Course detail

Signals and Systems Analysis

FEKT-BASSAcad. year: 2016/2017

One-dimensional (1D) and two-dimensional (2D) signals and systems with continuous time and their mathematical models. One-dimensional (1D) and two-dimensional (2D) signals and discrete-time systems and their mathematical models. Examples of real signals. Representation in the time and frequency domains, Fourier representation of signals, mutual respect. Definition and method of calculation of FFT. Transformation Z, unilateral and bilateral transform, direct and inverse transform and its applications to differential equations. Random signals and their description, probability theory, the definition of power spectral density. Communication signals and communication systems definition analog and digital. Analog and digital modulations in communication technology. Methods of implementation of communication systems in microprocessors and digital signal processors. The issue is illustrated by the examples of specific signals and systems, and these examples are presented in Matlab. In the laboratory measurements and run a simulation of signals and systems on spectrum analyzer with FFT and using appropriate measurement products for specific measuring instruments.

Language of instruction


Number of ECTS credits


Mode of study

Not applicable.

Learning outcomes of the course unit

The students become thoroughly familiar with the continuous-time signals and systems, discrete-time signals and systems, and their representation in the frequency domain, continuous-time and discrete-time random signals, sampling and signal recovery, analog and, in particular, digital communication systems. Students will learn how to use Matlab for signal processing applications in different practical areas. Students will have an idea of how to implement signals and systems in microprocessors and digital signal processors.


The subject knowledge on the secondary school level especially of maths and physics is required. Furthermore, a general knowledge of programming and computer skills are important. Emphasis is placed on the knowledge of complex numbers and their application.


Not applicable.

Planned learning activities and teaching methods

The subject includes both lectures, and numerical and laboratory exercises. The teaching methods are chosen according to type of education. The lectures combine modern audiovisual Power Point presentation with more detailed derivation of fundamental methods and algorithms on the board. Applets are also used to increase the clarity of instruction. All lectures and examples are given in e-learning. In numeric exercises, the students learn the basic mathematical methods for the analysis of signals and systems. In laboratory exercises method in Matlab are implemented for direct verification of signal and system properties.

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 lab work and numerical exercise tests they can get 40 out of 100 points. The remaining 60 points can be obtained by successfully passing the final written examination.

Course curriculum

1. Signals and systems and their mathematical models
Real signals and their mathematical continuous-time models. Basic signal operations (time scaling, flipping, time shifting translation, time shifting translation and flipping, convolution, correlation). Signal classification, unit impulse, unit step, harmonic signal. Real systems with continuous- and discrete-time. Dynamic system, its input and output, status. Linear time-invariant system. Impulse response. Response of LTI system using convolution, superposition.
2. Periodic signals and their spectrum
Function substitution by functional series. Periodic continuous-time signal, harmonic signal and its representation by phasors. Periodic and harmonic discrete-time signals. Fourier series, spectrum of periodic rectangular pulses, spectrum theorems.
3. Fourier representation of aperiodic continuous-time signals
Definition of the Fourier transform of aperiodic continuous-time signals. Spectra of selected signals. Spectrum theorems. Definition of the inverse Fourier transform. The inverse Fourier transform of rectangular spectral impulse. Relationship between the Fourier series and the Fourier transform.
4. Continuous-time systems
The characteristics of a linear time-invariant (non-parametric) system (frequency response, hodograf). System transfer function, zero-pole plot. Ideal transfer circuit. Frequency filters. Non-linear systems. Superheterodyne.
5. Sampling of continuous-time signals
Ideal sampling of continuous-time signal and its reconstruction. Sampling theorem. Amplitude quantization. A/D and D/A conversions. Aliasing. Sampling of bandpass signals.
6. Discrete-time signals
Discrete time axis. Basic discrete signals. Signal theorems. Discrete linear, periodic and circular convolutions. Using FFT for convolution calculation.
7. Fourier transform of discrete-time signals.
The discrete Fourier series and the discrete Fourier transform. The fast Fourier transform (FFT). Decimation-in-Time (DIT) and Decimation-in-Frequency (DIF) algorithms, FFT algorithm properties.
8. Z transform and its properties
Definition of the Z transform and its properties. The inverse Z transform and its calculation. The relationship between the Z transform and the discrete Fourier transform.
9. Modulation signals in base-band and transition-band
Communication system and its properties, modulation and transmission rates, spectrum of communication channel. Amplitude, frequency, and phase analog modulations and their spectra. Digital modulations.
10. Stochastic variables and processes and their properties
Continuous and discrete time variables. Definition of stochastic processes with continuous- and discrete-time and their representations. Cumulative distribution function, probability density function. Moments (mean, variance, standard deviation, etc.). Stationarity and ergodicity.
11. Power spectral density and its calculation
Power spectral density of continuous- and discrete-time stochastic processes. Periodogram, using FFT for its calculation. White noise. Processing of stochastic signal by linear system. Non-parametric and parametric models.
12. Discrete-time systems
Linear time-invariant discrete system, impulse response. System transfer function, frequency response, zero-pole plot. Systems of the type of IIR and FIR. Connection of LTI systems. Series, parallel and feedback connections of partial sections.
13. Realization of LTI discrete system
Design of LTI discrete system based on analog prototype. Structures of realization. Mason’s gain rule. Implementation of LTI system on microprocessor. Calculation of frequency response based on time responses.

