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Most books on linear systems for undergraduates cover discrete and continuous systems material together in a single volume. Such books also include topics in discrete and continuous filter design, and discrete and continuous state-space representations. However, with this magnitude of coverage, the student typically gets a little of both discrete and continuous linear systems but not enough of either. Minimal coverage of discrete linear systems material is acceptable provided that there is ample coverage of continuous linear systems. On the other hand, minimal coverage of continuous linear systems does no justice to either of the two areas. Under the best of circumstances, a student needs a solid background in both these subjects.

Continuous linear systems and discrete linear systems are broad topics and each merit a single book devoted to the respective subject matter. The objective of this set of two volumes is to present the needed material for each at the undergraduate level, and present the required material using MATLAB® (The MathWorks Inc.).

Volume 1

Chapter 1 Signal Representation

1.1 Examples of Continuous Signals

1.2 The Continuous Signal

1.3 Periodic and Nonperiodic Signals

1.4 General Form of Sinusoidal Signals

1.5 Energy and Power Signals

1.6 The Shifting Operation

1.7 The Reflection Operation

1.8 Even and Odd Functions

1.9 Time Scaling

1.10 The Unit Step Signal

1.11 The Signum Signal

1.12 The Ramp Signal

1.13 The Sampling Signal

1.14 The Impulse Signal

1.15 Some Insights: Signals in the Real World

1.16 End-of-Chapter Examples

1.17 End-of-Chapter Problems

References

Chapter 2 Continuous Systems

2.1 Definition of a System

2.2 Input and Output

2.3 Linear Continuous System

2.4 Time-Invariant System

2.5 Systems without Memory

2.6 Causal Systems

2.7 The Inverse of a System

2.8 Stable Systems

2.9 Convolution

2.10 Simple Block Diagrams

2.11 Graphical Convolution

2.12 Differential Equations and Physical Systems

2.13 Homogeneous Differential Equations and Their Solutions

2.14 Nonhomogeneous Differential Equations and Their Solutions

2.15 The Stability of Linear Continuous Systems: The Characteristic Equation

2.16 Block Diagram Representation of Linear Systems

2.17 From Block Diagrams to Differential Equations

2.18 From Differential Equations to Block Diagrams

2.19 The Impulse Response

2.20 Some Insights: Calculating y(t)

2.21 End-of-Chapter Examples

2.22 End-of-Chapter Problems

References

Chapter 3 Fourier Series

3.1 Review of Complex Numbers

3.2 Orthogonal Functions

3.3 Periodic Signals

3.4 Conditions for Writing a Signal as a Fourier Series Sum

3.5 Basis Functions

3.6 The Magnitude and the Phase Spectra

3.7 Fourier Series and the Sin-Cos Notation

3.8 Fourier Series Approximation and the Resulting Error

3.9 The Theorem of Parseval

3.10 Systems with Periodic Inputs

3.11 A Formula for Finding y(t) When x(t) Is Periodic: The Steady-State Response

3.12 Some Insight: Why the Fourier Series?

3.13 End-of-Chapter Examples

3.14 End-of-Chapter Problems

References

Chapter 4 The Fourier Transform and Linear Systems

4.1 Definition

4.2 Introduction

4.3 The Fourier Transform Pairs

4.4 Energy of Non-Periodic Signals

4.5 The Energy Spectral Density of a Linear System

4.6 Some Insights: Notes and a Useful Formula

4.7 End-of-Chapter Examples

4.8 End-of-Chapter Problems

References

Chapter 5 The Laplace Transform and Linear Systems

5.1 Definition

5.2 The Bilateral Laplace Transform

5.3 The Unilateral Laplace Transform

5.4 The Inverse Laplace Transform

5.5 Block Diagrams Using the Laplace Transform

5.6 Representation of Transfer Functions as Block Diagrams

5.7 Procedure for Drawing the Block Diagram from the Transfer Function

5.8 Solving LTI Systems Using the Laplace Transform

5.9 Solving Differential Equations Using the Laplace Transform

5.10 The Final Value Theorem

5.11 The Initial Value Theorem

5.12 Some Insights: Poles and Zeros

5.13 End-of-Chapter Examples

5.14 End-of-Chapter Problems

References

Chapter 6 State-Space and Linear Systems

6.1 Introduction

6.2 A Review of Matrix Algebra

6.3 General Representation of Systems in State-Space

6.4 General Solution of State-Space Equations Using the Laplace Transform

