Signals and Systems with MATLAB Computing and Simulink Modeling – Fourth Edition

Signals and Systems with MATLAB Computing and Simulink Modeling – Fourth Edition
اسم المؤلف
Steven T. Karris
التاريخ
23 فبراير 2024
المشاهدات
106
التقييم
(لا توجد تقييمات)
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Signals and Systems with MATLAB Computing and Simulink Modeling – Fourth Edition
Steven T. Karris
Table of Contents
Elementary Signals 1-1
1.1 Signals Described in Math Form 1-1
1.2 The Unit Step Function 1-2
1.3 The Unit Ramp Function 1-10
1.4 The Delta Function 1-11
1.4.1 The Sampling Property of the Delta Function 1-12
1.4.2 The Sifting Property of the Delta Function 1-13
1.5 Higher Order Delta Functions 1-14
1.6 Summary 1-22
1.7 Exercises 1-23
1.8 Solutions to End-of-Chapter Exercises 1-24
MATLAB Computing
Pages 1-20, 1-21
Simulink Modeling
Page 1-18
The Laplace Transformation 2-1
2.1 Definition of the Laplace Transformation 2-1
2.2 Properties and Theorems of the Laplace Transform 2-2
2.2.1 Linearity Property 2-3
2.2.2 Time Shifting Property 2-3
2.2.3 Frequency Shifting Property 2-4
2.2.4 Scaling Property 2-4
2.2.5 Differentiation in Time Domain Property 2-4
2.2.6 Differentiation in Complex Frequency Domain Property 2-6
2.2.7 Integration in Time Domain Property 2-6
2.2.8 Integration in Complex Frequency Domain Property 2-8
2.2.9 Time Periodicity Property 2-8
2.2.10 Initial Value Theorem 2-9
2.2.11 Final Value Theorem 2-10
2.2.12 Convolution in Time Domain Property 2-11
2.2.13 Convolution in Complex Frequency Domain Property 2-12
2.3 The Laplace Transform of Common Functions of Time 2-14
2.3.1 The Laplace Transform of the Unit Step Function u0(t) 2-14
2.3.2 The Laplace Transform of the Ramp Function u1(t) 2-14
2.3.3 The Laplace Transform of tnu0(t) 2-152.3.4 The LaplaceTransform of the Delta Function 8(t) 2-18
2.3.5 The LaplaceTransformof the Delayed DeltaFunction 5(t – a) 2-18
2.3.6 The LaplaceTransform of e-atu0(t) 2-19
2.3.7 The LaplaceTransformof t ne-a u0(t) . 2-19
2.3.8 The LaplaceTransform of sino t u0t .2-20
2.3.9 The LaplaceTransform of cosot u0t .2-20
2.3.10 The LaplaceTransformof e-atsinot u0(t) .2-21
2.3.11 The LaplaceTransformof e-atcosot u0(t) . 2-22
2.4 The Laplace Transform of Common Waveforms . 2-23
2.4.1 The Laplace Transform of a Pulse 2-23
2.4.2 The Laplace Transform of a Linear Segment 2-23
2.4.3 The Laplace Transform of a Triangular Waveform . 2-24
2.4.4 The Laplace Transform of a Rectangular Periodic Waveform 2-25
2.4.5 The Laplace Transform of a Half-Rectified Sine Waveform . 2-26
2.5 Using MATLAB for Finding the Laplace Transforms of Time Functions . 2-27
2.6 Summary 2-28
2.7 Exercises . 2-31
The Laplace Transform of a Sawtooth Periodic Waveform . 2-32
The Laplace Transform of a Full-Rectified Sine Waveform . 2-32
2.8 Solutions to End-of-Chapter Exercises . 2-33
3 The Inverse Laplace Transform 3-1
3.1 The Inverse Laplace Transform Integral 3-1
3.2 Partial Fraction Expansion 3-1
3.2.1 Distinct Poles . 3-2
3.2.2 Complex Poles .3-5
3.2.3 Multiple (Repeated) Poles 3-8
3.3 Case where F(s) is Improper Rational Function 3-13
3.4 Alternate Method of Partial Fraction Expansion 3-15
3.5 Summary .3-19
3.6 Exercises 3-21
3.7 Solutions to End-of-Chapter Exercises . 3-22
MATLAB Computing
Pages 3-3, 3-4, 3-5, 3-6, 3-8, 3-10, 3-12, 3-13, 3-14, 3-22
4 Circuit Analysis with Laplace Transforms 4-1
4.1 Circuit Transformation from Time to Complex Frequency 4-1
4.1.1 Resistive Network Transformation . 4-1
4.1.2 Inductive Network Transformation 4-1
4.1.3 Capacitive Network Transformation 4-14.2 Complex Impedance Z(s) .
4.3 Complex Admittance Y(s) .
4.4 Transfer Functions
4.5 Using the Simulink Transfer Fcn Block .
4.6 Summary .
4.7 Exercises .
4.8 Solutions to End-of-Chapter Exercises .
. 4-8
.4-11
.4-13
.4-17
.4-20
.4-21
.4-24
MATLAB Computing
Pages 4-6, 4-8, 4-12, 4-16, 4-17, 4-18, 4-26, 4-27, 4-28, 4-29, 4-34
Simulink Modeling
Page 4-17
5 State Variables and State Equations 5-1
5.1 Expressing Differential Equations in State Equation Form
5.2 Solution of Single State Equations
5.3 The State Transition Matrix
5.4 Computation of the State Transition Matrix .
5.4.1 Distinct Eigenvalues
5.4.2 Multiple (Repeated) Eigenvalues
5.5 Eigenvectors
5.6 Circuit Analysis with State Variables
5.7 Relationship between State Equations and Laplace Transform
5.8 Summary .
5.9 Exercises .
5.10 Solutions to End-of-Chapter Exercises .
5-1
5-6
5-9
5-11
5-11
5-15
5-18
5-22
5-30
5-38
5-41
5-43
MATLAB Computing
Pages 5-14, 5-15, 5-18, 5-26, 5-36, 5-48, 5-51
Simulink Modeling
Pages 5-27, 5-37, 5-45
6 The Impulse Response and Convolution 6-1
6.1 The Impulse Response in Time Domain .
6.2 Even and Odd Functions of Time
6.3 Convolution .
6.4 Graphical Evaluation of the Convolution Integral .
6.5 Circuit Analysis with the Convolution Integral .
6.6 Summary
6.7 Exercises .
6-1
6-4
6-7
6-8
6-18
6-21
6-236.8 Solutions to End-of-Chapter Exercises 6-25
MATLAB Applications
Pages 6-12, 6-15, 6-30
7 Fourier Series 7-1
7.1 Wave Analysis 7-1
7.2 Evaluation of the Coefficients 7-2
7.3 Symmetry in Trigonometric Fourier Series 7-6
7.3.1 Symmetry in Square Waveform 7-8
7.3.2 Symmetry in Square Waveform with Ordinate Axis Shifted 7-8
7.3.3 Symmetry in Sawtooth Waveform 7-9
7.3.4 Symmetry in Triangular Waveform 7-9
7.3.5 Symmetry in Fundamental, Second, and Third Harmonics 7-10
