Springer Texts in Electrical Engineering

1 Introduction.- 1.1 The Cardinal Series.- 1.2 History.- 2 Fundamentals of Fourier Analysis and Stochastic Processes.- 2.1 Signal Classes.- 2.2 The Fourier Transform.- 2.2.1 The Fourier Series.- 2.2.1.1 Convergence.- 2.2.1.2 Orthogonal Basis Functions.- 2.2.2 Some Elementary Functions.- 2.2.3 Some Transforms of Elementary Functions.- 2.2.4 Other Properties.- 2.3 Stochastic Processes.- 2.3.1 First and Second Order Statistics.- 2.3.2 Stationary Processes.- 2.3.2.1 Power Spectral Density.- 2.3.2.2 Some Stationary Noise Models.- 2.3.2.3 Linear Systems with Stationary Stochastic Inputs.- 2.4 Exercises.- 3 The Cardinal Series.- 3.1 Interpretation.- 3.2 Proofs.- 3.2.1 Using Comb Functions.- 3.2.2 Fourier Series Proof.- 3.2.3 Papoulis' Proof.- 3.3 Properties.- 3.3.1 Convergence.- 3.3.1.1 For Finite Energy Signals.- 3.3.1.2 For Bandlimited Functions with Finite Area Spectra.- 3.3.2 Trapezoidal Integration.- 3.3.2.1 Of Bandlimited Functions.- 3.3.2.2 Of Linear Integral Transforms.- 3.3.2.3 Parseval's Theorem for the Cardinal Series.- 3.3.3 The Time-Bandwidth Product.- 3.4 Application to Spectra Containing Distributions.- 3.5 Application to Bandlimited Stochastic Processes.- 3.6 Exercises.- 4 Generalizations of the Sampling Theorem.- 4.1 Generalized Interpolation Functions.- 4.1.1 Oversampling.- 4.1.1.1 Sample Dependency.- 4.1.1.2 Relaxed Interpolation Formulae.- 4.1.2 Criteria for Generalized Interpolation Functions.- 4.1.2.1 Interpolation Functions.- 4.1.2.2 Reconstruction from a Filtered Signal's Samples.- 4.2 Papoulis' Generalization.- 4.2.1 Derivation.- 4.2.2 Interpolation Function Computation.- 4.2.3 Example Applications.- 4.2.3.1 Recurrent Nonuniform Sampling.- 4.2.3.2 Interlaced Signal-Derivative Sampling.- 4.2.3.3 Higher Order Derivative Sampling.- 4.2.3.4 Effects of Oversampling.- 4.3 Derivative Interpolation.- 4.3.1 Properties of the Derivative Kernel.- 4.4 A Relation Between the Taylor and Cardinal Series.- 4.5 Sampling Trigonometric Polynomials.- 4.6 Sampling Theory for Bandpass Functions.- 4.6.1 Heterodyned Sampling.- 4.6.2 Direct Bandpass Sampling.- 4.7 A Summary of Sampling Theorems for Directly Sampled Signals.- 4.8 Lagrangian Interpolation.- 4.9 Kramer's Generalization.- 4.10 Exercises.- 5 Sources of Error.- 5.1 Effects of Additive Data Noise.- 5.1.1 On Cardinal Series Interpolation.- 5.1.1.1 Interpolation Noise Level.- 5.1.1.2 Effects of Oversampling and Filtering.- 5.1.2 Interpolation Noise Variance for Directly Sampled Signals.- 5.1.2.1 Interpolation with Lost Samples.- 5.1.2.2 Bandpass Functions.- 5.1.3 On Papoulis' Generalization.- 5.1.3.1 Examples.- 5.1.3.2 Notes.- 5.1.4 On Derivative Interpolation.- 5.1.4.1 A Lower Bound on the NINV.- 5.1.4.2 Examples.- 5.2 Jitter.- 5.2.1 Filtered Cardinal Series Interpolation.- 5.2.2 Unbiased Interpolation from Jittered Samples.- 5.2.3 In Stochastic Bandlimited Signal Interpolation.- 5.2.3.1 NINV of Unbiased Restoration.- 5.2.3.2 Examples.- 5.3 Truncation Error.- 5.3.1 An Error Bound.- 5.3.2 Noisy Stochastic Signals.- 5.4 Exercises.- 6 The Sampling Theorem in Higher Dimensions.- 6.1 Multidimensional Fourier Analysis.- 6.1.1 Properties.- 6.1.1.1 Separability.- 6.1.1.2 Rotation, Scale and Transposition.- 6.1.1.3 Polar Representation.- 6.1.2 Fourier Series.- 6.1.2.1 Multidimensional Periodicity.- 6.1.2.2 The Fourier Series Expansion.- 6.2 The Multidimensional Sampling Theorem.- 6.2.1 The Nyquist Density.- 6.2.2 Generalized Interpolation Functions.- 6.2.2.1 Tightening the Integration Region.- 6.2.2.2 Allowing Slower Roll Off.- 6.3 Restoring Lost Samples.- 6.3.1 Restoration Formulae.- 6.3.2 Noise Sensitivity.- 6.3.2.1 Filtering.- 6.3.2.2 Deleting Samples from Optical Images.- 6.4 Periodic Sample Decimation and Restoration.- 6.4.1 Preliminaries.- 6.4.2 First Order Decimated Sample Restoration.- 6.4.3 Sampling Below the Nyquist Density.- 6.4.4 Higher Order Decimation.- 6.5 Raster Sampling.- 6.6 Exercises.- 7 Continuous Sampling.- 7.1 Interpolation From Periodic Continuous Samples.- 7.1.1 The Restoration Algorithm.- 7.1.2 Noise Sensitivity.- 7.1.2.1 White Noise.- 7.1.2.2 Colored Noise.- 7.1.3 Observations.- 7.1.3.1 Comparison with the NINV of the Cardinal Series.- 7.1.3.2 In the Limit as an Extrapolation Algorithm.- 7.1.4 Application to Interval Interpolation.- 7.2 Prolate Spheroidal Wave Functions.- 7.2.1 Properties.- 7.2.2 Application to Extrapolation.- 7.2.3 Application to Interval Interpolation.- 7.3 The Papoulis-Gerchberg Algorithm.- 7.3.1 The Basic Algorithm.- 7.3.2 Proof of the PGA using PSWF's.- 7.3.3 Geometrical Interpretation in a Hilbert Space.- 7.3.4 Remarks.- 7.4 Exercises.