Nov 09 2013
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Education

DFT x FFT

The Fast Fourier Transform (FFT) is an essential algorithm for modern DSP applications. Basically, FFT is a fast implementation of the DFT achieved by exploiting the properties of the periodic complex exponential function e^{jk\frac{2\pi n}{N}}. A nice lecture on FFT is given in a video from Prof. Barry Van Veen.

Oct 29 2013
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Education

Edsger Wybe Dijkstra view on education

Electrical, science and computer engineers have a great debt to Prof. E. W. Dijkstra.  He solved, among other important problems, the single source shortest-path problem, essential to address  communications networks, airline flight plans, and many other practical applications. His letter on teaching computer science is inspiring and applies to other fields: http://www.cs.utexas.edu/users/EWD/ewd10xx/EWD1036.PDF. As for his biography, one may learn from: http://en.wikipedia.org/wiki/Edsger_W._Dijkstra. I personally liked two of his quotes:  i) “A formula is worth a thousand pictures.”, ii) “if 10 years from now, when you are doing something quick and dirty, you
suddenly visualize that I am looking over your shoulders and say to yourself,
Dijkstra would not have liked this, well that would be enough immortality for
me.”

Oct 21 2013
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Education

Articles and Citations at Google Scholar

My research includes a variety of applications of Digital Signal Processing theory. For those interested in reading some of my scientific papers on DSP, please check at Google Scholar.

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Oct 16 2013
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Education

Signal Processing and Automatic Speech Recognition

Noticia_UFSC

Signal Processing and Pattern Recognition fields are combined to address a very interesting problem: Automatic Speech Recognition (ASR). An application of our research in ASR for people with hearing impairments: http://noticias.ufsc.br/2013/10/projeto-busca-viabilizar-a-comunicacao-de-deficientes-auditivos-pelo-telefone/

Texto

Sep 11 2013
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Education Link

8th International Symposium on Image and Signal Processing and Analysis (ISPA 2013)

8th International Symposium on Image and Signal Processing and Analysis (ISPA 2013)

One of the best workshops on Signal Processing I ever attended, special thanks to Alberto Carini, Co-Chair of the Program Committee.

Aug 16 2013
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Education

Plano de Ensino EEL7065

This page provides info in Portuguese to the UFSC course: EEL 7065 – Sinais e Sistemas Discretos.

EEL 7065  –  Semestre 2014_1

Professor: Joceli Mayer, Ph.D.
Carga Horária: 72 HA, disciplina obrigatória.
Pré-requisitos: EEL7052 – Sistemas Lineares.

Objetivos:

Ao final do semestre o(a) aluno(a) deverá ter conhecimentos e habilidades para entender, analisar e projetar sinais e sistemas discretos básicos, os quais se apresentam em inúmeras áreas da Engenharia Elétrica: Telecomunicações, Processamento de Sinais, Instrumentação, Engenharia Biomédica, Controle e Automação, Eletrônica, etc.

Ementa:

Sinais e sistemas discretos: sinais discretos básicos, propriedades de sistemas discretos. Sistemas discretos lineares e invariantes no tempo (LIT): a soma de convolução discreta, propriedades de sistemas discretos LIT, sistemas LIT descritos por equações de diferenças finitas. Análise de Fourier para sinais discretos: Série de Fourier, Transformada de Fourier, Transformada Discreta de Fourier. Caracterização de sinais e sistemas discretos no domínio da freqüência. Amostragem de sinais: amostragem de sinais contínuos, processamento digital de sinais contínuos, amostragem de sinais discretos.