7.4 Trigonometric Form of Fourier Series for Common Waveforms 7-10
7.4.1 Trigonometric Fourier Series for Square Waveform 7-11
7.4.2 Trigonometric Fourier Series for Sawtooth Waveform 7-14
7.4.3 Trigonometric Fourier Series for Triangular Waveform 7-16
7.4.4 Trigonometric Fourier Series for Half-Wave Rectifier Waveform . 7-17
7.4.5 Trigonometric Fourier Series for Full-Wave Rectifier Waveform 7-20
7.5 Gibbs Phenomenon . 7-24
7.6 Alternate Forms of the Trigonometric Fourier Series . 7-24
7.7 Circuit Analysis with Trigonometric Fourier Series 7-28
7.8 The Exponential Form of the Fourier Series 7-31
7.9 Symmetry in Exponential Fourier Series 7-33
7.9.1 Even Functions . 7-33
7.9.2 Odd Functions 7-34
7.9.3 Half-Wave Symmetry . 7-34
7.9.4 No Symmetry . 7-34
7.9.5 Relation of C-n toCn . 7-34
7.10 Line Spectra . 7-36
7.11 Computation of RMS Values from Fourier Series 7-41
7.12 Computation of Average Power from Fourier Series . 7-44
7.13 Evaluation of Fourier Coefficients Using Excel® 7-46
7.14 Evaluation of Fourier Coefficients Using MATLAB® . 7-47
7.15 Summary . 7-50
7.16 Exercises 7-53
7.17 Solutions to End-of-Chapter Exercises 7-55
MATLAB Computing
Pages 7-38, 7-47Simulink Modeling
Page 7-31
8 The Fourier Transform 8-1
8.1 Definition and Special Forms 8-1
8.2 Special Forms of the Fourier Transform 8-2
8.2.1 Real Time Functions 8-3
8.2.2 Imaginary Time Functions 8-6
8.3 Properties and Theorems of the Fourier Transform 8-9
8.3.1 Linearity . 8-9
8.3.2 Symmetry . 8-9
8.3.3 Time Scaling 8-10
8.3.4 Time Shifting . 8-11
8.3.5 Frequency Shifting . 8-11
8.3.6 Time Differentiation 8-12
8.3.7 Frequency Differentiation . 8-13
8.3.8 Time Integration 8-13
8.3.9 Conjugate Time and Frequency Functions . 8-13
8.3.10 Time Convolution 8-14
8.3.11 Frequency Convolution . 8-15
8.3.12 Area Under f(t) 8-15
8.3.13 Area Under F(ro) .8-15
8.3.14 Parseval’s Theorem . 8-16
8.4 Fourier Transform Pairs of Common Functions 8-18
8.4.1 The Delta Function Pair . 8-18
8.4.2 The Constant Function Pair . 8-18
8.4.3 The Cosine Function Pair . 8-19
8.4.4 The Sine Function Pair 8-20
8.4.5 The Signum Function Pair . 8-20
8.4.6 The Unit Step Function Pair . 8-22
8.4.7 The e jra0tu0(t) Function Pair 8-24
8.4.8 The (coso0t)(u0t) FunctionPair 8-24
8.4.9 The (sino0t)(u0t) FunctionPair 8-25
8.5 Derivation of the Fourier Transform from the Laplace Transform . 8-25
8.6 Fourier Transforms of Common Waveforms . 8-27
8.6.1 The Transform of f(t) = A[u0(t + T) – u0(t – T)] .8-27
8.6.2 The Transform of f(t) = A[u0(t) – u0(t – 2T)] 8-28
8.6.3 The Transform of f(t) = A[u0(t + T) + u0(t) – u0(t – T) – u0(t – 2T)] .8-298.6.4 The Transform of f(t) = Acos©0t[u0(t + T) -u0(t- T)] 8-30
8.6.5 The Transform of a Periodic Time Function with Period T 8-31
8.6.6
TO
The Transform of the Periodic Time Function f(t) = A ^ S(t- nT) 8-32
n = -TO
9
8.7 Using MATLAB for Finding the Fourier Transform of Time Functions 8-33
8.8 The System Function and Applications to Circuit Analysis 8-34
8.9 Summary 8-42
8.10 Exercises 8-47
8.11 Solutions to End-of-Chapter Exercises . 8-49
MATLAB Computing
Pages 8-33, 8-34, 8-50, 8-54, 8-55, 8-56, 8-59, 8-60
Discrete-Time Systems and the Z Transform 9-1
9.1 Definition and Special Forms of the Z Transform 9-1
9.2 Properties and Theorems of the Z Transform 9-3
9.2.1 Linearity 9-3
9.2.2 Shift of f[n]u0[n] in the Discrete-Time Domain . 9-3
9.2.3 Right Shift in the Discrete-Time Domain . 9-4
9.2.4 Left Shift in the Discrete-Time Domain 9-5
9.2.5 Multiplication by an in the Discrete-Time Domain . 9-6
-naT
9.2.6 Multiplication by e in the Discrete-Time Domain . 9-6
9.2.7 Multiplication by n and n2 in the Discrete-Time Domain . 9-6
9.2.8 Summation in the Discrete-Time Domain 9-7
9.2.9 Convolution in the Discrete-Time Domain . 9-8
9.2.10 Convolution in the Discrete-Frequency Domain 9-9
9.2.11 Initial Value Theorem 9-9
9.2.12 Final Value Theorem 9-10
9.3 The Z Transform of Common Discrete-Time Functions . 9-11
9.3.1 The Transform of the Geometric Sequence . 9-11
9.3.2 The Transform of the Discrete-Time Unit Step Function 9-14
9.3.3 The Transform of the Discrete-Time Exponential Sequence .9-16
9.3.4 The Transform of the Discrete-Time Cosine and Sine Functions .9-16
9.3.5 The Transform of the Discrete-Time Unit Ramp Function 9-18
9.4 Computation of the Z Transform with Contour Integration 9-20
9.5 Transformation Between s- and z-Domains .9-22
9.6 The Inverse Z Transform .9-259.6.1 Partial Fraction Expansion .9-25
9.6.2 The Inversion Integral . .9-32
9.6.3 Long Division of Polynomials .9-36
9.7 The Transfer Function of Discrete-Time Systems . .9-38
9.8 State Equations for Discrete-Time Systems .9-45
9.9 Summary . .9-48
9.10 Exercises . .9-53
9.11 Solutions to End-of-Chapter Exercises . .9-55
MATLAB Computing
Pages 9-35, 9-37, 9-38, 9-41, 9-42, 9-59, 9-61
Simulink Modeling
Page 9-44
Excel Plots
Pages 9-35, 9-44
10
11
The DFT and the FFT Algorithm 10-1
10.1 The Discrete Fourier Transform (DFT) . . 10-1
10.2 Even and Odd Properties of the DFT . 10-9
10.3 Common Properties and Theorems of the DFT . 10-10
10.3.1 Linearity . . 10-10
10.3.2 Time Shift . 10-11
10.3.3 Frequency Shift . . 10-12
10.3.4 Time Convolution . 10-12
10.3.5 Frequency Convolution . . 10-13
10.4 The Sampling Theorem . 10-13
10.5 Number of Operations Required to Compute the DFT . . 10-16
10.6 The Fast Fourier Transform (FFT) . 10-17
10.7 Summary . 10-28
10.8 Exercises . 10-31
