Practical Approach To Signals and Systems

D. Sundararajan is a consultant in digital signal processing at NewTech Software Pt. Ltd in Bangalore India. He has taught undergraduate and graduate classes in digital signal processing, engineering mathematics, programming, operating systems and digital logic design at Concordia University, Canada and Nanyang Technological Institute, Singapore. Sundararajan is the principal inventor of the latest family of Discrete Fourier Transform (DFT) algorithms, and has published two books on DFT and related areas. He also wrote a lab manual for the Digital Electronics Course at Concordia University, and has three patents in fast Fourier transforms. Over the course of his engineering career he has held positions at the National Aerospace Laboratory, Bangalore, the National Physical Laboratory, New Delhi, working on the design of digital and analog signal processing systems. Sundararajan holds a B.E. in Electrical Engineering from Madras University, an M.Tech in Electrical Engineering from IIT Chennai, and a Ph.D. in the same from Concordia University.

Preface. Abbreviations. 1 Introduction. 1.1 The Organization of this Book. 2 Discrete Signals. 2.1 Classification of Signals. 2.1.1 Continuous, Discrete, and Digital Signals. 2.1.2 Periodic and Aperiodic Signals. 2.1.3 Energy and Power Signals. 2.1.4 Even- and Odd-Symmetric Signals. 2.1.5 Causal and Noncausal Signals. 2.1.6 Deterministic and Random Signals. 2.2 Basic Signals. 2.2.1 Unit-Impulse Signal. 2.2.2 Unit-Step Signal. 2.2.3 Unit-Ramp Signal. 2.2.4 Sinusoids and Exponentials. 2.3 Signal Operations. 2.3.1 Time Shifting. 2.3.2 Time Reversal. 2.3.3 Time Scaling. 2.4 Summary. References. Exercises. 3 Continuous Signals. 3.1 Classification of Signals. 3.1.1 Continuous Signals. 3.1.2 Periodic and Aperiodic Signals. 3.1.3 Energy and Power Signals. 3.1.4 Even- and Odd-Symmetric Signals. 3.1.5 Causal and Noncausal Signals. 3.2 Basic Signals. 3.2.1 The Unit-Step Signal. 3.2.2 The Unit-Impulse Signal. 3.2.3 The Unit-Ramp Signal. 3.2.4 Sinusoids. 3.3 Signal Operations. 3.3.1 Time Shifting. 3.3.2 Time Reversal. 3.3.3 Time Scaling. 3.4 Summary. Reference. Exercises. 4 Time-Domain Analysis of Discrete Systems. 4.1 Difference Equation Model. 4.1.1 System Response. 4.1.2 Impulse Response. 4.1.3 Characterization of Systems by their Responses to Impulse and Unit-Step Signals. 4.2 Classification of Systems. 4.2.1 Linear and Nonlinear Systems. 4.2.2 Time-Invariant and Time-Varying Systems. 4.2.3 Causal and Noncausal Systems. 4.2.4 Instantaneous and Dynamic Systems. 4.2.5 Inverse Systems. 4.2.6 Continuous and Discrete Systems. 4.3 Convolution-Summation Model. 4.3.1 Properties of Convolution-Summation. 4.3.2 The Difference Equation and the Convolution-Summation. 4.3.3 Response to Complex Exponential Input. 4.4 System Stability. 4.5 Realization of Discrete Systems. 4.5.1 Decomposition of Higher-Order Systems. 4.5.2 Feedback Systems. 4.6 Summary. References. Exercises. 5 Time-Domain Analysis of Continuous Systems. 5.1 Classification of Systems. 5.1.1 Linear and Nonlinear Systems. 5.1.2 Time-Invariant and Time-Varying Systems. 5.1.3 Causal and Noncausal Systems. 5.1.4 Instantaneous and Dynamic Systems. 5.1.5 Lumped-Parameter and Distributed-Parameter Systems. 5.1.6 Inverse Systems. 