Course Notes, January 14, 2009

ECE438, Spring 2009


1)Definitions

ECE438 is about digital signals and systems

2) Digital Signal = a signal that can be represented by a sequence of 0's and 1's.

so the signal must be DT X(t) = t, i.e. need x(n), n belongs to Z

Signal values must be discrete

-$ x(n) \in {0,1} $ <-- binary valued signal
$ x(n) \in {0,1,2,...,255} $ <-- gray scale valued signal


Another example of digital signal

-the pixels in a bitmap image (grayscale) can have a value of 0,1,2,...,255 for each individual pixel. --If you concatenate all the rows of the image you can convert it to a 1 dimensional signal. i.e. $ x = (row1,row2,row3) $

2D Digital signal = signal that can be represented by an array of 0's and 1's

example: 128x128 gray scale image
$ p_{ij} \in {0,...,255} $

matrix $ A_{ij} = p_{ij} $ of size 128x128

Vip logo.jpg

Digital signals play an important roll in forensics applications such as: watermarking, image identification, and forgery detection among many others. Go to PSAPF and VIP's Sensor Forensics to find out more information about these applications.

Digital Systems = system that can process a ditital signal.
E.g.

  • Software (MATLAB,C, ...)
  • Firmware
  • Digital Hardware

Advantages of Digital Systems

  • precise,reproducable
  • easier to store data
  • easier to build:
    • just need to represent 2 states instead of a continuous range of values

Software based digital systems

  • easier to build
  • cheap to build
  • adaptable
  • easy to fix/upgrade

Hardware-based digital systems

  • fast.

Continuous time world

  • most natural signals live here
  • things are easy to write, understand, conceptualize

Digital World

  • digital media signals live here along with computers, MATLAB, digital circuits

These world are brought together using sampling & quantization, as well as reconstruction

Signal Characteristics

  • Deterministic vs. random
    • x(t) well defined , s.a. $ x(t) = e^{j\pi t} $
    • x(n) well defined , s.a. $ x(n) = j^{n} $
      ex: Lena's image
  • Random
    • x(t) drawn according to some distribution
    • example: x(t) white noise
      x = rand(10) (almost) random
  • Periodic vs. non-periodic
    • if $ \exists $ positive T such that x(t+T) = x(t),$ \forall t $ then we say that x(t) is periodic with period T

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