10.9 Solutions to End-of-Chapter Exercises . 10-33
MATLAB Computing
Pages 10-5, 10-7, 10-34
Excel Analysis ToolPak
Pages 10-6, 10-8
Analog and Digital Filters
11.1 Filter Types and Classifications 11-1
11.2 Basic Analog Filters 11-211.2.1 RC Low-Pass Filter 11-2
11.2.2 RC High-Pass Filter 11-4
11.2.3 RLC Band-Pass Filter 11-7
11.2.4 RLC Band-Elimination Filter 11-8
11.3 Low-Pass Analog Filter Prototypes 11-10
11.3.1 Butterworth Analog Low-Pass Filter Design 11-14
11.3.2 Chebyshev Type I Analog Low-Pass Filter Design 11-25
11.3.3 Chebyshev Type II Analog Low-Pass Filter Design 11-38
11.3.4 Elliptic Analog Low-Pass Filter Design 11-39
11.4 High-Pass, Band-Pass, and Band-Elimination Filter Design 11-41
11.5 Digital Filters 11-51
11.6 Digital Filter Design with Simulink 11-70
11.6.1 The Direct Form I Realization of a Digital Filter 11-70
11.6.2 The Direct Form II Realization of a Digital Filter 11-71
11.6.3 The Series Form Realization of a Digital Filter 11-73
11.6.4 The Parallel Form Realization of a Digital Filter 11-75
11.6.5 The Digital Filter Design Block 11-78
11.7 Summary 11-87
11.8 Exercises 11-91
11.9 Solutions to End-of-Chapter Exercises 11-97
MATLAB Computing
Pages 11-3, 11-4, 11-6, 11-7, 11-9, 11-15, 11-19, 11-23, 11-24, 11-31,
11-35, 11-36, 11-37, 11-38, 11-40, 11-41, 11-42, 11-43, 11-45, 11-46,
11-48, 11-50, 11-55, 11-56, 11-57, 11-60, 11-62, 11-64, 11-67, 11-68,
and 11-97 through 11-106
Simulink Modeling
Pages 11-71, 11-74, 11-77, 11-78, 11-80, 11-82, 11-83, 11-84
A Introduction to MATLAB A-1
A.1 MATLAB® and Simulink® . A-1
A.2 Command Window A-1
A.3 Roots of Polynomials A-3
A.4 Polynomial Construction from Known Roots . A-4
A.5 Evaluation of a Polynomial at SpecifiedValues . A-6
A.6 Rational Polynomials A-8
A.7 Using MATLAB to Make Plots . A-10
A.8 Subplots A-18
A.9 Multiplication, Division, and Exponentiation A-18
A.10 Script and Function Files . A-26
A.11 Display Formats . A-31MATLAB Computing
Pages A-3 through A-8, A-10, A-13, A-14, A-16, A-17,
A-21, A-22, A-24, A-27
B Introduction to Simulink B-1
B.1 Simulink and its Relation to MATLAB B-1
B.2 Simulink Demos B-20
MATLAB Computing
Page B-4
Simulink Modeling
Pages B-7, B-12, B-14, B-18
C A Review of Complex Numbers C-1
C.1 Definition of a Complex Number C-1
C.2 Addition and Subtraction of Complex Numbers C-2
C.3 Multiplication of Complex Numbers C-3
C.4 Division of Complex Numbers C-4
C.5 Exponential and Polar Forms of Complex Numbers C-4
MATLAB Computing
Pages C-6, C-7, C-8
Simulink Modeling
Page C-7
D Matrices and Determinants D-1
D.1 Matrix Definition . D-1
D.2 Matrix Operations D-2
D.3 Special Forms of Matrices . D-6
D.4 Determinants . D-10
D.5 Minors and Cofactors . D-12
D.6 Cramer’s Rule D-17
D.7 Gaussian Elimination Method . D-19
D.8 The Adjoint of a Matrix . D-21
D.9 Singular and Non-Singular Matrices . D-21
D.10 The Inverse of a Matrix D-22
D.11 Solution of Simultaneous Equations with Matrices . D-24
D.12 Exercises D-31E
MATLAB Computing
Pages D-3, D-4, D-5, D-7, D-8, D-9, D-10,
D-12, D-19, D-23, D-27, D-29
Simulink Modeling
Page D-3
Excel Spreadsheet
Page D-28
Window Functions E-1
E.1 Window Function Defined E-1
E.2 Common Window Functions E-1
E.2.1 Rectangular Window Function E-2
E.2.2 Triangular Window Function E-5
E.2.3 Hanning Window Function E-7
E.2.4 Hamming Window Function E-9
E.2.5 Blackman Window Function E-12
E.2.6 Kaiser Family of Window Functions E-14
E.3 Other Window Functions E-15
E.4 Fourier Series Method for Approximating an FIR Amplitude Response E-17
References R-1
Index IN-1
Index
Symbols
% (percent) symbol in MATLAB A-2
A
abs(z) in MATLAB A-23
active analog filter – see filter
adjoint of a matrix – see matrix
admittance
capacitive 4-2
inductive 4-2
complex input 4-11
algebraic constrain block in Simulink B-18
aliasing 10-14
all-pass filter – see filter
all-pole approximation 11-21
all-pole low-pass filter
see filter – low-pass
alternate form of the trigonometric
Fourier series – see Fourier series
alternate method of partial
fraction expansion – see partial
fraction expansion
angle(z) MATLAB function A-23
argument 11-2
attenuation rate 11-12
autoscale icon in Simulink B-12
axis MATLAB command A-16
B
band-elimination filter – see filter
band-elimination filter design
see filter design
band-limited signal 10-13
band-pass filter – see filter
band-pass filter design
see filter design
band-stop filter – see filter
Bessel filter – see filter
bilinear MATLAB function 11-59
bilinear transformation – see
transformation methods for mapping
analog prototype filters to digital filters
bode MATLAB function 11-24
box MATLAB command A-12
buttap MATLAB function 11-17
buttefly operation 10-23
Butterworth analog low-pass
filter design – see filter design
C
c2d MATLAB function 9-46
capacitive admittance – see admittance
capacitive impedance – see impedance
cascade form realization – see digital filter
Category I FFT algorithm – see
FFT algorithm
Category II FFT algorithm – see
FFT algorithm
Cauchy’s residue theorem
see residue theorem
Cauer filter – see elliptic filter
Cayley-Hamilton theorem 5-11
characteristic equation 5-19
cheb1ap MATLAB function 11-35
cheb2ap MATLAB function 11-38
Chebyshev filters – see filter
Chebyshev Type I analog low-pass
filter design – see filter design
Chebyshev Type I filters – see filter
Chebyshev Type I low-pass filter
magnitude-square function 11-26

  • see filter
    Chebyshev Type II analog low-pass
