The Digital signal processing I course introduces the fundamental techniques of digital signal processing (discrete signals and discrete linear systems representation and characterization) and gives the necessary background to study the advanced processing techniques proposed in the Digital signal processing II subject. It is a subject that combines theory and practice and gives the basic background understanding the digital linear processing of signals, basically represented with digital filters based on linear constant-coefficient difference equations. First, characterization of signals and systems basic theory is presented, specifically the analysis of linear time invariant systems in time domain (system´s properties, impulse response, discrete convolution), its frequency characterization (Fourier Transform of Discrete Signals) and the relation between discrete and continuous time signals and systems. After studying the resampling techniques of discrete signals, the frequency analysis tool for time limited discrete signals is analyzed in detail (Discrete Fourier Transform - DFT), its properties and theorems and a computationally efficient algorithm to perform its computation is presented (Fast Fourier Transform - FFT). The techniques to perform real-time digital filtering of discrete signals in the frequency domain through the use of DFT are studied (Overlap & Add, Overlap & Save). Finally, the Z-Transform of discrete signals is studied and is seen as a powerful tool for the analysis and the design of digital filters (FIR and IIR).
Type Subject
Previous Knowledge

Sampling theorem for bandlimited signals and time and frequency characterization for analog signals and systems.


Digital signal and image processing students learn the following knowledge and develop the following skills:

1. Acquire the basic knowledge from the study and characterization of discrete signals and systems, specially the study of linear and time invariant systems, to allow the implementation, analysis and design of digital signal and image processing systems.
2. Identify, formulate and solve digital signal and image processing problems in a multidisciplinary environment, individually or as a member of a team.
3. Analyze, design and make use of systems, procedures and algorithms in order to achieve the proposed goals in a specific digital signal processing problem, making use of open source simulation, analysis and application development tools in this area (SciLab), and to analyze and understand the given results.
4. To use new e-learning techniques and tools (virtual campus, study guide, sharing documents, forums)


1 Characterization of discrete linear time invariant systems. Impulse Response
1.1 Introduction
1.2 Linear and time invariant systems (S.L.I.T.)
1.3 Impulse response and linear discrete convolution
1.4 Linear constant-coefficient difference equations

2 Fourier analysis of discrete time signals and systems
2.1 Introduction: TFSD
2.2 Frequency representation of sequences by means of TFSD
2.3 TFSD properties
2.4 Digital processing of bandlimited analog signals
2.5 Discrete domain sampling frequency modification (decimation and interpolation)

3 The Discrete Fourier Transform (D.F.T.)
3.1 Periodic sequences representation: Discrete Fourier Series (D.F.S.)
3.2 Discrete Fourier Series properties
3.3 Fourier transform of periodic sequences
3.4 Sampling of the TFSD
3.5 Frequency representation of finite length sequences. The Discrete Fourier Transform (D.F.T.)
3.6 D.F.T properties
3.7 Linear convolution using the DFT
3.8 The Fast Fourier Transform (F.F.T.)

4 The Z transform
4.1 Introduction
4.2 Definition of the Z transform
4.3 Region of convergence properties (ROC)
4.4 Z transform properties
4.5 The inverse Z transform
4.6 Linear and time invariant study through the Z transform


Two methodologies are applied depending on the profile selected by the student at the time of registering. In the attended methodology theory is given through the teacher´s lectures, providing the basic knowledge to enable the student to develop the complete program of activities. In the semi-attended methodology, theory is developed by the student with the aid of an electronic study guide. In this case the student has a more active paper. In the study guide basic contents and bibliographic references are pointed out to allow the student to progress in his own learning.

Aside from the theoretical background, both learning methodologies share the following aspects: practice demonstrations, problems classes, practices classes, and the all the individual work the student has to do as homework (advanced problems and practice demonstrations)

Teacher´s lectures and work with the study guide are complemented with problems classes and practical demonstrations. Therefore, theoretical comprehension is improved through visual examples with the simulation software SciLab and also with the discussion of key concepts, allowing students to develop some practical skills, the ability to solve problems, the ability to be more creative in order to face new situations or to work as a member of a team.

During the course some theoretical and practical problems are given as homework. Students have a specific software to solve some of these problems, where they can evaluate and share their results with other students. Auxiliary practices teachers are available for asking questions about these practices. Students receive continuous support through meetings with the teacher, where they get some advice concerning their whole learning process and achievement. Some self-evaluation tests are also available in the electronic study guide in order to evaluate one´s degree of comprehension after every chapter.

Finally, both students and teachers plan virtual meetings in order to promote the discussion of certain key concepts during the course. In these meetings the teacher can interact with a reduced number of interested students, sharing specific application examples in SciLab or to solve some questions by sharing electronic documents.


The student evaluation will be completed with:
A. Exams
D. Homework
G. Computer assignments
K. Laboratory reports
M. Participation in the virtual campus

The subject mark will have a possible contribution of a continuous assessment mark where the student´s effort during the whole course will be reflected. The final mark will have a contribution of the 40% of this continuous assessment mark, if the normal mark is at least 3.5 and this contribution is positive with respect to the normal mark solely.

The normal mark will have a theoretical part (6075) and a practical part (225%). The mark for the theory part will be given from the exam problems (75%). The mark for the practices part will be given through the exam practice problems (15%), and the result of practice problems in a partial control (10%).

The continuous assessment mark will be computed with the results of different activities: homework assignments and participation in discussion forums.

Evaluation Criteria

Objective 1:
-Students should prove to have a general and basic knowledge of digital signal processing (time and frequency characterization of discrete time signals and systems), and show an ability to connect the different conceptual blocks [A, D, G].

Objective 2:
-Student should show abilities for analysis and synthesis in solving exercises: to posing different ways of reaching the desired objectives and choosing the simpler, faster and better way of achieving the goals planned with the given restrictions [A, D].

Objective 3:
-Students should prove to have elementary computing skills in practical development software (SciLab) and the given modules and functions during the posed practices problems [G].
-Student should show ability to work as a member of a interdisciplinary team and ability to put the acquired theoretical knowledge into practice.[G,K].

Objective 4:
-Student should be capable of working in a e-learning environment with several documents and knowledge sources (problems, study guide, specific bibliography, transparencies, discussion forums) and should show an ability for self-learning and autonomous work, ability to adapt to new situations, ability to communicate with non expert persons and ability for information management [M].

Basic Bibliography

Oppenheim, Alan V; Schafer, Ronald W., Discrete-Time Signal Processing, Prentice-Hall
New Jersey, 1999

Morán, José Antonio; Socoró, Joan Claudi; Cobo, Germán; Sevillano, Xavier; Guia d´estudi de Processament digital del senyal I, Enginyeria La Salle, 2011

Socoró, Joan Claudi; Cobo, Germán; Morán, José Antonio; Calzada, Àngel; Monzó, Carlos; Sevillano, Xavier; Problemes de Processament digital del senyal I, Enginyeria La Salle, 2011

Trilla, Alexandre; Sevillano, Xavier; Pràctiques de Processament digital del senyal I, Enginyeria La Salle, 2011

Additional Material

Proakis, John G.; Manolakis, Dimitris G., Digital Signal Processing, Macmillan Publishing Company, New York, 1992