# Signals, Systems, and Transforms, 4th Edition

Published Date: Jul 12, 2007

## Description

For sophomore/junior-level signals and systems courses in Electrical and Computer Engineering departments.

Signals, Systems, and Transforms, Fourth Edition is ideal for electrical and computer engineers. The text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

Contents

Preface

1          Introduction

1.1        Modeling

1.2        Continuous-Time Physical Systems

Electric Circuits,

Operational Amplifier Circuits,

Simple Pendulum,

DC Power Supplies,

Analogous Systems,

1.3        Samplers and Discrete-Time Physical Systems

Analog-to-Digital Converter,

Numerical Integration,

Picture in a Picture,

Compact Disks,

Sampling in Telephone Systems,

Data-Acquisition System,

2          Continuous-Time Signals and Systems

2.1        Transformations of Continuous-Time Signals

Time Transformations,

Amplitude Transformations,

2.2        Signal Characteristics

Even and Odd Signals,

Periodic Signals,

2.3        Common Signals in Engineering

2.4        Singularity Functions

Unit Step Function,

Unit Impulse Function,

2.5        Mathematical Functions for Signals

2.6        Continuous-Time Systems

Interconnecting Systems,

Feedback System,

2.7        Properties of Continuous-Time Systems

Stability

Linearity

Summary

Problems

3          Continuous-Time Linear Time-Invariant Systems

3.1        Impulse Representation of Continuous-Time Signals

3.2        Convolution for Continuous-Time LTI Systems

3.3        Properties of Convolution

3.4        Properties of Continuous-Time LTI Systems

Memoryless Systems,

Invertibility,

Causality,

Stability,

Unit Step Response,

3.5        Differential-Equation Models

Solution of Differential Equations,

General Case,

Relation to Physical Systems,

3.6        Terms in the Natural Response

Stability,

3.7        System Response for Complex-Exponential Inputs

Linearity,

Complex Inputs for LTI Systems,

Impulse Response,

3.8        Block Diagrams

Direct Form I,

Direct Form II,

nth-Order Realizations,

Practical Considerations,

Summary

Problems

4          Fourier Series

4.1        Approximating Periodic Functions

Periodic Functions,

Approximating Periodic Functions,

4.2        Fourier Series

Fourier Series,

Fourier Coefficients,

4.3        Fourier Series and Frequency Spectra

Frequency Spectra,

4.4        Properties of Fourier Series

4.5        System Analysis

4.6        Fourier Series Transformations

Amplitude Transformations,

Time Transformations,

Summary

Problems

5          The Fourier Transform

5.1        Definition of the Fourier Transform

5.2        Properties of the Fourier Transform

Linearity,

Time Scaling,

Time Shifting,

Time Transformation,

Duality,

Convolution,

Frequency Shifting,

Time Differentiation,

Time Integration,

Frequency Differentiation,

Summary,

5.3        Fourier Transforms of Time Functions

DC Level,

Unit Step Function,

Switched Cosine,

Pulsed Cosine,

Exponential Pulse,

Fourier Transforms of Periodic Functions,

Summary,

5.4        Sampling Continuous-Time Signals

Impulse Sampling,

Shannon’s Sampling Theorem,

Practical Sampling,

5.5        Application of the Fourier Transform

Frequency Response of Linear Systems,

Frequency Spectra of Signals,

Summary,

5.6        Energy and Power Density Spectra

Energy Density Spectrum,

Power Density Spectrum,

Power and Energy Transmission,

Summary,

Summary

Problems

6          Applications of the Fourier Transform

6.1        Ideal Filters

6.2        Real Filters

RC Low-Pass Filter,

Butterworth Filter,

Chebyschev and Elliptic Filters,

Bandpass Filters,

Summary,

6.3        Bandwidth Relationships

6.4        Reconstruction of signals from sample data

Interpolating Function,

Digital-to-analog Conversion,

6.5        Sinusoidal Amplitude Modulation

Frequency-Division Multiplexing,

6.6        Pulse-Amplitude Modulation

Time-Division Multiplexing,

Flat-Top PAM,

Summary

Problems

7          The Laplace Transform

7.1        Definitions of Laplace Transforms

7.2        Examples

7.3        Laplace Transforms of Functions

7.4        Laplace Transform Properties

Real Shifting,

Differentiation,

Integration,

Multiplication by t,

Initial Value,

Final Value,

Time Transformation,

7.6        Response of LTI Systems

Initial Conditions,

Transfer Functions,

Convolution,

Transforms with Complex Poles,

Functions with Repeated Poles,

7.7        LTI Systems Characteristics

Causality,