    filter design – see filter design
    Chebyshev Type II filters – see filter
    circuit analysis with Laplace transforms 4-1
    circuit analysis with state variables 5-22
    circuit transformation from time
    to complex frequency 4-1
    clc MATLAB command A-2
    clear MATLAB command A-2
    cofactor of a matrix – see matrix
    collect(s) MATLAB symbolic function 3-12
    column vector in MATLAB A-19
    command screen in MATLAB A-1
    Command Window in MATLAB A-1
    commas in MATLAB A-8
    comment line in MATLAB A-2
    Commonly Used Blocks in Simulink B-7
    complex admittance – see admittance
    complex conjugate in MATLAB A-4
    complex conjugate pairs 3-5
    complex impedance – see impedance
    complex number C-2
    complex numbers in MATLAB A-3
    complex poles 3-5
    Complex to Magnitude-Angle
    block in Simulink C-7
    computation of the state
    transition matrix 5-11
    computation of the Z Transform with
    contour integration – see Z transform
    Configuration Parameters
    in Simulink B-12
    congugate of a matrix – see matrix
    conj(A) MATLAB function D-9
    conjugate of a complex number C-3
    conjugate time and frequency functions
    of the Fourier transform – see
    Fourier transform – properties of
    constant function – Fourier transform of
    see Fourier transform – properties of
    Contents Pane in Simulink B-7
    contour integral 9-20
    conv MATLAB function A-7
    convolution in the complex frequency
    domain – see Laplace transform
    properties of
    convolution in the discrete-frequency
    domain – see Z transform – properties of
    convolution in the discrete-time
    domain – see Z transform – properties of
    convolution in the time domain property
    of the Laplace transform – see
    Laplace transform – properties of
    convolution integral defined 6-8
    graphical evaluation of 6-8
    convolution property of the Fourier
    transform – see Fourier transform
    properties of
    Cooley and Tukey 10-18
    cosine function – Fourier transform of
    see Fourier transform of
    common functions
    cosa>0t u0(t) function – Fourier transform of
    see Fourier transform of
    common functions
    Cramer’s rule D-17
    D
    d2c MATLAB function 9-47
    data points in MATLAB A-14
    decade – definition of 11-12
    decimation in frequency – see FFT algorithm
    decimation in time – see FFT algorithm
    deconv MATLAB function A-6
    default color in MATLAB A-15
    default line in MATLAB A-15
    default marker in MATLAB A-15
    default values in MATLAB A-12
    delta (impulse) function
    definition of 1-11
    doublet 1-14
    Fourier transform of – see Fourier
    transform of common functions
    higher order 1-14
    nth-order 1-14
    sampling property of 1-12
    sifting property of 1-13
    triplet 1-14
    demo in MATLAB A-2
    DeMoivre’s theorem 11-15
    derivation of the Fourier transform from
    the Laplace Transform 8-25
    determinant of a matrix – see matrix
    determinant of order 2 – see matrix
    DFT – common properties of
    even time function 10-9
    even frequency function 10-9
    frequency convolution 10-13
    frequency shift 10-12
    linearity 10-10
    odd time function 10-9
    odd frequency function 10-9
    time convolution 10-12
    time shift 10-11
    DFT – definition of 10-1
    N-point 10-2, 10-16
    diagonal elements of a matrix – see matrix
    diagonal matrix – see matrix
    IN-1differentiation in complex frequency
    domain property of the Laplace
    transform – see Laplace transform
  • properties of
    differentiation in time domain property
    of the Laplace transform – see
    Laplace transform – properties of
    differentiation property of the Fourier
    transform 8-12 – see Fourier
    transform – properties of
    digital filter 11-1, 11-51, 11-70
    Finite Impulse Response (FIR) 11-52
    FIR 11-52
    IIR 11-51
    Infinite Impulse Response (IIR) 11-51
    realization of
    Direct Form I 11-70
    Direct Form II 11-71
    cascade (series) form 11-73
    non-recursive 11-52
    parallel form 11-70
    recursive 11-52
    series (cascade) form 11-73
    Digital Filter Design Simulink block 11-78
    digital filter design with Simulink 11-70
    dimpulse MATLAB function 9-29
    Dirac MATLAB function 1-20
    Direct Form I realization – see digital filter
    realization of
    Direct Form II realization – see digital filter
    realization of
    direct term in MATLAB 3-4
    discontinuous function – definition of 1-2
    Discrete Fourier Transform (DFT) 10-1
    discrete impulse response 9-40
    discrete-time system transfer function 9-40
    discrete unit step function 9-3
    discrete-time exponential sequence 9-16
    discrete-time systems 9-1
    discrete-time unit ramp function 9-18
    discrete-time unit step function 9-14
    Display block in Simulink B-18
    display formats in MATLAB A-31
    distinct eigenvalues – see eigenvalues
    distinct poles – see poles
    division of complex numbers C-4
    dot operator in MATLAB
    division with A-21
    exponentiation with A-21
    multiplication with A-20
    double-memory technique
    see FFT Algorithm
    doublet – see delta function
    E
    Editor Window in MATLAB A-1
    Editor/Debugger in MATLAB A-1
    eig(x) MATLAB function 5-17
