Contents. Acknowledgments

Transcription

1 Table of Preface Acknowledgments Notation page xii xx xxi 1 Signals and systems Continuous and discrete signals Unit step and nascent delta functions Relationship between complex exponentials and delta functions Attributes of signals Signal arithmetics and transformations Linear and time-invariant systems Signals through continuous LTI systems Signals through discrete LTI systems Continuous and discrete convolutions Homework problems 29 2 Vector spaces and signal representation Inner product space Vector space Inner product space Bases of vector space Signal representation by orthogonal bases Signal representation by standard bases An example: the Fourier transforms Unitary transformation and signal representation Linear transformation Eigenvalue problems Eigenvectors of D 2 as Fourier basis Unitary transformations Unitary transformations in N-D space Projection theorem and signal approximation Projection theorem and pseudo-inverse 70

2 Table of vii Signal approximation Frames and biorthogonal bases Frames Signal expansion by frames and Riesz bases Frames in finite-dimensional space Kernel function and Mercer s theorem Summary Homework problems Continuous-time Fourier transform The Fourier series expansion of periodic signals Formulation of the Fourier expansion Physical interpretation Properties of the Fourier series expansion The Fourier expansion of typical functions The Fourier transform of non-periodic signals Formulation of the CTFT Relation to the Fourier expansion Properties of the Fourier transform Fourier spectra of typical functions The uncertainty principle Homework problems Discrete-time Fourier transform Discrete-time Fourier transform Fourier transform of discrete signals Properties of the DTFT DTFT of typical functions The sampling theorem Reconstruction by interpolation Discrete Fourier transform Formulation of the DFT Array representation Properties of the DFT Four different forms of the Fourier transform DFT computation and fast Fourier transform Two-dimensional Fourier transform Two-dimensional signals and their spectra Fourier transform of typical 2-D functions Four forms of 2-D Fourier transform Computation of the 2-D DFT Homework problems 215

3 Table of viii 5 Applications of the Fourier transforms LTI systems in time and frequency domains Solving differential and difference equations Magnitude and phase filtering Implementation of 1-D filtering Implementation of 2-D filtering Hilbert transform and analytic signals Radon transform and image restoration from projections Orthogonal frequency-division modulation (OFDM) Homework problems The Laplace and z-transforms The Laplace transform From Fourier transform to Laplace transform The region of convergence Properties of the Laplace transform The Laplace transform of typical signals Analysis of continuous LTI systems by Laplace transform First-order system Second-order system The unilateral Laplace transform The z-transform From Fourier transform to z-transform Region of convergence Properties of the z-transform The z-transform of typical signals Analysis of discrete LTI systems by z-transform First- and second-order systems The unilateral z-transform Homework problems Fourier-related orthogonal transforms The Hartley transform Continuous Hartley transform Properties of the Hartley transform Hartley transform of typical signals Discrete Hartley transform The 2-D Hartley transform The discrete sine and cosine transforms The continuous cosine and sine transforms From DFT to DCT and DST Matrix forms of DCT and DST Fast algorithms for the DCT and DST 366

4 Table of ix DCT and DST filtering The 2-D DCT and DST Homework problems The Walsh-Hadamard, slant, and Haar transforms The Walsh-Hadamard transform Hadamard matrix Hadamard-ordered Walsh-Hadamard transform (WHT h ) Fast Walsh-Hadamard transform algorithm Sequency-ordered Walsh-Hadamard matrix (WHT w ) Fast Walsh-Hadamard transform (sequency ordered) The slant transform Slant matrix Slant transform and its fast algorithm The Haar transform Continuous Haar transform Discrete Haar transform Computation of the discrete Haar transform Filter bank implementation Two-dimensional transforms Homework problems Karhunen-Loève transform and principal component analysis Stochastic process and signal correlation Signals as stochastic processes Signal correlation Karhunen-Loève transform (KLT) Continuous KLT Discrete KLT Optimalities of the KLT Geometric interpretation of the KLT Principal component analysis (PCA) Comparison with other orthogonal transforms Approximation of the KLT by the DCT Applications of the KLT Image processing and analysis Feature extraction for pattern classification Singular value decomposition transform Singular value decomposition Application in image compression Homework problems Continuous- and discrete-time wavelet transforms 461

5 Table of x 10.1 Why wavelet? Short-time Fourier transform and Gabor transform The Heisenberg uncertainty Continuous-time wavelet transform (CTWT) Mother and daughter wavelets The forward and inverse wavelet transforms Properties of the CTWT Typical mother wavelet functions Discrete-time wavelet transform (DTWT) Discretization of wavelet functions The forward and inverse transform A fast inverse transform algorithm Wavelet transform computation Filtering based on wavelet transform Homework problems Multiresolution analysis and discrete wavelet transform Multiresolution analysis (MRA) Scale spaces Wavelet spaces Properties of the scaling and wavelet filters Relationship between scaling and wavelet filters Wavelet series expansion Construction of scaling and wavelet functions Discrete wavelet transform (DWT) Discrete wavelet transform (DWT) Fast wavelet transform (FWT) Filter bank implementation of DWT and inverse DWT Two-channel filter bank and inverse DWT Two-dimensional DWT Applications in filtering and compression Homework problems 542 Appendices 546 A Review of linear algebra 546 A.1 Basic definitions 546 A.2 Eigenvalues and eigenvectors 551 A.3 Hermitian matrix and unitary matrix 552 A.4 Toeplitz and circulant matrices 554 A.5 Vector and matrix differentiation 554 B Review of random variables 556

6 Table of xi B.1 Random variables 556 B.2 Multivariate random variables 558 B.3 Stochastic models 562 Bibliography 565 Index 566