Master
2019/2020

## Digital Signal Processing

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type:
Elective course (Cognitive Sciences and Technologies: From Neuron to Cognition)

Area of studies:
Psychology

Delivered by:
School of Psychology

When:
1 year, 3 module

Mode of studies:
offline

Master’s programme:
Cognitive Sciences and Technologies: From Neuron to Cognition

Language:
English

ECTS credits:
4

### Course Syllabus

#### Abstract

In modern science system's based approach is frequently exercised as it allows to formalize the problems encountered in the real world by representing them in the well studied framework that provides for efficient analysis and solution. In order to get a full grip of his powerful machinery researchers need to understand the fundamentals principles of the theory. The existing classes are either too specialized and mathematically detailed or too much of a cook-book nature. The goal of this class is to provide master students of non-mathematical background with a unifying view on the theory behind the system-oriented approach. The main goal is to develop the intuition behind the complex concepts. We will attempt to do it by showing the similarity of discrete and continuous treatments of the system's theory and interpret the results using the natural concepts of linear algebra. The second part of the class is primarily dedicated to the fundamentals of estimation theory. We start from the basic concepts and estimation of model parameters in the white and colored noise cases. We will then introduce the Maximum Likelihood and Bayesian approaches to the problem of parameter estimation. We will illustrate the Bayesian methodology via examples of solving the inverse problem in neuroimaging and spectroscopy. We will use the Bayesian concept to develop Kalman filters - advanced model-based estimators taking into account the dynamical properties of the signals.

#### Learning Objectives

- Learn the unifying view on the theory behind system oriented approach in the modern science and to demonstrate the equivalence of “continuous” and “discrete” approaches.
- The student will be able to reflect developed mathematical methods to psychological fields and problems.
- The student will obtain necessary basic knowledge in signals, harmonic analysis and systems theory.
- The student will be able to describe psychological problems in terms of computational mathematics.
- The student will be able to identify systems theory aspects in psychological and neurobiological research tasks and suggest a method to tackle the problem and rank several available techniques in the order of applicability in the current situation.

#### Expected Learning Outcomes

- Know the definition of signal, basic model signal types, understand the difference between the continuous and discrete signals, be able to formally write an expression of an arbitrary signal, be able to show the link between the continuous and discrete treatments, be able to explain the role linear algebra plays in signal processing.
- Understand the notion of a transform, be able to interpret a transform in the linear algebra framework, be familiar with basic transforms used (Fourier transform, Wavelet transform), be able to formulate one’s own transform and write expressions for the coefficients and synthesis equation, be able to connect statistical properties of signals on both sides of the transform operation
- Understand the notion of system, understand why LTI systems is an important class of systems, know what convolution operation is, be able to calculate the output of an arbitrary LTI system, be able to relate statistical properties of the output signal to those of the input.
- Understand methods for statistical description of random processes, relation between times and frequency domain representation of second order statistical descriptors of random processes.
- Be able to perform z-domain analysis of digital filters, predict their frequency response, assess their characteristics e.g. group-delay, pass-band ripples, stop-band attenuation.

#### Course Contents

- Signals1. Deterministic Signals (discrete, continuous); 2. Signals as vectors; 3. Hilbert space, infinite dim Eucledian space, notion of distance; 4. Demonstrated equivalence of discrete and continuous forms of treatment.
- Linear systems1. Input-output relation, a system as an operator; 2. Calculation of an output of a system for an arbitrary input, pulse response; 3. Linear time invariant (LTI) systems; 4. Transient processes; 5. Passing random process through an LTI system; 6. Time domain; 7. Frequency domain, Transfer function or why a linear system cannot “bring in new frequencies”; 8. Z- transform as a method for analysis of discrete-time LTIs; 9. Digital Temporal filters as most frequently used LTIs; 10. FIR, IIR filters; 11. Stability, implementation details.
- Statistical description of signals in the original and transformed spaces1. Second order statistical description of random signals revisited; 2. Periodogram, Power spectral density (PSD); 3. Wiener-Hinchin theorem; 4. Methods to compute PSD.
- Random signals, random processes1. Mean, std, autocorrelation function, distribution function; 2. Properties of random signals (stationarity, ergodicity); 3. Scalar and vector random processes.
- Transforms and decompositions1. General idea of a transform; 2. A transform as a change of basis; 3. Calculating transform coefficients or why do we need an orthogonal basis? 4. Fourier transform: i. Periodic signals; ii. Aperiodic signals; iii. Discrete Fourier transform; 5. State-space representation and discrete transforms as a matrix-vector multiplication; 6. Time-frequency representation of signals; 7. Short-time Fourier transform; 8. Wavelet transform.

#### Bibliography

#### Recommended Core Bibliography

- Kundu, D., & Nandi, S. (2012). Statistical Signal Processing : Frequency Estimation. New Delhi: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=537977
- Miller, S. L., & Childers, D. (2012). Probability and Random Processes : With Applications to Signal Processing and Communications (Vol. 2nd ed). Burlington: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=453841
- Miller, S. L., & Childers, D. G. (2004). Probability and Random Processes : With Applications to Signal Processing and Communications. Amsterdam: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=187186
- Silver, N. (2012). The Signal and the Noise : Why So Many Predictions Fail-but Some Don’t. New York: Penguin Books. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1122593

#### Recommended Additional Bibliography

- Peña, D., Tiao, G. C., & Tsay, R. S. (2001). A Course in Time Series Analysis. New York: Wiley-Interscience. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=535798