5.2 Difference Equation Model. 5.3 Convolution-Integral Model. 5.3.1 Properties of Convolution-Integral. 5.4 System Response. 5.4.1 Impulse Response. 5.4.2 Response to Unit-Step Input. 5.4.3 Characterization of Systems by their Responses to Impulse and Unit-Step Signals. 5.4.4 Response to Complex Exponential Input. 5.5 System Stability. 5.6 Realization of Continuous Systems. 5.6.1 Decomposition of Higher-Order Systems. 5.6.2 Feedback Systems. 5.7 Summary. Reference. Exercises. 6 The Discrete Fourier Transform. 6.1 The Time-Domain and Frequency-Domain. 6.2 The Fourier Analysis. 6.2.1 Versions of Fourier Analysis. 6.3 The Discrete Fourier Transform. 6.3.1 The Approximation of Arbitrary Waveforms with Finite Number Samples. 6.3.2 The DFT and the IDFT. 6.3.3 DFT of Some Basic Signals. 6.4 Properties of the Discrete Fourier Transform. 6.4.1 Linearity. 6.4.2 Periodicity. 6.4.3 Circular Shift of a Sequence. 6.4.4 Circular Shift of a Spectrum. 6.4.5 Symmetry. 6.4.6 Circular Convolution of Time-Domain Sequences. 6.4.7 Circular Convolution of Frequency-Domain Sequences. 6.4.8 Parseval's Theorem. 6.5 Applications of the Discrete Fourier Transform. 6.5.1 Computation of the Linear Convolution Using the DFT. 6.5.2 Interpolation and Decimation. 6.6. Summary. References. Exercises. 7 Fourier Series. 7.1 Fourier Series. 7.1.1 FS as the Limiting Case of the DFT. 7.1.2 The Compact Trigonometric Form of the FS. 7.1.3 The Trigonometric Form of the FS. 7.1.4 Periodicity of the FS. 7.1.5 Existence of the FS. 7.1.6 Gibbs Phenomenon. 7.2 Properties of the Fourier Series. 7.2.1 Linearity. 7.2.2 Symmetry. 7.2.3 Time-Shifting. 7.2.4 Frequency-Shifting. 7.2.5 Convolution in the Time-Domain. 7.2.6 Convolution in the Frequency-Domain. 7.2.7 Duality. 7.2.8 Time-Scaling. 7.2.9 Time-Differentiation. 7.2.10 Time-Integration. 7.2.11 Parseval's Theorem. 7.3 Approximation of the Fourier Series. 7.3.1 Aliasing Effect. 7.4 Applications of the Fourier Series. 7.5 Summary. References. Exercises. 8 The Discrete-Time Fourier Transform. 8.1 The Discrete-Time Fourier Transform. 8.1.1 The DTFT as the Limiting Case of the DFT. 8.1.2 The Dual Relationship Between the DTFT and the FS. 8.1.3 The DTFT of a Discrete Periodic Signal. 8.1.4 Determination of the DFT from the DTFT. 8.2 Properties of the Discrete-Time Fourier Transform. 8.2.1 Linearity. 8.2.2 Time-Shifting. 8.2.3 Frequency-Shifting. 8.2.4 Convolution in the Time-Domain. 8.2.5 Convolution in the Frequency-Domain. 8.2.6 Symmetry. 8.2.7 Time-Reversal. 8.2.8 Time-Expansion. 8.2.9 Frequency-Differentiation. 8.2.10 Difference. 8.2.11 Summation. 8.2.12 Parseval's Theorem and the Energy Transfer Function. 8.3 Approximation of the Discrete-Time Fourier Transform. 8.3.1 Approximation of the Inverse DTFT by the IDFT. 8.4 Applications of the Discrete-Time Fourier Transform. 8.4.1 Transfer Function and the System Response. 8.4.2 Digital Filter Design Using DTFT. 8.4.3 Digital Differentiator. 8.4.4 Hilbert Transform. 8.5 Summary. References. Exercises. 9 The Fourier Transform. 9.1 The Fourier Transform. 9.1.1 The FT as the Limiting Case of the DTFT. 9.1.2 Existence of the FT. 9.2 Properties of the Fourier Transform. 9.2.1 Linearity. 9.2.2 Duality. 9.2.3 Symmetry. 9.2.4 Time-Shifting. 9.2.5 Frequency-Shifting. 9.2.6 Convolution in the Time-Domain. 