# Digital signal processing (course master's degree program, Moscow State University )

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## Содержание

• Master degree program, 1 year, autumn, department MMP ( ММП),
• Lectures — Thursday, 4:20 pm, room 605
• Control — exam, 14 June, 9:00, room 637
• InstructorKrasotkina O.
• Office hours — Tuesday, Thursday, 11:00 am : 19:00 pm, or by appointment, room 532

## Description:

The main goal of this course is to expose students to the mathematical theory of signal analysis, and at the same time, to some of its many applications in the finance, geophysic, image understanding, bioinformatics, and etc. Course begins with a discussion of the analysis and representation of discrete-time signal systems, including discrete-time convolution, difference equations, the z-transform, and the discrete-time Fourier transform. The signal is meant an experimentally acquired or mathematically simulated function of spatial coordinates and time which is to be analyzed with the purpose of studying behavior of the respective distributed dynamical system.

It was even Leonard Euler who marked that  everything what happens in the world  bears the sense of  some minimum or maximum. The second part of course consider a wide class of signal analysis problems, which allow for treating them in unified terms of respective standard mathematical optimization problem for which there exist or can be created effective methods of solving. A signal is considered as a set of experimentally acquired values of a number of variables each of which is associated with respective node of an undirected adjacency graph that presents the fixed structure of the data set. The proposed theoretical approach is illustrated with its applications to the problems of segmentation, smoothing, fine texture analysis, matching of visual images, multi-alignment of long molecular sequences. .

## Prerequisite:

Some prerequisites include linear algebra, functional analysis, optimization and probability theory. Practical work would typically involve a fair amount of scientific programming in Python or Matlab (Scilab).

## Syllabus:

### Introduction

Applied signal analysis problem . What is a signal? Signal analysis problems: examples .

### Part 1 Signal representation. Classical signal analysis

• Theme 1. Signal representation. Signal Spaces.
• Theme 2. Dynamic Representation of Signals (Time Domain)
• Theme 3. Spectral representation (Frequency domain). History. Generalized Fourier series. Trigonometric basis. Examples. Gib_bs effect. Fourier Transform. Fourier Transform Properties
• Theme 4. From Physical to Digital Signal. Measuring Noise. Sampling. Aliasing. Quantization
• Theme 5. Spectral Analysis The Discrete Fourier Transform. The inverse DFT. Power Leakage. Tradeoff Between Time and Frequency Resolution
• Theme 6 Filtering and Feedforward Filters. Delaying. Z-plane. Phase Response. Digital Filters.
• Theme 7 Z-Transform. Examples of z-trsnsform. Convolution. Properties of z-transform. Impulse Response and the Transfer Function.
• Theme 8 Feedback filters. Resonance. Bandwidth. Mixing Feedback and Feedforward filters. Implementation.
• Theme 9. Compression. Entropy Compression. Source Compression.
• Theme 10 Audio and Video compression. Audio Coding Standards. Image coding Standards. Video Coding Standards.

### Part 2 Statistical signal analysis

• Theme 1. The beyesian framework for the estimation of signal models.
• Theme 2. Hidden Markov Model of the Signal.
• Theme 3 Hidden Markov Model Estimation for Additive Loss Function
• Theme 4 Hidden Markov Model Estimation for Singular Loss Function
• Theme 5 Structural Parameters Estimation.

## Selected applications:

• A mathematical and algorithmic framework for dynamic returns-based style analysis of investment portfolios
• A new method of seismic explorations for oil and gas in crystalline basement rocks
• An Automatic Procedure for Matching Ultrasonic Railway Defectograms

## Practical work:

• Sounds and signals. Harmonics
• Non-periodic signals. Noise
• Autocorrelation
• Discrete Fourier Transform
• Filtering
• Differentiation and Integration
• Modulation and sampling

## Students oral presentation:

Fast Fourier Transform

Filtering and Feedforward filters

Z-Tranform and Convolution

Feedback filters

Compression

Audio- Video Codecs

Statistical Signal processing

Adaptive Filtering

Inverse problem and Signal reconstruction

Time Frequency and Multirate signal processing

Speech Processing

Image and Video Processing

Nonlinear and Fractal Signal Processing