Discrete Signals and Systems with MATLAB® (Hardcover, 3 ed)

1. SIGNAL REPRESENTATION 1.1 INTRODUCTION1.2 WHY DO WE DISCRETIZE CONTINUOUS SIGNALS?1.3 PERIODIC AND NONPERIODIC DISCRETE SIGNALS1.4 THE UNIT STEP DISCRETE SIGNAL1.5 THE IMPULSE DISCRETE SIGNAL1.6 THE RAMP DISCRETE SIGNAL1.7 THE REAL EXPONENTIAL DISCRETE SIGNAL1.8 THE SINUSOIDAL DISCRETE SIGNAL1.9 THE EXPONENTIALLY MODULATED SINUSOIDAL SIGNAL1.10 THE COMPLEX PERIODIC DISCRETE SIGNAL1.11 THE SHIFTING OPERATION1.12 REPRESENTING A DISCRETE SIGNAL USING IMPULSES1.13 THE REFLECTION OPERATION1.14 TIME SCALING1.15 AMPLITUDE SCALING1.16 EVEN AND ODD DISCRETE SIGNAL1.17 DOES A DISCRETE SIGNAL HAVE A TIME CONSTANT?1.18 BASIC OPERATIONS ON DISCRETE SIGNALS1.18.1 Modulation 1.18.2 Addition and Subtraction 1.18.3 Scalar Multiplication1.18.4 Combined Operations1.19 ENERGY AND POWER DISCRETE SIGNALS1.20 BOUNDED AND UNBOUNDED DISCRETE SIGNALS1.21 SOME INSIGHTS: SIGNALS IN THE REAL WORLD1.21.1 The Step Signal1.21.2 The Impulse Signal 1.21.3 The Sinusoidal Signal1.21.4 The Ramp Signal1.21.5 Other Signals1.22 END OF CHAPTER EXAMPLES1.23 END OF CHAPTER PROBLEMS 2THE DISCRETE SYSTEM 2.1 DEFINITION OF A SYSTEM2.2 INPUT AND OUTPUT 2.3 LINEAR DISCRETE SYSTEMS2.4 TIME INVARIANCE AND DISCRETE SYSTEMS2.5 SYSTEMS WITH MEMORY2.6 CASUAL SYSTEMS2.7 THE INVERSE OF A SYSTEM2.8 STABLE SYSTEM2.9 CONVOLUTION2.10 DIFFERENCE EQUATIONS OF PHYSICAL SYSTEMS2.11 THE HOMOGENOUS DIFFERENCE EQUATION AND ITS SOLUTION2.11.1 Case When Roots Are All Distinct2.11.2 Case When Two Roots Are Real And Equal2.11.3 Case When Two Roots Are Complex 2.12 NONHOMOGENOUS DIFFERENCE EQUATIONS AND THEIR SOLUTIONS2.12.1 How Do We Find The Particular Solution?2.13 THE STABILITY OF LINEAR DISCRETE SYSTEMS: THE CHARACTERISTICEQUATION2.13 .1 Stability Depending On The Values Of The Poles2.13.2 Stability From The Jury Test2.14 BLOCK DIAGRAM REPRESENTATION OF LINEAR DISCRETE SYSTEMS2.14.1 The Delay Element 2.14.2 The Summing/Subtracting Junction2.14.3 The Multiplier2.15 FROM THE BLOCK DIAGRAM TO THE DIFFERENCE EQUATION2.16 FROM THE DIFFERENCE EQUATION TO THE BLOCK DIAGRAM: A FORMAL PROCEDURE2.17 THE IMPULSE RESPONSE2.18 CORRELATION2.18.1 Cross-correlation2.18.2 Auto-correlation2.19 SOME INSIGHTS2.19.1 How Can We Find These Eigenvalues?2.19.2 Stability and Eigenvalues2.20 END OF CHAPTER EXAMPLES2.21 END OF CHAPTER PROBLEMS 3THE FOURIER SERIES AND THE FOURIER TRANSFORM OF DISCRETE SIGNALS 3.1INTRODUCTION3.2REVIEW OF COMPLEX NUMBERS3.2.1 Definition3.2.2 Addition3.2.3 Subtraction3.2.4 Multiplication3.2.5 Division3.2.6 From Rectangular To Polar3.2.7 From Polar To Rectangular3.3THE FOURIER SERIES OF DISCRETE PERIODIC SIGNALS3.4 THE DISCRETE SYSTEM WITH PERIODIC INPUTS: THE STEADY-STATE RESPONSE3.4.1 The General Form For 3.5THE FREQUANCY RESPONSE OF DISCRETE SYSTEMS3.5.1 Properties Of The Frequency Response3.5.1.1 The Periodicity Property3.5.1.2 The Symmetry Property3.6 THE FOURIER TRANSFORM OF DISCRETE SIGNALS3.7 CONVERGENCE CONDITIONS3.8PROPERTIES OF THE FOURIER TRANSFORM OF DISCRETE SIGNALS3.8.1 The Periodicity Property3.8.2 The Linearity Property3.8.3 The Discrete-Tine Shifting Property3.8.4 The Frequency Shifting Property3.8.5 The Reflection Property3.8.6 The Convolution Property3.9PARSEVAL’S RELATION AND ENERGY CALCULATIONS3.10 NUMERICAL EVALUATION OF THE FOURIER TRANSFORM OF DISCRETE