PDS 10/11 - Processamento Digital de Sinais - Digital Signal Processing

Pedro M. Q. Aguiar
. Office 7.24, north tower, 7th floor. Office hours: door is always open but please use e-mail (aguiar at isr dot ist dot utl dot pt) to schedule appointments.


Course webpage
(this page)



The course deals with computer processing of discrete-time signals. It is tailored to Eng&Sc MSc students (but also attended by a few PhD candidates), in particular of Electrical and Computer Eng., Biomedical Eng., Physics, and Informatics. Students will have hands-on experience, through laboratory assignments. Course topics:

& Discrete-time signal representations (transforms) and linear systems

ü  Discrete-time signals and systems. Discrete-time signals: sequences. Discrete-time systems. Linear time-invariant (LTI) systems. Properties of LTI systems. Linear constant-coefficient difference equations. Frequency-domain representation of discrete-time signals and systems. Fourier transform. Properties of the Fourier transform.

ü  Sampling of continuous-time signals. Periodic sampling. Frequency-domain representation of sampling. Reconstruction. Discrete-time processing of continuous-time signals.

ü  The z-transform. Definition. Properties of the region of convergence. The inverse z-transform. Properties of the z-transform.

ü  The discrete Fourier transform (DFT). Representation of periodic sequences: the discrete Fourier series (DFS). Properties of the DFS. The Fourier transform of periodic signals. Sampling the Fourier transform. Fourier representation of finite-duration sequences: the DFT. Properties of the DFT. Linear filtering using the DFT.

ü  Fourier analysis of signals using the DFT. DFT analysis of sinusoidal signals. The time-dependent Fourier transform. Fourier analysis of nonstationary signals.

ü  Digital filters. Block diagram representation. Direct form I and II. Canonic forms. Flow graph representation. Transposed forms. Digital filter design techniques. Design of IIR filters from continuous-time filters. Design of FIR filters using windows.

& Statistical signal processing

ü  Extracting information from signals and LS. Estimation in signal processing. The Least Squares (LS) approach. Linear LS. Geometrical interpretations. Weighted LS. Order-recursive LS.

ü  Random signals and parameter estimation. Random variables. Gaussian distribution. Random vectors. Multivariate Gaussian. Conditional probability and independency. Random processes. Gaussian random processes. White noise. Time series models. The estimation problem. Assessing estimator performance.

ü  Minimum variance unbiased (MVU) estimation. Unbiased estimators. Minimum variance criterion. Existence of MVU estimator. Finding the MVU estimator.

ü  Cramer-Rao (CR) lower bound. Estimator accuracy considerations. CR lower bound (CRLB) and its derivation. CRLB for signals in white Gaussian noise. CRLB for transformed parameters. CRLB bound for parameter vectors. The Fisher information matrix. Signal processing examples.

ü  Maximum likelihood (ML) estimators. Definition of the ML estimator (MLE). Finding the MLE. MLE for signals in white Gaussian noise and relation to LS. Properties of the MLE. MLE for transformed parameters. MLE for parameter vectors. Numerical determination of MLE. Signal processing examples.

ü  Bayesian estimation. Prior knowledge and estimation. Choosing a prior. The Gaussian case. Bayesian linear model. Risk functions. Minimum mean square error (MMSE) estimators. Maximum a posteriori (MAP) estimators. Linear MMSE estimation. Geometrical interpretations. Signal processing examples.



Class meets twice a week: Tuesdays, 9:30-11:00, room EA3, and Thursdays, 11:00-12:30, room EA2. Clearing doubts meeting: Tuesdays, 11:00-12:30 and 15:30-17:00 (please send e-mail in advance).



& [OS] "Discrete-Time Signal Processing", A. Oppenheim and R. Schafer, Prentice Hall, 2nd Edition, 1999, (1st Edition, 1989), chapters 2, 3, 4, 6, 7, 8, and 10 (11), particularly the sections pointed out in the schedule below.

& [K] "Fundamentals of Statistical Signal Processing - Estimation Theory", S. Kay, Prentice Hall, 1993, chapters A1.2, 1, 2, 3, 7, 10, 11, 12, particularly the sections pointed out in the schedule below.

Whenever possible, before each lecture, students should read the corresponding book sections. Good practice is at least read them after the lecture. A very important complement is to solve the problems at the end of the corresponding book chapter (some of them will be solved during the lectures). Good practice is at least solve the problems pointed out in the schedule below.



Students should team up (teams of two) at the beginning of the semester and register for a weekly lab session. There will be an introductory lab session, followed by 8 lab assignments. Each team of two students should deliver a lab report corresponding to each assignment, at the end of the corresponding lab session. Lab room: LSDC1 (north tower, 5th floor).



