ECE438 Course Notes January 14, 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
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.
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Continuous time world
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Digital World
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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