    eigenvalues
    distinct 15-1
    multiple (repeated) 5-15
    eigenvector 5-19
    e-j“t u0(t) Fourier transform of – see Fourier
    transform of common functions
    element-by-element operation in MATLAB
    division A-21
    exponentiation A-21
    multiplication A-20
    elements of the matrix – see matrix
    ellip MATLAB function 11-40
    elliptic filter – see filter
    elliptic filter design – see filter design
    eps in MATLAB A-22
    Euler’s identities C-5
    even functions 6-4, 7-33
    even symmetry – see Fourier
    series – symmetry
    Excel’s Analysis ToolPak 10-5
    exit MATLAB command A-2
    expand MATLAB symbolic function 3-10
    exponential form of complex numbers C-5
    exponential form of the Fourier series –
    see Fourier series
    exponential order function –
    definition of 2-2
    eye(n) MATLAB function D-7
    eye(size(A)) MATLAB function D-7
    F
    factor(s) MATLAB symbolic function 3-4
    Fast Fourier Transform (FFT) 10-1, 10-17
    FDA Tool Digital Filter Design
    Simulink block 11-82
    FFT algorithm
    Category I 10-19
    Category II 10-20
    decimation in frequency 10-20
    decimation in time 10-20
    double-memory technique 10-20
    in-place 10-20
    natural input-output 10-20
    FFT definition of 10-1, 10-17
    fft(x) MATLAB function 10-5, 11-68
    Figure Window in MATLAB A-13
    filter – see also digital filter
    active
    high-pass 4-22, 4-32
    low-pass 4-22, 4-32, 4-23, 4-35
    all-pass 11-1, 11-94
    all-pole 11-21
    band-elimination 11-1, 11-8
    band-pass 11-1, 11-7
    band-stop – see band-elimination
    Bessel 11-95
    Chebyshev 11-10
    Inverted 11-38
    magnitude-square function 11-26
    prototype 11-10
    Type I 11-25
    Type II 11-38
    elliptic 11-39
    high-pass 4-22, 4-30, 11-1, 11-4
    low-pass 4-22, 4-30, 11-1, 11-2
    low-pass analog filter
    prototypes 11-10
    maximally flat 11-14 (footnote)
    notch (band-elimination) 11-8
    phase shift 11-1, 11-94
    RC high-pass 11-4
    RC low-pass 11-2
    RLC band-elimination 4-22, 4-31, 11-8
    RLC band-pass 4-22, 4-31, 11-7
    filter design – see also digital filter
    band-elimination 11-41
    band-pass 11-41
    Butterworth analog low-pass 11-14
    Chebyshev
    Type I 11-25
    Type II 11-38
    elliptic 11-39
    high-pass 11-41
    low-pass 11-14
    filter MATLAB function 11-63
    final value theorem in Z transform
    see Z transform – properties of
    final value theorem in Laplace transform
    see Laplace transform – properties of
    find MATLAB function 11-68
    Finite Impulse Response (FIR) digital filter
    see digital filter
    FIR – see digital filter
    first harmonic – see Fourier series
    harmonics of
    first-order circuit 5-1
    first-order simultaneous
    differential equations 5-1
    Flip Block command in Simulink B-11
    format in MATLAB A-31
    fourier MATLAB command 8-33
    Fourier series
    exponential form of 7-31
    method used in window functions E-17
    trigonometric form of 7-2, 7-10
    alternate form of 7-25
    Fourier series coefficients – evaluation of
    numerical evaluation using Excel 7-46
    numerical evaluation using MATLAB® 7-47
    Fourier series of common waveforms
    full-wave rectifier 7-20, 7-24
    half-wave rectifier 7-17, 7-20
    square waveform
    with even symmetry 7-9, 7-13
    with odd symmetry 7-8, 7-12
    sawtooth 7-9, 7-15
    triangular 7-9, 7-16
    Fourier series – harmonics of
    first 7-1, 7-10
    second 7-1, 7-10
    third 7-1, 7-10
    Fourier series – symmetry
    even 7-7
    half-wave 7-7, 7-34
    odd 7-7
    quarter-wave 7-7 (footnote)
    types of 7-7
    Fourier integral – see Fourier transform
    Fourier transform
    definition of 8-1
    inverse of 8-1
    special forms of 8-2
    IN-2Fourier transform – properties of
    area under f(t) 8-15
    area under F(ro) 8-15
    conjugate time and frequency
    functions 8-13
    constant function 8-18
    frequency convolution 8-15
    frequency differentiation 8-13
    frequency shifting 8-11
    imaginary time functions – Fourier
    transform of 8-6
    linearity 8-9
    Parseval’s theorem 8-16
    real time functions – Fourier
    transform of 8-3
    symmetry 8-9
    time convolution 8-14
    time differentiation 8-12
    time integration 8-13
    time scaling 8-10
    time shifting 8-11
    Fourier Transform derivation from
    Laplace transform 8-25
    Fourier transform of common functions
    cosro 0t 8-19
    cosro0t u0(t) 8-24
    delta (8(t) and 8(t-a)) 8-18
    e-jro0t 8-19
    e-jro0t u0(t) 8-24
    signum (sgn(t)) 8-20
    sinro0t 8-20
    sinro0t u0(t) 8-25
    unit step (u0(t)) 8-22
    Fourier transform of common waveforms
    combined rectangular pulses 8-29
    cosine within a rectangular pulse 8-30
    shifted rectangular pulse 8-28
    symmetrical rectangular pulse 8-27
    periodic time functions 8-31, 8-32
    fourth-order all-pole low-pass filter
    see filter, low-pass
    fplot in MATLAB A-27
    frequency convolution in DFT
    see DFT – common properties of
    frequency convolution in Fourier
    transform of – see Fourier transform
  • properties of
    frequency differentiation in Fourier
    transform of – see Fourier transform
  • properties of
    frequency shift in DFT
    see DFT – common properties of
    frequency shift in Fourier transform