9.2.7 Convolution in the Frequency-Domain. 9.2.8 Conjugation. 9.2.9 Time-Reversal. 9.2.10 Time-Scaling. 9.2.11 Time-Differentiation. 9.2.12 Time-Integration. 9.2.13 Frequency-Differentiation. 9.2.14 Parseval's Theorem and the Energy Transfer Function. 9.3 Fourier Transform of Mixed Class Signals. 9.3.1 The FT of a Continuous Periodic Signal. 9.3.2 Determination of the FS from the FT. 9.3.3 The FT of a Sampled Signal and the Aliasing Effect. 9.3.4 The FT of a Sampled Aperiodic Signal and the DTFT of the Corresponding Discrete Signal. 9.3.5 The FT of a Sampled Periodic Signal and the DFT of the Corresponding Discrete Signal. 9.3.6 Approximation of the Continuous Signal from its Sampled Version. 9.4 Approximation of the Fourier Transform. 9.5 Applications of the Fourier Transform. 9.5.1 Transfer Function and the System Response. 9.5.2 Ideal Filters and their Unrealizability. 9.5.3 Modulation and Demodulation. 9.6 Summary. References. Exercises. 10 The z-Transform. 10.1 Fourier Analysis and the z-Transform. 10.2 The z-Transform. 10.3 Properties of the z-Transform. 10.3.1 Linearity. 10.3.2 Left Shift of a Sequence. 10.3.3 Right Shift of a Sequence. 10.3.4 Convolution. 10.3.5 Multiplication by n. 10.3.6 Multiplication by an. 10.3.7 Summation. 10.3.8 Initial Value. 10.3.9 Final Value. 10.3.10 Transform of Semiperiodic Functions. 10.4 The Inverse z-Transform. 10.4.1 Finding the Inverse z-Transform. 10.5 Applications of the z-Transform. 10.5.1 Transfer Function and the System Response. 10.5.2 Characterization of a System by its Poles and Zeros. 10.5.3 System Stability. 10.5.4 Realization of Systems. 10.5.5 Feedback Systems. 10.6 Summary. References. Exercises. 11 The Laplace Transform. 11.1 The Laplace Transform. 11.1.1 Relationship Between the Laplace Transform and the z-Transform. 11.2 Properties of the Laplace Transform. 11.2.1 Linearity. 11.2.2 Time-Shifting. 11.2.3 Frequency-Shifting. 11.2.4 Time-Differentiation. 11.2.5 Integration. 11.2.6 Time-Scaling. 11.2.7 Convolution in Time. 11.2.8 Multiplication by t. 11.2.9 Initial Value. 11.2.10 Final Value. 11.2.11 Transform of Semiperiodic Functions. 11.3 The Inverse Laplace Transform. 11.4 Applications of the Laplace Transform. 11.4.1 Transfer Function and the System Response. 11.4.2 Characterization of a System by its Poles and Zeros. 11.4.3 System Stability. 11.4.4 Realization of Systems. 11.4.5 Frequency-Domain Representation of Circuits. 11.4.6 Feedback Systems. 11.4.7 Analog Filters. 11.5 Summary. Reference. Exercises. 12 State-Space Analysis of Discrete Systems. 12.1 The State-Space Model. 12.1.1 Parallel Realization. 12.1.2 Cascade Realization. 12.2 Time-Domain Solution of the State Equation. 12.2.1 Iterative Solution. 12.2.2 Closed-Form Solution. 12.2.3 The Impulse Response. 12.3 Frequency-Domain Solution of the State Equation. 12.4 Linear Transformation of State Vectors. 12.5 Summary. Reference. Exercises. 13 State-Space Analysis of Continuous Systems. 13.1 The State-Space Model. 13.2 Time-Domain Solution of the State Equation. 13.3 Frequency-Domain Solution of the State Equation. 13.4 Linear Transformation of State Vectors. 13.5 Summary. Reference. Exercises. Appendix A Transform Pairs and Properties. Appendix B Useful Mathematical Formulas. Answers to Selected Exercises. Index.