Measurement Theory

Week 1

Chapter 1. Introduction. Objective of the subject. Data types. Measurement accuracy, measurement uncertainty. The measurement procedure: Observation in the case of a deterministic channel. Observation in the case of a noisy channel. Chapter 2. Basics of decision theory: two-hypothesis Bayesian decision. Examples: constant signal detection, variable amplitude signal detection.

Week 2

Chapter 2 (cont.). Basics of decision theory. Examples: detection of a random amplitude signal in noise. Chapter 3. Basics of estimation theory: Bayesian estimators. Minimum mean square error, minimum mean absolute error, maximum posterior estimate. Bayesian estimator for Gaussian distributions. Maximum likelihood estimator. Gauss-Markov estimator.

Week 3

Chapter 3 (cont.). Basics of estimation theory: Estimators for parameters characterized by a deterministic model. Qualification of Estimates. Minimum variance, unbiased estimators. Cramer-Rao lower bound. Examples of scalar and vector parameter cases.

Week 4

Chapter 3 (cont.). Basics of estimation theory: the case of Gaussian distributed linear models loaded with white noise. Examples: polynomial of the discrete time index, discrete Fourier series expansion, FIR filter, linear model in the case of coloured noise, linear model in the case of a known component. The best linear unbiased estimator (BLUE). Maximum Likelihood (ML) estimators. Least Squared (LS) Estimators.

Week 5

Chapter 3 (cont.). The basics of the estimation theory: Complex examples: target tracking, measurement of azimuth. Summary.

Week 6

Chapter 4. Model fitting: regression calculation. Fully specified fully or with partially specified statistical characteristics, linear regression, linear regression based on measurement data. Adaptive linear combiner: Wiener-Hopf equation. Examination of the regression matrix: eigenvalue, eigenvector problem. Iterative model fitting methods: Newton, steepest descent, LMS, alpha-LMS, LMS-Newton, LMS-Newton with iterative estimation of the regression matrix. Iterative model fitting based on Taylor expansion of the criterion function. Adaptive IIR systems. Stability theory approach.

Week 7

Chapter 5. Basics of filter theory. Optimal non-recursive estimator: scalar Wiener filter. Recursive estimator from the optimal non-recursive estimator. Optimal recursive estimator: scalar Kalman estimator. Example. Optimal recursive predictor. Kalman filter in the vector case.

Week 8

Chapter 6. Recursive computation of LS estimators: The case of a linear observation model. LS estimators under constraints. The case of a nonlinear observation model.

Week 9

Chapter 7. Model-based signal processing. Recalling the basics. Simple averaging, exponential averaging, sliding window averaging, time, and frequency domain behaviour. Representation of signals in signal spaces: linear vector space, linear space, integral transformation. Observer for signal processing tasks. Bandpass filtering instead of frequency transposition-integration-frequency transposition. Derivation and properties of the resonator structure. Relation to the Lagrange structure and the frequency sampling procedure.

Week 10

Chapter 7 (cont.). Model-based signal processing. Recursive generation of an arbitrary discrete transformation. The resonator-based discrete Fourier transformer. The resonator-based observer as a universal signal processing device. Relationship with interpolation structures. Quadratic real coefficient resonator basic elements: direct, orthogonal, wave-digital. Passivity is a condition for resonator-based observers.

Week 11

Chapter 7 (cont.). Model-based signal processing. The boundedness condition for a resonator-based observer. The process of design that preserves passivity properties. Energy relations of signal processing algorithms. Example of energy relations for all-pass networks/computations. Efficiently implementable orthogonal transformation. The (formal) relationship between the recursive DFT and the LMS procedure.

Week 12

Chapter 7 (cont.). Model-based signal processing. Structure dependence of transient switching phenomena. Passivity in control technology: regulation through a network. Orthogonal structures in general. Orthogonal transformation for data reduction (Principal component analysis.).

Week 13

Chapter 8. The basics of non-linear signal processing: special test signals, special structures, homomorphic signal processing, application of sequence analysis. Polynomial filter. Outlook: measurement theory methods in complex tasks. Summary of the material of the subject.