Work placements

Not applicable.


The aim of the course is to acquaint students with one-dimensional (1D) and two-dimensional (2D) signals and systems with continuous-time signals and discrete-time systems with pulse and digital signals and systems. It is also necessary to introduce the concept of spectrum 1D and 2D signals and emphasize its difference from the frequency characteristics of 1D and 2D system. Consequently, the aim is to provide students with basic information about random signals and their impact on systems bring analog and digital modulation and define description of the characteristics of communication systems.

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

SMÉKAL, Z.: Analýza signálů a soustav-BASS. FEKT, 2012.
SMÉKAL, Z.: Deterministické a náhodné signály pro integrovanou výuku VUT a VŠB-TUO. Elektronické texty, VUT Brno, 2013. ISBN 978-80-2014-4826-1
SMÉKAL, Z.: Signals and Syztems Analysis for joint teaching programme of BUT and VSB-TUO (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme AUDIO-J Bachelor's

    branch J-AUD , 2. year of study, winter semester, compulsory

  • Programme EEKR-B Bachelor's

    branch B-MET , 2. year of study, winter semester, compulsory
    branch B-TLI , 2. year of study, winter semester, compulsory

  • Programme EEKR-CZV lifelong learning

    branch ET-CZV , 1. year of study, winter semester, compulsory

Type of course unit



39 hours, optionally

Teacher / Lecturer


Signals. Examples, definition. Harmonic signal.
Periodic signals. Fourier series expansion. Properties.
Fourier Transform. Properties of the Fourier transform.
Linear time-invariant systems and their description. Filtering.
Continuous-time random signals.
Sampling and signal recovery. Digital signals.
Discrete-time signals. Discrete Fourier series.
Discrete Fourier transform. FFT.
Random sequences. Pseudorandom sequences. PSD.
Discrete-time systems. Z transform. Examples.
Communication systems. Baseband signals.
Amplitude modulation. Frequency modulation. ASK, FSK, PSK.
I-Q modulations. Multiplexing and multiple-access techniques.

Fundamentals seminar

13 hours, optionally

Teacher / Lecturer


Description of signals.
Fourier series expansion. Examples.
Properties of the Fourier transform.
Random signals. Sampling. Quantisation noise.
Discrete Fourier transform.
Amplitude and frequency modulation, keying.

Laboratory exercise

13 hours, compulsory

Teacher / Lecturer


Spectral analysis of the periodic signals.
Amplitude and frequency modulation. Analysis of the random signal.
Sampling, aliasing.
Digital signal processing of the own speech.
The frequency response of the discrete-time system.