470
    Antenna, 6, 124, 322
    Antialiasing filter, 460
    Application-specific integrated circuit (ASIC), 4
    Armstrong, E., 418
    Attenuation, 113, 233, 338, 444, 627, 729
    B
    Basic analog signal operations
    adding, 65
    advancing and delaying, 65
    constant multiplying, 65
    modulation, 93
    reflection, 65
    time scaling, 93
    Basic analog signals
    complex exponentials, 78
    ramp, 82
    sinusoids, 79
    triangular pulse, 88
    809810 Index
    unit-impulse, 80
    unit-step, 81
    Basic discrete-time signals
    complex exponentials, 500
    sinusoids, 503
    unit-step and unit-sample signals, 506
    Bilinear transformation, 738
    warping, 738
    Biological systems, 4, 489
    C
    Capability of storing energy, 116, 127, 383
    Capacitance, 116, 159
    Capacitors, 8, 17, 30, 115, 125, 155, 423
    Carrier, 124, 410
    Cascade connections, 144, 365, 410, 521
    Chemical plant, 363
    Chirp signal, 43, 114, 715
    Circuit
    open, 17, 163, 340
    short, 163, 340
    Circuit theory, 72, 332, 412
    Circular shifting, 673
    Classical control, 158, 368, 402
    Clock frequencies, 9
    Closed-loop transfer function, 364
    Comb filters, 730, 799
    Complex conjugate poles, 188, 205, 237, 776
    Complex variables, 20
    function, 27
    Continuous-time LTI systems
    complete response, 223
    eigenfunction property, 169
    represented by ordinary differential equations, 220
    steady-state response, 223
    transient response, 223
    unit-step response, 225
    zero-input response, 222
    zero-state, zero-input responses, 222
    zero-state response, 222
    Continuous-time signals
    absolutely integrable, 73
    advanced, 66
    aperiodic, 60, 70
    basic signals, 77
    causal sinusoid, 74
    complex exponentials, 77
    definition, 62
    delayed, 66
    even, 60, 66, 68
    even and odd decomposition, 69
    finite energy, 60, 72
    finite power, 60, 72, 73
    finite support, 60
    full-wave rectified, 88
    odd, 60, 66, 68
    periodic, 60, 70
    real and imaginary parts, 78
    shifting and reflecting, 67
    sinusoids, 64
    speech, 62
    square integrable, 73
    train of rectangular pulses, 89
    windowing, 65
    Conversion, 5, 23, 61, 360, 474
    Convolution integral
    commutative property, 145
    distributive property, 145
    Fourier, 332
    graphical computation, 150
    Laplace, 213
    Convolution sum, 519, 574
    commutative property, 521
    deconvolution, 521
    graphical, 576
    length, 575
    noncausal signals, 580
    Cooley, J., 694
    Cramer’s rule, 396, 611
    Cut-off frequencies, 335
    D
    D/A converter, 62
    Damped sinusoid, 78, 95, 501
    Damping factor, 20, 168, 189, 560
    Dc gain, 299, 425, 629, 746, 794
    Dc loss, 425, 433, 733, 747, 799
    Dc source, 20, 50, 267, 294, 335, 359, 546
    De Moivre, A., 563
    Differentiation, 12, 81, 190, 349
    Digital communications, 477
    PCM, 477
    time-division multiplexing, 481
    Digital signal processing
    FFT, 693
    inverse FFT, 698Index 811
    Digital-to-analog converter
    D/A converter, 62, 470
    Dirac, P., 178
    Discrete Cosine Transform (DCT)
    1D-DCT, 704
    2D-DCT, 706
    transform coding, 704
    Discrete filtering
    analog signals, 723
    averaging filter, 526
    bilinear transformation, 736
    Butterworth LPF, 740
    Chebyshev LPF, 748
    eigenfunction property of LTI systems, 723
    FIR realizations, 779
    FIR window design, 761
    frequency scales, 734
    frequency selective filters, 723
    frequency specifications, 731
    group delay, 725
    IIR and FIR, 726
    IIR realizations, 772
    linear phase, 724
    loss function, 732
    median filter, 527
    rational frequency transformations, 753
    realization, 771
    time specifications, 735
    windows for FIR design, 763
    Discrete filters
    FIR, 582
    FIR filters and convolution sum, 575
    IIR, 582
    Discrete Fourier series, 663