Final grade is 30% on labs and 70% on exam (or two tests: midterm and final), with the requirement for approval of the minimum grade of 9.5 for both components. The lab grade is the average of the 7 best grades of the lab assignments (each assignment grade is based on both the report and lab class participation).


Schedule (tentative) and summary

Lecture, date




Labs (week)

#1, Feb 15

Course presentation

#2, Feb 17

Discrete signals and systems

[OS] ch 2 (2.1-2.5)

[OS] 2.4,2.24,2.39 (2.3,2.7,2.14,2.16)

#3, Feb 22

Discrete signals and systems

[OS] ch 2 (2.6-2.9)

[OS] 2.6, (2.18,2.24,2.37)

#4, Feb 24


[OS] ch4 (4.1-4.4) (3.1-3.4)

[OS] 4.2,4.8,4.25 (3.1,3.4,3.11)

Feb 28

Registration MEBiom

Mar 1

Registration MEEC/MEIC

#5, Mar 1


[OS] ch 3 (3.1,3.2) (4.1,4.2)

[OS] 3.1,3.4 (4.1,4.6)

#6, Mar 3


[OS] ch 3 (3.3,3.4) (4.3-4.7)

[OS] 3.6,3.8,3.36 (4.8,4.24,4.26)

#7, Mar 10


[OS] ch 8 (8.1-8.4) (8.1-8.5)

[OS] 8.2,8.4 (8.2,8.9)

#8, Mar 15


[OS] ch 8 (8.5,8.6) (8.6-8.8)

[OS] 8.5 (8.11,8.13)

Lab #0  (reportEx)

#9, Mar 17

DFT - consolidation

[OS] ch 8 (8.5,8.6) (8.6-8.8)

[OS] 8.7,8.32,8.57 (8.17,8.18,8.36)

#10, Mar 22


[OS] ch 8 (8.7) (8.9)

[OS] 8.27,8.36,8.39 (8.34,8.39,8.40)

Lab #1  romanzasmall

#11, Mar 24

Signal analysis using DFT

[OS] ch 10 (10.1-10.3,10.5) (11.1-11.3,11.5)

[OS] 10.1,10.2,10.3 (11.1,11.2,11.7)

#12, Mar 29

Digital filters

[OS] ch 6 (6.0-6.4),  ch 7 (7.0-7.2) (7.0,7.1,7.4)

[OS] 6.1,6.23,7.23,7.32 (6.1,6.7,7.7,7.23)

Lab #2  touchtone

#13, Mar 31

Extracting info from signals, LS

[K] ch 1 (1.1), ch 8 (8.3,8.4)

pLS (1,5,6)

#14, Apr 5

Extracting info from signals, LS

[K] ch 8 (8.5,8.6)

[K] 8.1,8.5

Lab #3  fugee

#15, Apr 7

Random signals

Random variables and signals

pPTCRB (PT1,2)

Apr 8

Test #1 (material of lectures #2- #12) (solEx)  (grades)

#16, Apr 12

Seminar - Brain signal analysis and computational models, by Prof. Lopes da Silva, University of Amsterdam

Lab #4  eogSig

#17, Apr 14

Random signals

[K] ch A1.2, ch 1 (1.2,1.3)

[K] 1.2,1.4

#18, Apr 19

MVU estimation

[K] ch 2

[K] 2.1,2.4,2.10

Lab #5  speech

#19, Apr 28

CR lower bound

[K] ch 3 (3.3,3.4,3.A,3.5)

pPTCRB (CRB1), [K] 3.3

#20, May 3

CR lower bound

[K] ch 3 (3.6,3.7,3.11)

pPTCRB (CRB4,5), [K] 3.9

Lab #6  ecg

#21, May 5

ML estimators

[K] ch 7 (7.4,7.5)

pML (1,3), [K] 7.2

#22, May 10

ML estimators

[K] ch 7 (7.6,7.8)

[K] 7.8,7.9,7.10,7.20,7.21

Lab #7  data

#23, May 12

ML estimators

[K] ch 7 (7.7,7.10)

[K] 7.13,7.24, pML (4,5,7)

#24, May 17

Bayesian estimation

[K] ch 10 (10.3-10.6)

[K] 10.11,10.12,10.14, pB (3,4)

Lab #8

#25, May 19

Bayesian estimation

[K] ch 11 (11.3-11.5)

[K] 11.1,11.4,11.16, pB (1,5,6,7)

#26, May 24


all the above

all the above

#27, May 26

Jun 3

Exam #1 and Test #2 (material of lectures #13 - #25) (solEx)  (grades)

Jun 25

Exam #2  (final grades)