We regularly measure/estimate distance, time, pressure, temperature, cost - and more. Others measure our blood sugar level, weight, satisfaction – and more. Measurements are an integral part of our cognitive processes. While each profession has its own measurement technology, there is also a common background and technical apparatus, the knowledge of which greatly helps in mastering the learning processes of the various fields and effective cooperation. Measurement theory undertakes to present this.

The subject presents the basics of the theoretical background of engineering methods that help to learn about the surrounding material world and to characterize it quantitatively and qualitatively. It reviews signal and system theory, estimation, and decision theory as well as data and signal processing methods with the aim of facilitating the solution of complex measurement, modelling, and information processing tasks. Primarily related to continuous and hybrid systems, it significantly develops conscious modelling and problem-solving skills. It achieves all of this by placing measurement and modelling problems in a unified framework. This framework also includes the basic concepts of signal transmission systems. The methods learned in the course serve as a foundation and background for solving research and development tasks.

Students who successfully fulfil the requirements of the subject are expected to:

1. Know the place, role and relationship of measurement and modelling in cognitive processes.

2. When solving practical problems, they can apply basic signal and system theory, as well as estimation and decision theory procedures.

3. They should be aware of the basic methods of model fitting (identification and adaptation), as well as the different techniques of optimization, regarding recursive procedures that can be implemented in real time.

4. Know the most frequently used recursive signal processing techniques and their implementation aspects.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.).

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.