    circular representation, 665
    circular shifting, 673
    complex exponential, 665
    periodic convolution, 673
    Z-transform, 668
    Discrete Fourier Transform (DFT), 680
    Fast Fourier Transform (FFT), 683
    linear and circular convolution, 688
    Discrete LTI systems
    autoregressive (AR), 515
    autoregressive moving average (ARMA), 517
    BIBO stability, 529, 581
    causality, 529
    convolution sum, 519
    difference equation representation, 514
    moving average (MA), 514
    recursive (IIR) and non-recursive (FIR), 514
    response to periodic signals, 671
    time-invariance, 512
    Discrete-Time Fourier Transform (DTFT), 639
    convergence, 639
    convolution sum, 662
    decimation, 648
    eigenfunctions, 640
    interpolation, 648
    Parseval’s theorem, 654
    periodic signals, 669
    sampled signal, 640
    symmetry, 656
    time and frequency shifts, 655
    time–frequency duality, 641
    time–frequency supports, 647
    Z-transform, 640
    Discrete-time signals
    absolutely summable, 494
    definition, 488
    discrete frequency, 500
    energy and power spectrum, 654
    Fibonacci sequence, 489
    finite energy, 494
    finite power, 494
    generic representation, 507
    inherently discrete-time, 489
    periodic, fundamental period, 490
    sample index, 489
    sinusoid, 489
    square summable, 494
    square summable, finite energy, 494
    Discrete-time sinusoids, 503, 544, 656
    periodic, 490
    Discretization, 11, 62, 560, 644
    Disturbance, 7, 368, 691
    Doppler frequency shift, 113, 444
    Double poles, 189, 572
    Double real poles, 208, 379
    Down-sampling, 500, 552, 649, 715, 800
    E
    Electrical circuit, 8
    Electronic noise, 7, 17
    Elliptic design, 439
    Elliptic filters, 445, 798812 Index
    Energy
    conservation property, 326
    consumption, 7
    continuous-time signals, 73
    discrete-time signals, 494
    spectrum, 326, 654
    Euler, L., 23
    Euler’s identity, 23, 78
    F
    Fast Fourier Transform (FFT), 693
    inverse, 698
    Fessenden, R., 124
    Filtering, 245
    analog basics, 334
    antialiasing, 455
    RC high-pass filter, 343
    RC low-pass filter, 342
    Filters
    band-stop, 340, 435, 753, 781
    bandpass, 340, 412, 440
    comb, 730
    denormalized, 426, 427
    frequency discriminating, 334
    gaussian, 720, 782
    ideal, 299, 336
    linear, 525, 785
    magnitude-normalized, 425
    median, 527
    non-linear, 525
    Finite calculus, 10
    finite-difference, 12
    summations, 14
    Fourier, J.B., 249
    Fourier series
    absolute uniform convergence, 274
    analog Parseval’s power relation, 252
    coefficients, 250
    coefficients from Laplace, 258
    complex exponential, 249
    convergence, 274
    dc component, 255
    even and odd signals, 260
    Fourier basis, 251
    fundamental frequency, 250
    fundamental period, 250
    Gibb’s phenomenon, 274
    harmonics, 255
    line spectrum, 253
    mean-square approximation, 275
    normality, 251
    orthogonality, 251
    periodic signals, 243
    product of periodic signals, 288
    spectrum analyzer, 252
    sum of periodic signals, 287
    time and frequency shifting, 278
    time reversal, 260
    trigonometric, 254
    Fourier transform
    amplitude modulation, 320
    analog spectrum and line spectrum, 323
    convolution integral, 332
    differentiation and integration, 349
    direct and inverse, 307
    duality, 317
    frequency shifting, 319
    Laplace ROC, 309
    linearity, 310
    periodic analog signals, 323
    shifting in time, 348
    symmetry, 327
    Frequencies
    carrier, 412
    instantaneous, 114, 419
    normalized, 423
    passband, 440
    stopband, 445, 731
    Frequency modulation (FM)
    definition, 94
    Frequency response
    poles and zeros, 341
    G
    Gauss, C., 15, 701
    Gaussian filter, 720, 782
    Gaussian noise, 526, 659
    Gibbs phenomenon
    filtering, 339