    see Fourier transform – properties of
    frequency shift in Laplace transform
    see Laplace transform – properties of
    freqz MATLAB function 11-57
    full rectification waveform – Laplace
    transform of – see Laplace transform
    of common waveforms
    full-wave rectifier – Fourier series of – see
    Fourier series of common waveforms
    Function Block Parameters in Simulink B-10
    function files in MATLAB A-26
    fundamental frequency 7-1
    fzero MATLAB function A-26
    G
    Gain block in Simulink B-9, B-18
    gamma function 2-15
    Gaussian elimination method D-19
    generalized factorial function 2-15
    Gibbs phenomenon 7-24
    grid MATLAB command A-12
    gtext MATLAB command A-13
    H
    half-wave rectifier – Fourier series of – see
    Fourier series of common waveforms
    half-wave symmetry
    see Fourier series – symmetry
    Heavyside MATLAB function 1-14
    help in MATLAB A-2
    Hermitian matrix – see matrix
    higher order delta functions – see delta
    function
    high-pass filter – see filter
    high-pass filter design – see filter design
    I
    identity matrix – see matrix
    ifft(x) MATLAB function 10-5
    ifourier MATLAB function 8-33
    IIR – see digital filter
    ilaplace MATLAB function 3-4
    imag(z) MATLAB function A-23
    imaginary axis – definition of C-2
    imaginary number – definition of C-2
    imaginary time functions 8-6
    see Fourier transform – properties of
    impedance
    capacitive 4-2
    inductive 4-2
    complex input 4-8, 4-9
    improper integral – definition of 2-15
    improper rational function –
    definition of 3-1, 3-13
    impulse function – see delta function
    impulse invariant method – see
    transformation methods for mapping
    analog prototype filters to
    digital filters
    increments between points
    in MATLAB A-14
    inductive admittance – see admittance
    inductive impedance – see impedance
    infinite impulse response – see digital filter
    initial value theorem in Z transform
    see Z transform – properties of
    initial value theorem in Laplace transform
    see Laplace transform – properties of
    in-place FFT algorithm 10-20
    see FFT algorithm
    integration in frequency in Laplace transform
    see Laplace transform – properties of
    integration in time in Laplace transform
    see Laplace transform – properties of
    Inverse Fourier transform
    see Fourier transform
    Inverse Laplace transform 2-1
    see Laplace transform
    inverse of a matrix – see matrix
    Inverse Z transform – see Z transform
    inversion integral 9-32
    Inverted Chebyshev filter – see filter
    J j
    operator C-1
    L L
    ’ Hopital’s rule 2-16
    laplace MATLAB function 2-27
    Laplace integral – see Laplace transform
    Laplace transform
    definition of 2-1
    Inverse of 2-1, 3-1
    Laplace transform – properties of
    convolution in the complex
    frequency domain 2-12
    convolution in the time domain 2-11
    differentiation in complex
    frequency domain 2-6
    differentiation in time domain 2-4
    final value theorem 2-10
    frequency shift 2-4
    initial value theorem 2-9
    integration in
    complex frequency domain 2-8
    time domain 2-6
    linearity 2-3
    scaling 2-4
    time periodicity 2-8
    time shift 2-3
    Laplace transform of common waveforms
    full-rectified 2-32
    half-rectified 2-26
    linear segment 2-23
    pulse 2-23
    rectangular periodic 2-25
    sawtooth 2-32
    triangular 2-24
    Laplace transform of common functions
    transform of e-at cosrot u0(t) 2-22
    transform of e-at sinrot u0(t) 2-21
    transform of e-at u0(t) 2-19
    transform of cosrot u0(t) 2-20
    transform of 8(t) 2-18
    transform of 8(t-a) 2-18
    transform of sinrot u0(t) 2-20
    transform of tn u0(t) function 2-15
    transform of tn e-at u0(t) 2-19
    transform of u0(t) 2-14
    transform of u1(t) 2-14
    leakage 10-13
    left shift in discrete-time domain
    see Z transform – properties of
    Leibnitz’s rule 6
    lims= MATLAB command A-27
    line spectra 7-36
    linear difference equation 9-38
    IN-3linearity in DFT
    see DFT – common properties of
    linearity in discrete-time domain
    see Z transform – properties of
    linearity property in Fourier transform
    see Fourier transform – properties of
    linearity property in Laplace transform
    see Laplace transform – properties of
    linspace MATLAB command A-14
    ln (natural log) A-13
    log in MATLAB A-13
    log(x) MATLAB function A-13
    log10(x) MATLAB function A-13
    log2(x) MATLAB function A-13
    loglog MATLAB command A-13
    long division of polynomials 9-36
    lower triangular matrix – see matrix
    low-pass analog filter prototypes – see filter
    low-pass filter – see filter
    lp2bp MATLAB function 11-43
    lp2bs MATLAB function 11-43
    lp2hp MATLAB function 11-43
    lp2lp MATLAB function 11-43
    M
    magnitude-squared function 11-10
    main diagonal of a matrix – see matrix
    Math Operations in Simulink B-10
    MATLAB Demos A-2
    MATLAB’s Editor/Debugger A-1
    matrix (matrices)
    adjoint of D-21
    cofactor of D-13
    conformable for addition D-2
    conformable for subtraction D-2
    conformable for multiplication D-4
    congugate of D-8
    definition of D-1
    determinant D-10
