Course detail

# Digital Signal Processing

FEKT-LCSIAcad. year: 2017/2018

Definition and classification of 1D and 2D discrete signals and systems. Signal and system examples. Spectral analysis using FFT. Spectrograms and moving spectra. The Hilbert transform. Representation of bandpass signals. Decimation and interpolation. Transversal and polyphase filters. Filter banks with perfect reconstruction. Quadrature mirror filters (QMF). The wavelet transform. Signal analysis with multiple resolution. Stochastic variables and processes, mathematical statistics. Power spectral density (PSD) and its estimation. Non-parametric methods for PSD calculation. Linear prediction analysis. Parametric methods for PSD calculation. Complex and real cepstra. In computer exercises students verify digital signal processing method in the Matlab environment in real time.

Learning outcomes of the course unit

The student will determine the spectral properties of 1D and 2D deterministic and stochastic signals using different base functions (Fourier analysis, wavelets) for multiple resolution. They will learn how to use multirate filter banks (for example, in methods of audio- and video-signal compression, ADSL transmission, etc.). They will also become familiar with linear prediction and cepstral analysis. They will program in the Matlab environment and use it for signal processing and system design.

Prerequisites

The subject knowledge on the Bachelor´s degree level with emphasis on digital signal processing is required. Furthermore, the basic ability to program in the Matlab environment is necessary.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

SMÉKAL, Z.: Číslicové zpracování signálů, FEKT, VUT v Brně

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in article 7 of the BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Lab exercises are mandatory for successfully passing this course and students have to obtain the required credits. For computer lab tests they can get 30 out of 100 points. The remaining of 70 points can be obtained by successfully passing the final written examination.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

1. Description of discrete signals and their division. Energy and power signals. Periodic discrete signals. Basic 1D and 2D signals (unit impulse, unit step, real and complex harmonic signals). The discrete Fourier series (DFS) and the discrete Fourier transform (DFT) spectra. The fast Fourier transform (FFT) algorithm. The Z transform.
2. External and internal (state-space) representations. Bounded Input Bounded Output (BIBO) stability, causality. Linear time-invariant 1D discrete system. Connection of partial sections. FIR and IIR systems. Frequency responses, fast convolution. Overlap-save and overlap-add methods. Linear shift-invariant 2D discrete system. The Fourier transform of 2D discrete signals, 2D frequency response.
3. Matrix representation of system equations and their solution. Semi-symbolic computer analysis. Signal flow graphs and Mason’s gain rule. Check of discrete system causality.
4. Definition of a periodic even sequence using an aperiodic sequence, definition of discrete cosine transform from DCT I to DCTIV. Relationship between DCT II and DFT. Definition of the discrete sine transform. Undersampling (decimation) and oversampling (interpolation) of discrete signal in an integer ratio. Description of the time and frequency domains. The transformation of sampling frequency in a rational number ratio. Optimization of the number of multiplier and memory registers of anti-aliasing low pass filter.
5. Zero-pole plot in the z domain, Minimal, maximal and mixed phases. All-pass filter, inverse discrete system. Sampling of bandpas signals. Real signal, analytical signal and complex lowpass signal. The Hilbert transform for continuous-time signals. Quadrature modulator and demodulator. The Hilbert transformer for discrete signals.
6. Analyzing part and synthesizing part of digital filter bank. Calculation of DFT spectrum of discrete signal using uniform-DFT filter banks. Sub-band coding. Quadrature mirror filters (QMF). Perfect signal reconstruction. Transmultiplexers.
7. The Gabor and the short-time Fourier transforms. Time-frequency resolution, The Heisenberg uncertainty principle. Orthogonal systems and their application to spectral analysis. Wavelets and their definition.
8. The continuous-time wavelet transform (CWT), the discrete wavelet transform (DWT). The discrete-time wavelet transform (DTWT). Relationship between DTWT and QMF digital filter banks.
9. Cumulative distribution function and probability density function, general and central moments. Stationary and ergodic continuous- and discrete-time stochastic processes. Estimates, consistent estimate. Random selection from probability distribution, statistics, statistical hypothesis testing, parametric and non-parametric tests, goodness of fit tests.
10. Forward and backward linear prediction. Calculation of linear prediction coefficients. Lattice structure of autoregressive (AR) and autoregressive moving average (ARMA) types and their application. Using linear predictive analysis for speech signal compression.
11. Definition of power spectral density and its properties. Definition of periodogram and its calculation. The Bartlett method of averaging periodograms. The Welch method of averaging modifmodified periodograms. The Blackman and Tukey method of smoothing the periodogram. Performance characteristics of nonparametric power spectral density estimators.
12. AR, MA or ARMA type stochastic processes. Model definition for power spectral density calculation. Relationship between autocorrelation coefficients and model parameters. The Yule-Walker and the Burg methods for AR type model. Selection of the order of type AR model.
13. Complex and real cepstra. Generalized superposition. Homomorphic filtering, definition and its application. Approximation of exponential function by continued fraction expansion.

Aims

The aim of the course is to present modern methods of digital signal processing that are based primarily on analyses of 1D and 2D discrete-time and digital signals and systems. Furthermore, the students will learn about parametric and non-parametric spectral analysis of stochastic signals and about mathematical statistics. They will use multirate digital filter banks for signal processing in practise.

Specification of controlled education, way of implementation and compensation for absences

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year.

Classification of course in study plans

• Programme EEKR-ML Master's

branch ML-TIT , 1. year of study, summer semester, 6 credits, compulsory

• Programme EEKR-ML1 Master's

branch ML1-TIT , 1. year of study, summer semester, 6 credits, compulsory

• Programme IBEP-VY Master's

branch VY-IBP , 1. year of study, summer semester, 6 credits, optional specialized
branch VY-IBP , 2. year of study, summer semester, 6 credits, optional specialized

• Programme EEKR-CZV lifelong learning

branch ET-CZV , 1. year of study, summer semester, 6 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Laboratory exercise

39 hours, compulsory

Teacher / Lecturer