    Global positioning system (GPS), 4
    H
    Harmonic frequencies, 253, 285, 323, 671
    Heaviside, O., 169
    Hertz, H., 28
    Human visual system, 781Index 813
    I
    Ideal filters
    band-pass, 336
    band-pass and high-pass, 336
    high-pass, 336
    linear phase, 336
    low-pass, 336
    zero-phase, 338
    Image processing
    2D filtering, 781
    bank of filters, 790
    binarization, 704
    data compression, 704, 790
    edge detection, 782
    non-linear filtering, 785
    Image processing, edge detection, 556
    Images
    digital, 534
    discretized, 534
    Inductors, 8, 30, 115, 125, 155, 423
    Instability, 375
    Integration, 14, 93, 100, 120, 193, 258, 272, 349, 361,
    638
    Inverse Laplace
    partial fraction expansion, 201
    two-sided, 227
    with exponentials, 212
    Inverse proportionality, 310
    Inverse Z-transform
    inspection, 586
    long-division method, 586
    partial fraction expansion, 587
    positive powers of z, 590
    proper rational functions, 587
    rational functions, 587
    J
    Jury, E.I., 563
    K
    Kalman filtering, 563
    L
    Laplace, P.S., 169
    Laplace transform
    convolution integral, 199
    derivative, 190
    integration, 193
    inverse, 171, 200
    linearity, 187
    poles and zeros, 174
    proper rational, 201
    region of convergence (ROC), 174
    sampled signals, 560
    two-sided, 171
    L’Hôpital’s rule, 264
    M
    Marconi, G., 124
    MATLAB
    analog Butterworth and Chebyshev filter design, 434
    analog Butterworth filtering, 433
    control toolbox, 379
    decimation and interpolation, 650
    DFT and FFT, 644
    discrete filter design, 749
    DTFT computation, 644
    FFT computation, 700
    filter design, 436
    Fourier series computation, 670
    general discrete filter design, 757
    numerical computations, 33
    phase computation, 659
    phase unwrapping, 659
    spectrogram, 421
    symbolic computations, 43
    window FIR filter design, 771
    Maximum frequency, 454
    Message, 124, 161, 172, 320, 410, 468, 477
    analog, 450, 477
    digital, 450, 477
    N
    Noise, 7, 17, 165, 334, 526, 659, 781
    Noisy signal, 13, 481, 527, 630
    Non-linear filtering
    median filter, 527
    Nyquist
    sampling rate, 464
    sampling theorem, 463
    Nyquist, H., 464
    P
    Parallel connections, 144, 364, 410
    Phase modulation (PM)
    definition, 94
    Phasor connection, 30814 Index
    Phasors
    sinusoidal steady state, 244
    Plant, 7, 363, 585
    Pole–zero cancellation, 186, 223, 374, 399, 761
    Power, 72, 73
    average, 73
    discrete-time signals, 494
    instantaneous, 72
    Power spectral density, 183
    Pulse
    amplitude modulation (PAM), 450
    code modulation (PCM), 477
    Q
    Quantization
    error, 473
    four-level quantizer, 472
    step, 473
    R
    Radio spectrum, 6, 416, 450, 477
    Real-time processing, 65
    Recursive system, 514, 540
    Reference signal, 7, 239, 368, 407, 585
    Resistance, 110, 116, 159
    Resistors, 8, 17, 72, 107, 115, 120, 147, 155, 246, 340,
    403, 423
    S
    Sampler
    time-varying system, 453
    Sampling
    antialiasing filter, 455, 460
    band-limited signal, 454
    Discrete-Time Fourier Transform, 453
    frequency aliasing, 455
    holder for D/A, 471
    ideal impulse, 451
    Nyquist rate, 464
    Nyquist–Shannon theorem, 463
    Parseval’s application, 458
    practical aspects, 469
    quantization, 472
    quantization error, 472, 473
    quantizer, 472
    sample-and-hold system, 470
    sampling frequency, 454
    sampling function, 451
    signal reconstruction, 459
    sinc interpolation, 462
    time-division multiplexing, 481
    uniform sampling, 450
    Shannon, C., 464
    Signal processing applications
    cognitive radio, 6
    compact-disc (CD) player, 5
    computer-control systems, 7
    software-defined radio, 6
    Signal processing system, 8
    analog-to-digital converter, 8