SIGNALS3.11 SOME INSIGHTS: WHY IS THIS FOURIER TRANSFORM?3.11.1 The Ease In Analysis And Design3.11.2 Sinusoidal Analysis3.12 END OF CHAPTER EXAMPLES3.13 END OF CHAPTER PROBLEMS 4 THE Z-TRANSFORM AND DISCRETE SYSTEMS 4.1 INTRODUCTION4.2 THE BILATERAL Z-TRANSFORM4.3 THE UNILATERAL Z-TRANSFORM4.4 CONVERGENCE CONSIDERATIONS4.5 THE INVERSE Z-TRANSFORM4.5.1 Partial Fraction Expansion4.5.2 Long Division4.6 PROPERTIES OF THE Z-TRANSFORM4.6.1 Linearity Property4.6.2 Shifting Property4.6.3 Multiplication By e-an4.6.4 Convolution4.7 REPRESENTATION OF TRANSFER FUNCTIONS AS BLOCK DIAGRAMS4.8 x(n), h(n), y(n), AND THE Z-TRANSFORM4.9 SOLVING DIFFERENCE EQUATION USING THE Z-TRANSFORM4.10 CONVERGENCE REVISISTED4.11 THE FINAL VALUE THEOREM4.12THE INITIAL VALUE THEOREM4.13SOME INSIGHTS : POLES AND ZEROS4.13.1 The Poles Of The System4.13.2 The Zeros Of The System4.13.3 The Stability Of The System4.14END OF CHAPTER EXAMPLES4.15END OF CHAPTER PROBLEMS 5 THE DISCRETE FOURIER TRANSFORM AND DISCRETE SYSTEMS 5.1 INTRODUCTION5.2 THE DISCRETE FOURIER TRANSFORM AND THE FINITE-DURATIONDISCRETE SIGNALS5.3 PROPERTIES OF THE DISCRETE FOURIER TRANSFORM5.3.1 How Does The Defining Equation Work?5.3.2 The DFT Symmetry 5.3.3 The DFT Linearity5.3.4 The Magnitude Of The DFT5.3.5 What Does k In X(k), The DFT, Means?5.4 THE RELATION THE DFT HAS WITH THE FOURIER TRANSFORM OF DISCRETE SIGNALS, THE Z-TRANSFORM, AND THE CONTINUOUS FOURIER TRANSFORM5.4.1 The DFT And The Fourier Transform Of x(n)5.4.2 The DFT And The z-transform Of x(n)5.4.3 The DFT And The Continuous Fourier Transform Of x(t)5.5 NUMERICAL COMPUTATION OF THE DFT5.6 THE FAST FOURIER TRANSFORM: A FASTER WAY OF COMPUTING THE DFT5.7 APPLICATIONS OF THE DFT5.7.1 Circular Convolution5.7.2 Linear Convolution5.7.3 Approximation To The Continuous Fourier Transform5.7.4 Approximation To The Coefficients Of The Fourier Series And The Average Power Of The Periodic Signal x(t) 5.7.5 Total Energy In The Signal x(n) and x(t)5.7.6 Block Filtering5.7.7 Correlation5.8 SOME INSIGHTS5.8.1 The DFT Is The Same As The fft5.8.2 The DFT Points Are The Samples Of The Fourier Transform Of x(n)5.8.3 How Can We Be Certain That Most Of The Frequency Contents Of x(t) Are In The DFT?5.8.4 Is The Circular Convolution The Same As The Linear Convolution?5.8.5 Is ?5.8.6 Frequency Leakage And The DFT5.9 END OF CHAPTER EXAMPLES5.10 END OF CHAPTER PROBLEMS 6 STATE-SPACE AND DISCRETE SYSTEMS 6.1 INTRODUCTION6.2 A REVIEW ON MATRIX ALGEBRA6.2.1 Definition, General Terms, And Notations6.2.2 The Identity Matrix6.2.3 Adding Two Matrices6.2.4 Subtracting Two Matrices6.2.5 Multiplying A Matrix By A Constant6.2.6 Determinant Of A Two-By-Two Matrix6.2.7 Transpose Of A Matrix6.2.8 Inverse Of A Matrix6.2.9 Matrix Multiplication6.2.10 Eigenvalues Of A Matrix6.2.11 Diagonal Form Of A Matrix6.2.11 Eigenvectors Of A Matrix6.3 GENERAL REPRESENTATION OF SYSTEMS IN STATE-SPACE6.3.1 Recursive Systems6.3.2 Nonrecursive Systems6.3.3 From The Block Diagram To State-Space6.3.4 From The Transfer Function H(z) To State-Space6.4 SOLUTION OF THE STATE-SPACE EQUATIONS IS THE Z-DOMAIN6.5 GENERAL SOLUTION OF THE STATE EQUATION IN REAL TIME6.6 PROPERTIES OF AND ITS EVALUATION6.7 TRANSFORMATIONS FOR IN STATE-SPACE REPRESENTATIONS6.8 SOME INSIGHTS: POLES AND STABILITY6.9 END OF CHAPTER EXAMPLES6.10 END OF CHAPTER PROBLEMS