Conteúdo Programático e Carga Horária Aproximada:

  • Aplicação de provas escritas e apresentação de gabaritos em aula, 8 ha.
  • Sinais e sistemas discretos, 14ha.
  • Introdução, 2ha.
  • Sinais discretos básicos, 6ha.
  • Propriedades de sistemas discretos, 6ha.
  • Sistemas discretos lineares e invariantes no tempo (LIT), 12ha.
  • A soma de convolução discreta, 4ha.
  • Propriedades de sistemas discretos LIT, 6ha.
  • Sistemas LIT descritos por equações de diferenças finitas, 2ha.
  • Análise de Fourier para sinais discretos, 18ha.
  • Série de Fourier, 4ha;
  • Transformada de Fourier, 4 ha.
  • Transformada Discreta de Fourier, 6 ha.
  • Caracterização de sinais e sistemas discretos na freqüência, 4ha.
  • Amostragem de sinais, 18 ha.
  • Amostragem de sinais contínuos, 6ha.
  • Processamento digital de sinais contínuos, 6ha.
  • Amostragem de sinais discretos, 6ha.

Sistema de Avaliação:

O desempenho do estudante será avaliado através de 3 provas escritas e uma prova de recuperação.
As provas escritas terão a duração máxima de duas horas-aula. As 3 listas de exercícios a serem disponibilizadas serão realizadas como atividades extra-classe, e devem ser entregues antes ou no dia da avaliação correspondente ao conteúdo da lista.  A média de avaliação nas 3 provas (M_P) e a média de avaliação das listas (M_L) correspondem à respectivamente 95% e 5% do peso da nota do semestre (N_S). Portanto, N_S = 0.95 M_P+ 0.05M_L .

Será permitido ao(a) aluno(a) realizar uma prova de recuperação. Para aqueles que realizarem a prova de recuperação, a média final será computada pela fórmula: 50% prova de recuperação + 50% da média do semestre (N_S).

Será considerado(a) aprovado(a) o(a) aluno(a) que obtiver média final igual ou superior a 60% e freqüência superior ou igual a 75%.

Organização:

O material do curso será ministrado através de aulas expositivas. As provas escritas e listas de exercícios são componentes essenciais ao processo de aprendizado pois servem como instrumento de realimentação ao estudante e avaliação do progresso dos estudantes no domínio do conteúdo.
Será oferecido atendimento extra classe aos alunos visando esclarecer dúvidas e complementar o conteúdo.

Literatura Básica:

  1. Signal and Systems, A. V. Openheim, A.S. Willsky, Prentice-Hall, 1983.
  2. Digital Signal Processing, Sanjit K. Mitra, Mc-GrawHill, 2001.
  3. Digital Signal Processing, Monson H. Hayes, Mc-GrawHill, 1999.
  4. Linear Systems and Signals, B. P. Lathi, Oxford, 2 ed., 2005.

Literatura Complementar:

  1. Signal Processing & Linear Sistems, B.P. Lathi, Berkeley-Cambridge, 1998.
  2. Digital Image Processing, Gonzalez & Woods, Prentice Hall, 2001.
  3. Discrete-Time Signal Processing, Alan Oppenheim & Shafer, Prentice Hall, 1989.

Páginas da Disciplina:

http://eel.ufsc.br/~mayer

EEL7065

Oct 19 2012
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Education

Analysis of a 2nd Order Filter

A simple IIR filter can model a variety of transfer functions, from lowpass, bandpass and highpass filters. Lets investigate this simple filter given as:

y[n] + a y[n-1] + b y[n-2] = x[n]

We find that that the resulting transfer function is:

H(\Omega) = \frac{Y(\Omega)}{X(\Omega)} = \frac{1}{[1-re^{j\theta}e^{-j\Omega}][ 1-re^{-j\theta}e^{-j\Omega} ]}

where a=-2r cos(\theta) and b= r^2 and it is stable only for |r|< 1 . The parameter \theta controls the central frequency of the filter while the parameter r  controls the selectivity and stability of the filter.

The resulting impulse response is given as :

h[n] = r^n \frac{sin((n+1)\theta)}{sin(\theta)} u[n]

The plots of the magnitude and impulse response are shown in: Sample and Frequency Response of a 2nd Order Filter (requires the Mathematica plugin).

In general, IIR filter design requires phase equalization and FIR designs are more popular due to its linear phase capabilities and other properties. However FIR design requires more coefficients, thus more computations than IIR filters, for the same selectivity.

