PDS 11/12 - 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.


Cláudia Soares. Please use e-mail (csoares 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  Random signals and parameter estimation. Estimation in signal processing. 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  Least squares (LS). The LS approach. Linear LS. Geometrical interpretations. Weighted LS. Order-recursive LS.

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 EA5. 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, chapters 2, 3, 4, 6, 7, 8, and 10, 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, 8, 10, 11, 12, particularly the sections pointed out in the schedule below.

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. Problems not included in the textbooks above, sample slides, and previous tests and exams are linked below.

·         [A] "Additional Problems", 2012.

·         [R] "Random signals", 2012.

·         Test #1 (2010/2011)   Example of solution

·         Test #2 and Exam (2010/2011)   Example of solution

·         Exam (2010/2011)



Students should team up (teams of two) at the beginning of the semester and register for one of three available weekly lab sessions (Tuesdays, 12:30-14:00 and 14:00-15:30, and Wednesdays, 14:00-15:30). There will be 6 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 grades of the lab assignments (each assignment grade is based on both the report and lab class participation).


Schedule (tentative) and summary

Lecture, date





#1, Feb 14

Course presentation

Registration (Feb 22)

#2, Feb 16

Discrete signals and systems

[OS] ch 2 (2.1-2.5)

[OS] 2.4,2.24,2.39

#3, Feb 23

Discrete signals and systems

[OS] ch 2 (2.6-2.9)

[OS] 2.6,

#4, Feb 28


[OS] ch4 (4.1-4.4)

[OS] 4.2,4.8,4.25

Lab #1 - Sampling

#5, Mar 1


[OS] ch 3 (3.1,3.2)

[OS] 3.1,3.4

#6, Mar 6


[OS] ch 3 (3.3,3.4)

[OS] 3.6,3.8,3.36

#7, Mar 8


[OS] ch 8 (8.1-8.4)

[OS] 8.2,8.4

#8, Mar 13


[OS] ch 8 (8.5,8.6)

[OS] 8.5

Lab #2 - Filtering

#9, Mar 15

DFT - consolidation

[OS] ch 8 (8.5,8.6)

[OS] 8.7,8.32,8.57

#10, Mar 20


[OS] ch 8 (8.7)

[OS] 8.27,8.36,8.39

Lab #3 - Analysis

#11, Mar 22

Signal analysis using DFT

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

[OS] 10.1,10.2,10.3

#12, Mar 27

Digital filters

[OS] ch 6 (6.0-6.4),  ch 7 (7.0-7.2)

[OS] 6.1,6.23,7.23,7.32

#13, Mar 29

Random signals and parameter estimation

[K] ch 1 (1.1), [R]

[A] 1,2

Apr 2, 20:00, Rooms E3 and E4, Test #1 (material of lectures #2- #12) Grades

Lab #4 - Random Signals 

#14, Apr 3

Random signals and parameter estimation

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

[K] 1.2,1.4

#15, Apr 12

MVU estimation

[K] ch 2

[K] 2.1,2.4,2.10

#16, Apr 17

CR lower bound

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

[A] 4, [K] 3.3

#17, Apr 19

CR lower bound

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

[A] 6,7, [K] 3.9

#18, Apr 24

ML estimators

[K] ch 7 (7.4,7.5)

[A] 10,11, [K] 7.2

Lab #5 - Estimation

#19, Apr 26

ML estimators

[K] ch 7 (7.6,7.8)

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

#20, May 3

ML estimators / Least squares

[K] ch 8 (8.3,8.4)

[A] 17,19,20

#21, May 8

ML estimators / Least squares

[K] ch 7 (7.7,7.10)

[K] 7.13,7.24, [A] 13,15,16

Lab #6 - ML and LS

#22, May 10

Least squares

[K] ch 8 (8.5,8.6)

[K] 8.1,8.5

#23, May 15

Bayesian estimation

[K] ch 10 (10.3-10.6)

[K] 10.11,10.12,10.14

#24, May 17

Bayesian estimation

[K] ch 11 (11.3-11.5)

[K] 11.1,11.4,11.16, [A] 21,23,24

#25, May 22


all the above

all the above

#26, May 24

Jun 5, 8:00, Rooms V1.08 and V1.09, Test #2 (material of lectures #13 - #24) and Exam #1  Grades

Jun 25, 8:00, Rooms E3 and E4, Exam #2  Grades

Jul 17, 15:00, Room F3, Special Exam ("Época Especial")  Grades