Transmission of Information by Orthogonal Functions

Historical Background and Motivation for the Use of Nonsinusoidal Functions 1.- Orthogonal Functions, Walsh Functions and Other Basic Mathematical Concepts 3.- Filtering of Time and Space Signals 6.- Direct and Carrier Transmission of Signals 6.- Nonsinusoidal Electromagnetic Waves 7.- Statistical Theory of Communication 8.- 1. Mathematical Foundations.- 1.1 Orthogonal Functions.- 1.1.1 Orthogonality and Linear Independence.- 1.1.2 Series Expansion by Orthogonal Functions.- 1.1.3 Invariance of Orthogonality to Fourier Transformation.- 1.1.4 Walsh Functions.- 1.1.5 Hadamard Matrices and General Two-Valued Functions.- 1.1.6 Haar Functions and General Three-Valued Functions.- 1.1.7 Functions with Several Variables.- 1.2 Generalized Fourier Analysis.- 1.2.1 Transition from Fourier Series to Fourier Transform.- 1.2.2 Generalized Fourier Transform.- 1.2.3 Invariance of Orthogonality to the Generalized Fourier Transform.- 1.2.4 Examples of the Generalized Fourier Transform.- 1.2.5 Fast Walsh-Fourier Transform.- 1.2.6 Fast Haar-Fourier Transform.- 1.2.7 Generalized Laplace Transform.- 1.2.8 Dyadic Differentiation, Integration and Correlation.- 1.3 Generalized Frequency.- 1.3.1 Physical Interpretation of the Generalized Frequency.- 1.3.2 Power Spectrum, Amplitude Spectrum, Filtering of Signals.- 1.3.3 Examples of Walsh-Fourier Transforms and Power Spectra.- 1.3.4 Dyadic Time.- 2. Sequency Filters for Time and Space Signals.- 2.1 Correlation Filters for Time Signals.- 2.1.1 Generation of Time Variable Walsh Functions.- 2.1.2 Sequency Low-Pass Filters.- 2.1.3 Sequency Band-Pass Filters.- 2.1.4 Multiplicity Filters and Asynchronous Filters.- 2.2 Resonance Filters for Time Signals.- 2.2.1 Series and Parallel LCS Resonance Filters.- 2.2.2 Low-Pass LCS Resonance Filters.- 2.2.3 Parametric Amplifiers.- 2.3 Instantaneous Filters for Space Signals.- 2.3.1 Filters for Signals with One Space Variable.- 2.3.2 Filters for Signals with Two Space Variables.- 2.3.3 Practical Implementation of Filters for Signals with Two Space Variables.- 2.3.4 Filters for Signals with Three Space Variables.- 2.4 Sampling Filters for Space Signals.- 2.4.1 Generators for Space Variable Walsh Functions.- 2.4.2 Sampling in Two or Three Space Dimensions by Block Pulses and Walsh Functions.- 2.4.3 Sampling Filters and Displays for Signals with Two or Three Space Variables.- 2.5 Digital Sequency Filters.- 2.5.1 Filters Based on the Generalized Fourier Transform.- 2.5.2 Filters Based on Difference Equations.- 3. Direct Transmission of Signals.- 3.1 Orthogonal Division as Generalization of Time and Frequency Division.- 3.1.1 Representation of Signals.- 3.1.2 Examples of Signals.- 3.1.3 Amplitude Sampling and Orthogonal Decomposition.- 3.2 Practical Problems of Transmission.- 3.2.1 Circuits for Orthogonal Division.- 3.2.2 Transmission of Digital Signals by Sine and Cosine Pulses.- 3.2.3 Conversion of Sequency Limited Signals into Frequency Limited Signals.- 3.3 Characterization of Communication Channels.- 3.3.1 Frequency Response of Attenuation and Phase Shift of a Communication Channel.- 3.3.2 Characterization of a Communication Channel by Crosstalk Parameters.- 4. Carrier Transmission of Signals.- 4.1 Amplitude Modulation (AM).- 4.1.1 Modulation and Synchronous Demodulation.- 4.1.2 Correction of Time Differences in Synchronous Demodulation.- 4.1.3 Methods of Single Sideband Modulation.- 4.2 Multiplexing of Time Variable Signals.- 4.2.1 Multiplex Systems.- 4.2.2 Examples of Mixed Sequency-Frequency Multiplex Systems.- 4.2.3 Digital Multiplexing.- 4.3 Time Base, Time Position and Code Modulation.- 4.3.1 Time Base Modulation (TBM).- 4.3.2 Time Position Modulation (TPM).- 4.3.3 Code Modulation (CM).- 5. Nonsinusoidal Electromagnetic Waves.- 5.1 Dipole Radiation of Walsh Waves.- 5.1.1 Hertzian Dipole Solution of Maxwell’s Equations.- 5.1.2 Near Zone-Wave Zone Effect.- 5.1.3 Magnetic Dipole Radiation.- 5.1.4 Implementation of Radiators.- 5.2 Multipole Radiation of Walsh Waves.- 5.2.1 Radiation of a One-Dimensional Quadrupole.- 5.2.2 Radiation of a Two-Dimensional Quadrupole.- 5.2.3 Multipole Radiation.- 5.3 Interference Effects, Doppler Effect.- 5.3.1 Radiation Diagram of a Row of Spherical Radiators.- 5.3.2 Doppler Effect.- 5.3.3 Circular Polarization, Interferometry, Shape Recognition.- 5.4 Signal Selection and Synchronization.- 5.4.1 Separation of Signals in Mobile Communication.- 5.4.2 Synchronous Reception of Walsh Waves.- 6. Application of Orthogonal Functions To Statistical Problems.- 6.1 Series Expansion of Stochastic Functions.- 6.1.1 Thermal Noise.- 6.1.2 Statistical Independence of the Components of an Orthogonal Expansion.- 6.2 Additive Disturbances.- 6.2.1 Least Mean Square Deviation of a Signal from Sample Functions.- 6.2.2 Examples of Circuits.- 6.2.3 Matched Filters.- 6.2.4 Compandors for Sequency Signals.- 6.3 Multiplicative Disturbances.- 6.3.1 Interference Fading.- 6.3.2 Diversity Transmission Using Many Copies.- 7. Signal Design for Improved Reliability.- 7.1 Transmission Capacity.- 7.1.1 Measures of Bandwidth.- 7.1.2 Transmission Capacity of Communication Channels.- 7.1.3 Signal Delay and Signal Distortions.- 7.2 Error Probability of Signals.- 7.2.1 Error Probability of Simple Signals due to Thermal Noise.- 7.2.2 Peak Power Limited Signals.- 7.2.3 Pulse-Type Disturbances.- 7.3 Coding.- 7.3.1 Coding with Binary Elements.- 7.3.2 Orthogonal, Transorthogonal and Biorthogonal Alphabets.- 7.3.3 Coding for Error-Free Transmission.- 7.3.4 Ternary Combination Alphabets.- 7.3.5 Combination Alphabets of Order 2r + 1.- References.