Jacek Cichoń

Katedra Podstaw Informatyki ● WIT ● PWr
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2023/24: Discrete Signals Processing

This is a lecture for students of Algebraic Computer Science at the Faculty of WIT. It takes place on Wednesdays from - in the room 215/D-1.

I am giving this lecture together with Dr. Przemysław Błaskiewicz.

Rules for passing the course

Completing this course involves receiving a pass for the exercises. There is no final exam.
The details will be determined by dr. Przemysław Błaśkiewicz.

Bibliography

  1. Thomas Holton, Digital Signal Processing, Cambridge University Press, 2021
  2. B.A. Shenoi, Introduction to Digital Signal Processing and Filter Design, John Wiley and Sons, 2006
  3. Lizhe Tan, Jean Jiang, Digital Signal Processing. Fundamentals and Applications, Elsevier, 2019
  4. Exercise list: DSPExercises.pdf
    Note: the list will expand.
  5. Soon ....
$ \def\RR{\mathbb{R}} \def\QQ{\mathbb{Q}} \def\ZZ{\mathbb{Z}} \def\CC{\mathbb{C}} \def\NN{\mathbb{N}} \def\BIND{\,>\!>\!=\,} \newcommand{\span}[1]{\mathrm{span}(#1)} \newcommand{\IS}[2]{\langle\,#1,#2\rangle} \newcommand{\sgn}[1]{\mathrm{sgn}(#1)} $

Topics discussed during the lecture

04.10.2023: Introduction I (P.B.)

  1. what is a signal and why, how, why do we process it
  2. signal in continuous domain, discrete signal
  3. what is a linear and non-linear system
  4. flow diagram block symbols

09.10.2023: Introduction II (P.B.)

  1. homogeneous sampling
  2. aliasing, anti-aliasing filters
  3. Nyquist limit

05.10.2023: Discrete signals (JCI)

12.10.2023: Introduction to Systems (JCI)

  1. lp spaces for p = 1, 2, $\infty$ and its basic properties
  2. Memory-less systems
  3. Homogenous systems
  4. Additive systems
  5. Linear systems
  6. Time invariant systems
  7. Basic theorem for LTI systems (unfortunatly, false in general setting): If F is LTI system than $$ T(x) = x \star h $$ where $h = T(\delta)$ and $$ (x \star y) [n] = \sum_{k\in\ZZ} x[k] y[n-k] ~. $$ Remark: $\star$ is called convolution.

19.10.2023: Convolutions (JCI)

  1. Basic property of convolution
    1. $x \star \delta = x$
    2. $x \star y = y \star x$
    3. $(\alpha x) \star y = \alpha(x \star y)$
    4. $(x_1+x_2)\star y = x_1\star y + x_2 \star y$
    5. $S_k(x)\star y = S_k (x\star y)$
  2. Def: System is causual iff its response at time n depends only of history of input up to time n
  3. Theorem. System $S(x) = x \star h$ is causual iff $(\forall n\lt 0)(h[n]=0)$
  4. Basic building blocks
  5. Beginning of analysis of system $y[n] = \frac12 y[n-1] + x[n]$.
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