    minor of D-13
    non-singular D-21
    singular D-21
    diagonal D-2, D-6
    diagonal elements of D-2
    elements of D-1
    Hermitian D-9
    identity D-7
    inverse of D-22
    left division in MATLAB D-25
    multiplication in MATLAB A-18
    power series of 5-9
    scalar D-7
    size of D-7
    skew-Hermitian D-9
    skew-symmetric D-9
    square D-1
    symmetric D-8
    trace of D-2
    transpose of D-8
    triangular
    lower D-6
    upper D-6
    zero D-2
    matrix left division in MATLAB – see matrix
    matrix multiplication in MATLAB – see matrix
    matrix power series – see matrix
    maximally flat filter – see filter
    mesh(x,y,z) MATLAB function A-17
    meshgrid(x,y) MATLAB command A-17
    m-file in MATLAB A-2, A-26
    minor of determinant – see matrix
    MINVERSE Excel function D-27
    MMULT Excel function D-27
    modulated signals 8-12
    multiple eigenvalues – see eigenvalues
    multiple poles – see poles
    multiplication by an in discrete-time domain
    see Z transform – properties of
    multiplication by e-naT in discrete-time
    domain – see Z transform – properties of
    multiplication by n in discrete-time domain
    see Z transform – properties of
    multiplication by n2 indiscrete-time domain
    see Z transform – properties of
    multiplication of complex numbers C-3
    N
    NaN in MATLAB A-26
    natural input-output FFT algorithm
    see FFT algorithm
    network transformation
    resistive 4-1
    capacitive 4-1
    inductive 4-1
    non-recursive realization digital filter
    see digital filter
    non-singular determinant – see matrix
    normalized cutoff frequency 11-15
    notch filter – see filter
    N-point DFT – see DFT – definition of
    nth-order delta function – see delta function
    numerical evaluation of Fourier coefficients
    see Fourier series coefficients
    Nyquist frequency 10-13
    O
    octave defined 11-12
    odd functions 6-5, 7-34
    odd symmetry – see Fourier
    series – symmetry
    orthogonal functions 7-2
    orthogonal vectors 5-19
    orthonormal basis 5-19
    P
    parallel form realization – see digital filter
    Parseval’s theorem – see
    Fourier transform – properties of
    partial fraction expansion 3-1, 3-2, 9-25
    alternate method of 3-15
    method of clearing the fractions 3-15
    phase angle 11-2
    phase shift filter – see filter
    picket-fence effect 10-14
    plot MATLAB command A-10
    polar form of complex numbers C-6
    polar plot in MATLAB A-24
    polar(theta,r) MATLAB function A-23
    poles 3-1
    complex 3-5
    distinct 3-2
    multiple (repeated) 3-8
    poly MATLAB function A-4
    polyder MATLAB function A-7
    polynomial construction from
    known roots in MATLAB A-4
    polyval MATLAB function A-6
    pre-sampling filter 10-13
    pre-warping 11-55
    proper rational function –
    definition of 3-1, 11-11
    properties of the DFT
    see DFT – common properties of
    properties of the Fourier Transform
    see Fourier transform – properties of
    properties of the Laplace Transform
    see Laplace transform – properties of
    properties of the Z Transform
    see Z transform – properties of
    Q
    quarter-wave symmetry 7-7 (footnote)
    quit MATLAB command A-2
    R
    radius of absolute convergence 9-3
    ramp function 1-9
    randn MATLAB function 11-68
    Random Source Simulink block 11-80
    rationalization of the quotient C-4
    RC high-pass filter – see filter
    RC low-pass filter – see filter
    real axis C-2
    real number C-2
    real(z) MATLAB function A-23
    rectangular form C-5
    rectangular pulse expressed in terms
    of the unit step function 1-4
    recursive realization digital filter
    see digital filter
    region of
    convergence 9-3
    divergence 9-3
    relationship between state equations
    and Laplace Transform 5-30
    residue 3-2, 9-41
    residue MATLAB function 3-3, 3-12
    residue theorem 9-20, 9-21
    right shift in the discrete-time domain
    see Z transform – properties of
    RLC band-elimination filter – see filter
    RLC band-pass filter – see filter
    roots of polynomials in MATLAB A-3
    roots(p) MATLAB function 3-6, A-3
    round(n) MATLAB function A-24
    row vector in MATLAB A-3
    Runge-Kutta method 5-1
    running Simulink B-7
    IN-4S
    sampling property of the delta function
    see delta function
    sampling theorem 10-13
    sawtooth waveform – see Laplace
    transform of common waveforms
    sawtooth waveform – Fourier series of
    see Fourier series of
    common waveforms
    scalar matrix – see matrix
    scaling property of the Laplace transform
    see Laplace transform – properties of
    Scope block in Simulink B-12
    script file in MATLAB A-2, A-26
    second harmonic – see Fourier series
    harmonics of
    semicolons in MATLAB A-8
    semilogx MATLAB command A-12
    semilogy MATLAB command A-12
    series form realization – see digital filter
    Shannon’s sampling theorem
    see sampling theorem
    shift of f[n] u0[n] in discrete-time domain
    see Z transform – properties of
    sifting property of the delta function
    see delta function
    signal flow graph 10-23
    signals described in math form 1-1
    signum function – see Fourier transform
    of common functions
    simout To Workspace block
    in Simulink B-12