Stability,

Invertibility,

Frequency Response,

7.8        Bilateral Laplace Transform

Region of Convergence,

Bilateral Transform from Unilateral Tables,

Inverse Bilateral Laplace Transform,

7.9        Relationship of the Laplace Transform to the Fourier Transform

Summary

Problems

8          State Variables for Continuous-Time Systems

8.1        State-Variable Modeling

8.2        Simulation Diagrams

8.3        Solution of State Equations

Laplace-Transform Solution,

Convolution Solution,

Infinite Series Solution,

8.4        Properties of the State Transition Matrix

8.5        Transfer Functions

Stability,

8.6        Similarity Transformations

Transformations,

Properties,

Summary

Problems

9          Discrete-Time Signals and Systems

9.1        Discrete-Time Signals and Systems

Unit Step and Unit Impulse Functions,

Equivalent Operations,

9.2        Transformations of Discrete-Time Signals

Time Transformations,

Amplitude Transformations,

9.3        Characteristics of Discrete-Time Signals

Even and Odd Signals,

Signals Periodic in n,

Signals Periodic in W

9.4        Common Discrete-Time Signals

9.5        Discrete-Time Systems

Interconnecting Systems,

9.6        Properties of Discrete-Time Systems

Systems with Memory,

Invertibility,

Inverse of a System,

Causality,

Stability,

Time Invariance,

Linearity,

Summary

Problems

10        Discrete-Time Linear Time-Invariant Systems

10.1      Impulse Representation of Discrete-Time Signals

10.2      Convolution for Discrete-Time Systems

Properties of Convolution,

10.3      Properties of Discrete-Time LTI Systems

Memory,

Invertibility,

Causality,

Stability,

Unit Step Response,

10.4      Difference-Equation Models

Difference-Equation Models,

Classical Method,

Solution by Iteration,

10.5      Terms in the Natural Response

Stability,

10.6      Block Diagrams

Two Standard Forms,

10.7      System Response for Complex-Exponential Inputs

Linearity,

Complex Inputs for LTI Systems,

Stability,

Sampled Signals,

Impulse Response,

Summary

Problems

11        The z-Transform

11.1      Definitions of z-Transforms

11.2      Examples

Two z-Transforms,

Digital-Filter Example,

11.3      z-Transforms of Functions

Sinusoids,

11.4      z-Transform Properties

Real Shifting,

Initial and Final Values,

Time Scaling,

Convolution in Time,

11.6      LTI System Applications

Transfer Functions,

Inverse z-Transform,

Complex Poles,

Causality,

Stability,

Invertibility,

11.7      Bilateral z-Transform

Bilateral Transforms,

Regions of Convergence,

Inverse Bilateral Transforms,

Summary

Problems

12        Fourier Transforms of Discrete-Time Signals

12.1      Discrete-Time Fourier Transform

z-Transform,

12.2      Properties of the Discrete-Time Fourier Transform

Periodicity,

Linearity,

Time Shift,

Frequency Shift,

Symmetry,

Time Reversal,

Convolution in Time,

Convolution in Frequency,

Multiplication by n,

Parseval’s Theorem,

12.3      Discrete-Time Fourier Transform of Periodic Sequences

12.4      Discrete Fourier Transform

Shorthand Notation for the DFT,

Frequency Resolution of the DFT,

Validity of the DFT,

Summary,

12.5      Fast Fourier Transform

Decomposition-in-Time Fast Fourier Transform Algorithm,

Decomposition-in-Frequency Fast Fourier Transform,

Summary,

12.6      Applications of the Discrete Fourier Transform

Calculation of Fourier Transforms,

Convolution,

Filtering,

Correlation,

Energy Spectral Density Estimation,

Summary,

12.7      The Discrete Cosine Transform,

Summary

Problems

13        State Variables for Discrete-Time Systems

13.1      State-Variable Modeling

13.2      Simulation Diagrams

13.3      Solution of State Equations

Recursive Solution,

z-Transform Solution,

13.4      Properties of the State Transition Matrix

13.5      Transfer Functions

Stability,

13.6      Similarity Transformations

Properties,

Summary

Problems

Appendices

A.        Integrals and Trigonometric Identities

Integrals,

Trigonometric Identities,

B.         Leibnitz’s and L’Hôpital’s Rules

Leibnitz’s Rule,

L’Hôpital’s Rule,

C.         Summation Formulas for Geometric Series

D.         Complex Numbers and Euler’s Relation

Complex-Number Arithmetic,

Euler’s Relation,

Conversion Between Forms,

E.         Solution of Differential Equations

Complementary Function,

Particular Solution,

General Solution,

Repeated Roots,

F.         Partial-Fraction Expansions

G.         Review of Matrices

Algebra of Matrices,

Other Relationships