Multirate Statistical Signal Processing

1. Introduction. 1.1. Multi-channel multirate signal measurement. 1.2. Multirate statistical signal processing. 1.3. Notation. 2. Background. 2.1. Second-order theory of stationary stochastic processes. 2.2. Statistical inference and information. 2.3. Theory of majorization. 2.4. Inverse problems, ill-posedness and Tikhonov’s theory of regularization. 3. Multirate Spectrum Estimation. 3.1. Introduction. 3.2. Formulating the inference problem. 3.3. The Maximum Entropy principle. 3.4. Solving Problem 2. 3.5. On well-posedness of the Maximum Entropy solution. 3.6. Practical considerations. 3.7. Examples. 3.8. Discussion on the Maximum Entropy formalism. 3.9. Concluding remarks. 4. Multirate Time Delay Estimation. 4.1. Introduction. 4.2. Time-Delay Estimation in Multirate Sensor Arrays. 4.3. Fusion of Low-rate Signals In The Presence Of Time Delay. 4.4. Designing The Synthesis Filters. 4.5. Procedure for designing multirate sensor arrays. 4.6. Concluding Remarks. 4.7. Proof of Theorem 1. 4.8. Proof of Theorem 2. 4.9. Perfect reconstruction linear-phase filter banks. 5. Multirate Signal Estimation. 5.1. Introduction. 5.2. Problem Statement. 5.3. Statistics of the non-observable vector X and the measurement vector V. 5.4. Estimating X given V. 5.5. Discussion. 5.6. Putting everything together. 5.7. Concluding remarks. 6. Algebraic Theory of Scalable Multirate Systems. 6.1. Introduction. 6.2. FIR analysis and synthesis systems. 6.3. Scalability in multirate systems. 6.4. Embedding partial ordering of scalability in a total ordering. 6.5. SC-Optimality and Subband Coding. 6.6. SC-Optimality and the Principal Component Filter Bank. 6.7. Concluding remarks. 6.8. Summary. 7. Information Theory of Multirate Systems. 7.1. Introduction. 7.2. The information content of a low-rate measurement. 7.3. Measuring statistical information in practice. 7.4.Scalability in terms of information. 7.5. Concluding remarks and open problems. 8. Distributed Algorithms. 8.1. Introduction. 8.2. Spectrum estimation using sensor networks. 8.3. Inverse and Ill-posed problems. 8.4. Spectrum estimation using generalized projections. 8.5. Distributed algorithms for calculating generalized projection. 8.6. Concluding remark. 8.7. Acknowledgements. 9. Epilogue.