Hilbert Transforms in Signal Processing

Theory of The One-Dimensional Hilbert Transformation: Concepts of Hilbert and Fourier Transformations. Analytic Functions. Cauchy Integral Representation of the Analytic Function. Examples of Derivation of Hilbert Transforms in the Time Domain. Fourier Transform of the Hilbert Transform. Derivation of Hilbert Transforms Using Fourier and Hartley Transforms. Hilbert Transforms of Periodic Signals and Bessel Functions of the First Kind. One-Sided Spectra of Analytic Signals and Analytic Spectra of Causal Signals. Definitions of Instantaneous Amplitude, Phase and Frequency of Analytic Signals. Negative Instantaneous Frequency of the Analytic Signal. Tables of Hilbert Transforms. References. Properties of The Hilbert Transformation-Derivations and Applications: Introduction. Table of Properties of the Hilbert Transformation. Linearity. Method of Generating Hilbert Transform Using Samples of a Function. Hilbert Transforms of Hyperbolic Functions. Iteration. Differentiation. Successive Iteration and Differentiation. Differentiation and Multiplication of Hermite Polynomials and Functions by Hilbert Transforms. Hilbert Transforms of Legendre Polynomials. Autoconvolution, Autocorrelation and Energy Equality. The N-Fold Autoconvolution. The Hilbert Transform of the Product of Two Signals. References. Distributions in The Theory of The Hilbert Transformation and Complex Signals: Introduction. Definition of a Distribution in Terms of Functionals. Complex Delta Distribution. Polar and Other Notations of the Complex Delta Distribution. Integral and Derivatives of The Complex Delta Distribution in Terms of the Cauchy Approximation Functions. Integral and Derivatives of the Complex Delta Distribution. Complex Sampling Sequence. 2D Complex Delta Distribution and Its Polar Representation. 2D Complex Sampling Sequence. 3D Complex Delta Distribution. Appendix. References. The Discrete Hilbert Transformation: Introduction. Discrete Fourier Transformation (DFT). Examples of Derivation of DFT for Selected Simple Signals. Z-Transformation. Elementary Properties of the DFT and Z-Transformation. Discrete Hilbert Transformation (DHT). DHT of Selected Sequences. Iteration of the DHT. System Theory of the DHT. Complex Analytic Discrete Sequence. Causal Discrete Time Sequences and Analytic Discrete Spectra. Bilinear Transformation and Cotangent Form of Hilbert Transformation. References. Appendices: Fourier Series Representation of the DFT. Calculation of DHT Using the Discrete Hartley Transformation. Computer Program Listings. Hilbert Transformers: General Features. Phase Splitter Hilbert Transformers. Simple Method of Design of Hilbert Phase Splitters. Hilbert Transformers with Tapped Delay Line. Band-Pass Hilbert Transformers. Generation of Hilbert Transforms Using SSB Filtering. Digital Hilbert Transformers. FIR Hilbert Transformers. FIR Hilbert Transformers with Half-Band Filters. Recapitulation of Design Procedure of FIR Hilbert Transformers. Digital All-Pass Hilbert Transformers. Design of Hilbert Transformers Using Bilinear Frequency Transformation. IIR Discrete-Time Hilbert Transformers. Differentiating Hilbert Transformers. References. Appendix: Design Details of IIR Hilbert Transformers. The Hilbert Transform in Modulation Theory: Introduction. Concept of The Modulation Function of a Harmonic Carrier. Classification. Test Signals in Modulation. Basic Theories of Amplitude and Angle Modulation. Single-Sideband Linear AM Modulation. General Forms of Single-Sideband Modulations. CSSB Signals for Linear Envelope and FM Modulators. CSSB Modulation for Square-Low AM Demodulator. References. Appendix: Derivation of the Spectrum of the CSSB Signal for Linear Amplitude Modulator. Modulation of The Periodic Test Signal. The Hilbert Transform in Signal and System Theory: Hilbert Transforms in the Theory of Linear Systems: Kramers-Kronig Relations. Amplitude-Phase Relations in DLTI Systems. Measurement Systems Using Amplitude Phase Relations of LTI Systems. Kramers-Kronig Relations in Linear Macroscopic Continuous Media. Concept of Signal Delay in Hilbertian Sense. Hilbert Transform in the Theory of Sampling. Sampling of Band-pass Signals. Quadrature Sampling of Band-pass Signal. Linear Transformations of Interpolatory Expansion. Generation of Random Signals Using Interpolation Expansion. Complex Analytic Random Signals. Instantaneous Spectral Moments Defined by the Wiegner-Ville Time-Frequency Distribution. Generation of 2-D Random Fields. Definition of Electrical Power in Terms of Hilbert Transforms and Analytic Signals. Generalization of Notion of Power For Signals with Finite Average Power. References. Multidimensional Complex Signals and Applications: Introduction. Definition of Two-Dimensional and Multidimensional Complex Signals. Separable and Conjugate 2-D Signals. Polar Notation of 2-D Complex Signals. Examples of Separable and Nonseparable 2-D Complex Signals. Image Decomposition and Reconstruction Using Amplitude and Phase Patterns. Three-Dimensional Complex Signals. Multidimensional Modulation Theory. References. Multidimensional Hilbert and Fourier Transformations: Introduction. Even and Odd Multidimensional Real Signals. Signal Domain Definition of the n-D and 2-D Hilbert Transformations. 2-D and Partial Hilbert Transformations. Derivation of n-D Hilbert Transforms Using n-D Fourier Transform. 2-D Discrete Hilbert Transformation. 2-D DHT. Stark's Extension of Bedrosian's Theorem. 2-D Hilbert Transforms Tables. References.