6.5 General Solution of the State-Space Equations in Real Time

6.6 Ways of Evaluating eAt

6.7 Some Insights: Poles and Stability

6.8 End-of-Chapter Examples

6.9 End-of-Chapter Problems

References

Volume 2

1. Signal Representation

1.1 Introduction

1.2 Why Do We Discretize Continuous Signals?

1.3 Periodic And Nonperiodic Discrete Signals

1.4 The Unit Step Discrete Signal

1.5 The Impulse Discrete Signal

1.6 The Ramp Discrete Signal

1.7 The Real Exponential Discrete Signal

1.8 The Sinusoidal Discrete Signal

1.9 The Exponentially Modulated Sinusoidal Signal

1.10 The Complex Periodic Discrete Signal

1.11 The Shifting Operation

1.12 Representing A Discrete Signal Using Impulses

1.13 The Reflection Operation

1.14 Time Scaling

1.15 Amplitude Scaling

1.16 Even And Odd Discrete Signal

1.17 Does A Discrete Signal Have A Time Constant?

1.18 Basic Operations On Discrete Signals

1.19 Energy And Power Discrete Signals

1.20 Bounded And Unbounded Discrete Signals

1.21 Some Insights: Signals In The Real World

1.22 End Of Chapter Examples

1.23 End Of Chapter Problems

2. The Discrete System

2.1 Definition of A System

2.2 Input and Output

2.3 Linear Discrete Systems

2.4 Time Invariance and Discrete Systems

2.5 Systems with Memory

2.6 Casual Systems

2.7 The Inverse of A System

2.8 Stable System

2.9 Convolution

2.10 Difference Equations of Physical Systems

2.11 The Homogenous Difference Equation and Its Solution

2.12 Nonhomogenous Difference Equations And Their Solutions

2.13 The Stability Of Linear Discrete Systems: The Characteristic Equation

2.14 Block Diagram Representation Of Linear Discrete Systems

2.15 From The Block Diagram To The Difference Equation

2.16 From The Difference Equation To The Block Diagram: A Formal Procedure

2.17 The Impulse Response

2.18 Correlation

2.19 Some Insights

2.20 End Of Chapter Examples

2.21 End Of Chapter Problems

3. The Fourier Series And The Fourier Transform Of Discrete Signals

3.1 Introduction

3.2 Review Of Complex Numbers

3.3 The Fourier Series Of Discrete Periodic Signals

3.4 The Discrete System With Periodic Inputs: The Steady-State Response

3.5 THE FREQUANCY RESPONSE OF DISCRETE SYSTEMS

3.6 THE FOURIER TRANSFORM OF DISCRETE SIGNALS

3.7 CONVERGENCE CONDITIONS

3.8 PROPERTIES OF THE FOURIER TRANSFORM OF DISCRETE SIGNALS

3.9 Parseval’s Relation And Energy Calculations

3.10 Numerical Evaluation Of The Fourier Transform Of Discrete Signals

3.11 Some Insights: Why Is This Fourier Transform?

3.12 End Of Chapter Examples

3.13 End Of Chapter Problems

4. The Z-Transform And Discrete Systems

4.1 Introduction

4.2 The Bilateral Z-Transform

4.3 The Unilateral Z-Transform

4.4 Convergence Considerations

4.5 The Inverse Z-Transform

4.6 Properties Of The Z-Transform

4.7 Representation Of Transfer Functions As Block Diagrams

4.8 X(N), H(N), Y(N), And The Z-Transform

4.9 Solving Difference Equation Using The Z-Transform

4.10 Convergence Revisisted

4.11 The Final Value Theorem

4.12 The Initial Value Theorem

4.13 Some Insights : Poles And Zeros

4.14 End Of Chapter Examples

4.15 End Of Chapter Problems

5. The Discrete Fourier Transform And Discrete Systems

5.1 Introduction

5.2 The Discrete Fourier Transform And The Finite-Duration Discrete Signals

5.3 Properties Of The Discrete Fourier Transform

5.4 The Relation The Dft Has With The Fourier Transform Of Discrete Signals, The Z-Transform, And The Continuous Fourier Transform

5.5 Numerical Computation Of The Dft

5.6 The Fast Fourier Transform: A Faster Way Of Computing The Dft

5.7 Applications Of The Dft

5.8 Some Insights

5.9 End Of Chapter Examples

5.10 End Of Chapter Problems

6. State-Space And Discrete Systems

6.1 Introduction

6.2 A Review On Matrix Algebra

6.3 General Representation Of Systems In State-Space

6.4 Solution Of The State-Space Equations Is The Z-Domain

6.5 General Solution Of The State Equation In Real Time

6.6 Properties Of And Its Evaluation

6.7 Transformations For In State-Space Representations

6.8 Some Insights: Poles And Stability

6.9 End Of Chapter Examples

6.10 End Of Chapter Problems

### Biography

Dr. Taan S. ElAli, PhD, is a full professor of electrical engineering. He is currently the Coordinator of the Engineering Program in the College of Aeronautics at Embry Riddle Aeronautical University.