Oct 17 2012
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Education

Interesting DSP links

  • Watch from You Tube how to generate speech from a piano and hardware => Link.
  • Free book: Signal Processing for Communications  => Link.
  • Free book:  The Scientist and Engineers’s Guide to DSP  => Link.
  • Free book on Digital Signal Processing along with basic implementation examples in C and Matlab => Link.
  • Lecture on Discrete Fourier Series from Oppenheim =>  Link.
  • Lectures on DSP from Oppenheim (MIT) => Link.
  • Properties of continuous Fourier Transform => Link.
  • Properties of discrete Fourier Transform => Link.
Oct 15 2012
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Education

Discrete Fourier Series and Discrete Fourier Transform

The discrete Fourier series is  very related to the Fourier transform.

This post shows that one can be derived from the other.

A periodic signal x[n], with fundamental period N, can be represented as a sum of complex exponentials with fundamental angular frequency \frac{2\pi}{N}:
x[n] = \sum_{k=<N>} a_k e^{jk\frac{2\pi}{N} n}
The coefficients a_k represent the Fourier Series. These coefficients can be obtained from the inner product of the periodic signal with the exponential basis:
a_k = \frac{1}{N} \sum_{n=<N>} x[n] e^{-jk\frac{2\pi}{N} n}

For a given period N, and letting W = e^{j\frac{2\pi}{N}} we can derive these equations in matrix forms:

\begin{bmatrix} x[0]\\ x[1]\\ x[2]\\ \vdots\\ x[N-1] \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1\\ 1 & W & W^2 & \cdots & W^{N-1}\\ 1 & W^2 & W^4 & \cdots & W^{2N-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ 1 & W^{N-1} & W^{2N-2} & \cdots & W^{N^2-2N+1} \end{bmatrix} \begin{bmatrix} a_0\\ a_1\\ a_2\\ \vdots\\ a_{N-1} \end{bmatrix}

The coefficients can be obtained similarly:
\begin{bmatrix} a_0\\ a_1\\ a_2\\ \vdots\\ a_{N-1} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1\\ 1 & W & W^2 & \cdots & W^{N-1}\\ 1 & W^2 & W^4 & \cdots & W^{2N-2}\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ 1 & W^{N-1} & W^{2N-2} & \cdots & W^{N^2-2N+1} \end{bmatrix}^{-1} \begin{bmatrix} x[0]\\ x[1]\\ x[2]\\ \vdots\\ x[N-1] \end{bmatrix}

Let us generate an aperiodic signal \hat{x}[n] by taking N samples from the periodic signal x[n] and setting to zero the remaining samples. By making N\rightarrow \infty, we can derive the Fourier transform of this aperiodic signal:

\hat{X}(\Omega) = \sum_{n=-\infty}^{+\infty} {\hat{x}[n]e^{-j\Omega n}}

From this equation we can easily see that the coefficients a_k of the periodic signal can be obtained by sampling the Fourier transform of the associated aperiodic signal at frequencies \Omega = k\frac{2\pi}{N}:
a_k = \frac{1}{N} \hat{X}(k\frac{2\pi}{N})

Moreover, any signal, periodic or aperiodic, written as:

x[n] = \sum_{k=0}^{N-1} a_k e^{jk\Omega_0 n}

has a Fourier transform, given as:

X(\Omega)=\sum_{k=0}^{N-1} a_k \sum_{r=-\infty}^{+\infty} 2\pi \delta(\Omega-k\Omega_0-2r\pi).

If x[n] is periodic with period N and \Omega_0 = \frac{2\pi}{N}, then

X(\Omega)=\sum_{k=-\infty}^{+\infty} a_k 2\pi \delta(\Omega-k\Omega_0).

From these results we may conclude:

  1. From the Fourier transform of an aperiodic signal \hat{x}[n] we derive the Fourier series coefficients of an equivalent periodic signal x[n] of chosen period N, by sampling the Fourier transform.
  2. From the Fourier series coefficients of a periodic signal we easily derive the Fourier transform of the same signal.
Oct 14 2012
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Education

DSP Demonstrations

Basic DSP principles in:

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