    digital signal processors (DSPs), 4, 8
    field-programmable gate arrays (FPGAs), 4, 8
    microprocessors and micro-controllers, 8
    real-time implementation, 8
    Signals
    absolutely summable, 494
    analog, 60
    band-limited, 454
    causal, anticausal, noncausal, 173
    causal discrete sinusoid, 494
    continuous-time, 60
    delayed, 66
    deterministic, 60
    digital, 60
    discrete finite energy, 494
    discrete sinusoid, 489
    discrete-time, 60
    finite energy discrete signal, 494
    full-wave rectified, 267
    modulation, 124
    piecewise smooth, 274
    radiated with antenna, 322
    random, 60
    sampled continuous-time, 489
    sinc, 264
    smooth, 274
    square summable, 494
    Sobel filters, 557, 782
    Spectrum, 327, 411, 446, 465, 640, 652, 724, 781
    Steinmetz, C., 28
    Superposition, 76, 118, 135, 203, 244, 333, 417, 453,
    512, 664, 682
    Support
    finite, 64, 93, 151, 179, 311, 321, 454, 479, 534,
    565, 630, 644, 702
    first-quadrant, 624Index 815
    infinite, 70, 173, 250, 308, 336, 523, 567, 664
    one-point, 311
    Switch, 126, 162, 450, 471, 481
    System
    analog, 117
    continuous-time, 117
    definition, 116
    digital, 117
    discrete-time, 117
    hybrid, 117
    System connections, 364
    Systems
    all-pass, 248
    amplitude modulation (AM), 124
    analog averager, 139
    discrete, 482, 560, 638, 693
    echo, 155, 161
    frequency modulation (FM), 125
    ideal communication system, 247
    linearity, 118
    multi-path channel, 172
    negative-feedback, 367, 403
    non-linear, 123
    positive-feedback, 367
    sample-and-hold sampling, 470
    single-input single-output, 117
    time-invariance, 123
    time-varying, 123
    unstable, 118, 154, 236, 373
    vocal system, 125
    zero-order-hold, 470
    T
    Tesla, N., 124
    Theorem, initial-value, 202
    Transfer function
    BIBO, 630
    constant-numerator, 609, 772
    discrete, 737
    feedback, 403
    feedforward, 376, 403
    normalized, 441
    one-dimensional, 626
    only-zeros, 391
    open-loop, 364
    rational, 341, 439, 727
    representation, 383, 560
    two-dimensional, 626
    Transient responses, 223, 600
    Transmitter, 112, 124, 172, 410, 478
    Triangular signal, 88, 142, 271, 292, 300
    Tsypkin, Y., 563
    Tukey, J., 694
    Two-Dimensional Discrete Fourier Transform, 701
    2D-FFT, 702
    aperiodic signals, 702
    Two-dimensional discrete signals, 532
    impulse, 533
    periodic, 534
    rectangular and block periodic, 536
    separable, 533
    unit-ramp, 533
    unit-step, 533
    Two-dimensional discrete systems, 532, 537
    convolution, 537
    FIR, 540
    IIR, 540
    impulse response, 537
    LSI, 537
    LSI BIBO, 540
    LSI causal, 542
    separable, 538
    Two-dimensional filtering, 781
    1D vs. 2D, 781
    circular filters, 788
    frequency domain, 786
    separable, 787
    spatial, 781
    Two-dimensional Z-transform, 618
    convolution sum, 622
    inverse, 621
    stability testing, 623
    U
    Unit-step function, 99
    V
    Voice signal, 60, 124, 724
    Voltage, 8, 18
    ac, 239, 299
    dc, 299
    Voltage follower, 147, 366, 403
    circuit, 147, 403
    Voltage source, 17, 127, 148, 154, 192, 216, 246, 339,
    376, 441
    Voltage–current relation, 120816 Index
    W
    Walsh functions, 303
    Windowed cosine, 480
    Wireless communications, 112, 124, 172, 477
    Z
    Z-transform
    connection with sampling, 563
    damping radius, 562
    discrete frequency, 562
    inverse, 586
    linearity, 569
    one-sided, 569
    poles and zeros, 587
    ROC and uniqueness, 568
    s-plane and Z-plane, 562
    sampled signals, 561
    significance of poles, 573
    solution of difference equations, 596, 602
    time-shifting, 595
    transfer function, 574
    two-sided transform, 563
    Zadeh, L., 563
    Zero-boundary conditions, 53

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