(New page: Category:ECE438Spring2009mboutin 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 ...) |
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| − | [[Category:ECE438Spring2009mboutin]] | + | [[Category:ECE438Spring2009mboutin]][[Category:ECE438Spring2009mboutin:CourseNotes]] |
| − | Course Notes January 14, 2009 | + | == ECE438 Course Notes January 14, 2009 == |
1)Definitions | 1)Definitions | ||
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Signal values must be discrete | Signal values must be discrete | ||
| − | -x(n) | + | -<math>x(n) \in {0,1}</math> <-- binary valued signal |
| − | + | <br/><math>x(n) \in {0,1,2,...,255}</math> <-- gray scale valued signal | |
| Line 20: | Line 20: | ||
-the pixels in a bitmap image (grayscale) can have a value of 0,1,2,...,255 for each individual pixel. | -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. | --If you concatenate all the rows of the image you can convert it to a 1 dimensional signal. | ||
| − | i.e. x = (row1,row2,row3) | + | i.e. <math>x = (row1,row2,row3)</math> |
| − | <math> | + | 2D Digital signal = signal that can be represented by an array of 0's and 1's |
| + | |||
| + | <u>example</u>: 128x128 gray scale image<br/> | ||
| + | <math>p_{ij} \in {0,...,255}</math> | ||
| + | |||
| + | matrix <math>A_{ij} = p_{ij}</math> of size 128x128 <br/> | ||
| + | |||
| + | <strong>Digital Systems</strong> = system that can process a ditital signal.<br/> | ||
| + | E.g. | ||
| + | <ul> | ||
| + | <li>Software (MATLAB,C, ...) </li> | ||
| + | <li>Firmware</li> | ||
| + | <li>Digital Hardware</li> | ||
| + | </ul> | ||
| + | |||
| + | == Advantages of Digital Systems == | ||
| + | <ul> | ||
| + | <li>precise,reproducable</li> | ||
| + | <li>easier to store data</li> | ||
| + | <li>easier to build: | ||
| + | <ul> | ||
| + | <li>just need to represent 2 states instead of a continuous range of values</li> | ||
| + | </ul> | ||
| + | </li> | ||
| + | </ul> | ||
| + | |||
| + | <strong>Software based digital systems</strong> | ||
| + | <ul> | ||
| + | <li>easier to build</li> | ||
| + | <li>cheap to build</li> | ||
| + | <li>adaptable</li> | ||
| + | <li>easy to fix/upgrade</li> | ||
| + | </ul> | ||
| + | |||
| + | <strong>Hardware-based digital systems</strong> | ||
| + | <ul> | ||
| + | <li>fast.</li> | ||
| + | |||
| + | </ul> | ||
| + | <table border="1px"> | ||
| + | <tr> | ||
| + | <td width="50%" align="center" valign="top"> | ||
| + | <strong>Continuous time world</strong> | ||
| + | <ul> | ||
| + | <li>most natural signals live here</li> | ||
| + | <li>things are easy to write, understand, conceptualize</li> | ||
| + | |||
| + | </ul> | ||
| + | </td> | ||
| + | <td width="50%" align="center" valign="top"> | ||
| + | <strong>Digital World</strong> | ||
| + | <ul> | ||
| + | <li>digital media signals live here along with computers, MATLAB, digital circuits</li> | ||
| + | </ul> | ||
| + | </td> | ||
| + | </tr> | ||
| + | </table> | ||
| + | <p>These world are brought together using sampling & quantization, as well as reconstruction</p> | ||
| + | |||
| + | == Signal Characteristics == | ||
| + | <ul> | ||
| + | <li>Deterministic vs. random | ||
| + | <ul> | ||
| + | <li>x(t) well defined , s.a. <math>x(t) = e^{j\pi t}</math></li> | ||
| + | <li>x(n) well defined , s.a. <math>x(n) = j^{n}</math> <br/>ex: Lena's image</li> | ||
| + | </ul> | ||
| + | </li> | ||
| + | <li>Random | ||
| + | <ul> | ||
| + | <li>x(t) drawn according to some distribution</li> | ||
| + | <li>example: x(t) white noise<br/>x = rand(10) (almost) random</li> | ||
| + | </ul> | ||
| + | </li> | ||
| + | </ul> | ||
| + | |||
| + | <ul> | ||
| + | <li>Periodic vs. non-periodic | ||
| + | <ul><li> if <math>\exists</math> positive T such that x(t+T) = x(t),<math>\forall t</math> then we say that x(t) is periodic with period T</li></ul> | ||
| + | </li> | ||
| + | </ul> | ||
Revision as of 10:21, 14 January 2009
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 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
|
Digital World
|
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