    simple MATLAB symbolic function 3-7
    Simulation drop menu in Simulink B-12
    simulation start icon in Simulink B-12
    Simulink icon B-7
    Simulink Library Browser B-8
    sine function – Fourier transform of
    see Fourier transform of
    common functions
    singular determinant – see matrix
    Sinks library in Simulink B-18
    sina>0t u0(t) Fourier transform of – see
    Fourier transform of common functions
    size of a matrix – see matrix
    skew-Hermitian matrix – see matrix
    skew-symmetric matrix – see matrix
    special forms of the Fourier transform
    see Fourier transform
    spectrum analyzer 7-36
    square matrix – see matrix
    square waveform with even symmetry – see
    Fourier series of common waveforms
    square waveform with odd symmetry – see
    Fourier series of common waveforms
    ss2tf MATLAB function 5-33
    stability 11-13
    start simulation in Simulink B-12
    state equations
    for continuous-time systems 5-1
    for discrete-time systems 9-45
    state transition matrix 5-9
    state variables
    for continuous-time systems 5-1
    for discrete-time systems 9-45
    State-Space block in Simulink B-12
    state-space equations
    for continuous-time systems 5-1
    for discrete-time systems 9-45
    step function – see unit step function
    step invariant method – see transformation methods for mapping analog
    prototype filters to digital filters
    stop-band filter – see filter
    string in MATLAB A-16
    subplots in MATLAB A-18
    summation in the discrete-time Domain
    see Z transform – properties of
    symmetric matrix – see matrix
    symmetric rectangular pulse expressed
    as sum of unit step functions 1-6
    symmetric triangular waveform expressed
    as sum of unit step functions 1-6
    symmetry – see Fourier series – symmetry
    symmetry property of the Fourier transform
    see Fourier transform – properties of
    system function – definition of 8-35
    T
    Taylor series 5-1
    text MATLAB command A-14
    tf2ss MATLAB function 5-33
    theorems of the DFT 10-10
    theorems of the Fourier Transform 8-9
    theorems of the Laplace transform 2-2
    theorems of the Z Transform 9-3
    third harmonic – see Fourier
    series – harmonics of
    time convolution in DFT
    see DFT – common properties of
    time integration property of the Fourier
    transform – see Fourier
    transform – properties of
    time periodicity property of the Laplace
    transform 2-8 – see Laplace
    transform – properties of
    time scaling property of the Fourier
    transform – see Fourier
    transform – properties of
    time shift in DFT
    see DFT – common properties of
    time shift property of the Fourier transform
    see Fourier transform – properties of
    time shift property of the Laplace transform
    see Laplace transform – properties of
    title(‘string’) MATLAB command A-12
    trace of a matrix – see matrix
    Transfer Fcn block in Simulink 4-17
    Transfer Fcn Direct Form II
    Simulink block 11-71
    transfer function of
    continuous-time systems 4-13
    discrete-time systems 9-38
    transformation between
    s and z domains 9-22
    transformation methods for mapping
    analog prototype filters to digital filters
    Impulse Invariant Method 11-52
    Step Invariant Method 11-52
    Bilinear transformation 11-53
    transpose of a matrix – see matrix
    Tree Pane in Simulink B-7
    triangular waveform expressed in terms
    of the unit step function 1-4
    triplet – see delta function
    Tukey – see Cooley and Tukey
    U
    unit eigenvectors 5-19
    unit impulse function (S(t)) 1-8
    unit ramp function (u1(t)) 1-8
    unit step function (u0(t)) 1-2
    upper triangular matrix – see matrix
    using MATLAB for finding the Laplace
    transforms of time functions 2-27
    using MATLAB for finding the Fourier
    transforms of time function 8-33
    V
    Vandermonde matrix 10-18
    Vector Scope Simulink block 11-84
    W
    warping 11-54
    window functions
    Blackman E-12
    Fourier series method for approximating
    an FIR amplitude response E-17
    Hamming E-9, E-31
    Hanning E-7, E-27
    Kaiser E-14, E-35
    other used as MATLAB functions E-15
    rectangular E-2
    triangular E-5, E-23
    Window Visualization Tool in MATLAB E-4
    X
    xlabel MATLAB command A-12
    Y
    ylabel MATLAB command A-12
    Z Z
    transform
    computation of with contour
    integration 9-20
    definition of 9-1
    Inverse of 9-1, 9-25
    Z transform – properties of
    convolution in the discrete
    frequency domain 9-9
    convolution in the discrete
    time domain 9-8
    final value theorem 9-10
    initial value theorem 9-9
    left shift 9-5
    linearity 9-3
    IN-5multiplication by an 9-6
    multiplication by e-naT 9-6
    multiplication by n 9-6
    multiplication by n2 9-6
    right shift 9-4
    shift of f[n] u0[n] 9-3
    summation 9-7
    Z Transform of discrete-time functions
    cosine function cosnaT 9-16
    exponential sequence e-naT u0[n]
    9-16, 9-21
    geometric sequence an 9-11
    sine function sinnaT 9-16
    unit ramp function nu0[n] 9-18, 9-21
    unit step function u0[n] 9-14, 9-20
    zero matrix – see matrix
    zeros 3-1, 3-2
    zp2tf MATLAB function 11-17

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