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[[Category:ECE438Spring2009mboutin]][[Category:ECE438Spring2009mboutin:CourseNotes:Lecture1]]
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[[Category:ECE]]
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[[Category:ECE438]]
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[[Category:signal processing]]
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[[Category:ECE438Spring2009mboutin]]
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[[Category:lecture notes]]
  
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== ECE438 Course Notes January 14, 2009 ==
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=Lecture Notes for [[ECE438]] Spring 2009, [[user:mboutin|Prof. Boutin]]=
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*[[CourseNotes1_(BoutinSpring2009)|Course Notes Lecture 1 Jan. 14, 2009]]
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1)Definitions
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*[[CourseNotes2_(BoutinSpring2009)|Course Notes Lecture 2 Jan. 16, 2009]]
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*[[CourseNotes3_(BoutinSpring2009)|Course Notes Lecture 3 Jan. 21, 2009]]
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ECE438 is about digital signals and systems
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*[[CourseNotes4_(BoutinSpring2009)|Course Notes Lecture 4 Jan. 23, 2009]]
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*[[CourseNotes6_(BoutinSpring2009)|Course Notes Lecture 6 Jan. 28, 2009]]
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2) Digital Signal = a signal that can be represented by a sequence of 0's and 1's.
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*[[CourseNotes16_(BoutinSpring2009)|Course Notes Lecture 16 Feb. 23, 2009]]
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so the signal must be DT X(t) = t, i.e. need x(n), n belongs to Z
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*[[CourseNotes20_(BoutinSpring2009)|Course Notes Lecture 20 Mar. 11, 2009]]
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*[[CourseNotes30_(BoutinSpring2009)|Course Notes Lecture 30 Apr. 17, 2009]]
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Signal values must be discrete
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----
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[[ECE438_(BoutinFall2009)|Back to ECE438, Spring 2009]]
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-<math>x(n) \in {0,1}</math> <-- binary valued signal
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<br/><math>x(n) \in {0,1,2,...,255}</math> <-- gray scale valued signal
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Another example of digital signal
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-the pixels in a bitmap image (grayscale) can have a value of 0,1,2,...,255 for each individual pixel.
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--If you concatenate all the rows of the image you can convert it to a 1 dimensional signal.
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i.e. <math>x = (row1,row2,row3)</math>
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2D Digital signal = signal that can be represented by an array of 0's and 1's
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<u>example</u>: 128x128 gray scale image<br/>
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<math>p_{ij} \in {0,...,255}</math>
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matrix <math>A_{ij} = p_{ij}</math> of size 128x128 <br/>
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[[Image:Vip_logo.jpg| 70px| left]]
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Digital signals play an important roll in forensics applications such as: watermarking, image identification, and forgery detection among many others. Go to [http://cobweb.ecn.purdue.edu/~prints/publications.shtml PSAPF] and [http://cobweb.ecn.purdue.edu/~vip/teams/sensor_forensics.html VIP's Sensor Forensics] to find out more information about these applications.
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<strong>Digital Systems</strong> = system that can process a ditital signal.<br/>
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E.g.
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<ul>
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<li>Software (MATLAB,C, ...) </li>
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<li>Firmware</li>
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<li>Digital Hardware</li>
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</ul>
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== Advantages of Digital Systems ==
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<ul>
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<li>precise,reproducable</li>
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<li>easier to store data</li>
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<li>easier to build:
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  <ul>
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    <li>just need to represent 2 states instead of a continuous range of values</li>
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  </ul>
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</li>
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</ul>
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<strong>Software based digital systems</strong>
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<ul>
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<li>easier to build</li>
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<li>cheap to build</li>
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<li>adaptable</li>
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<li>easy to fix/upgrade</li>
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</ul>
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<strong>Hardware-based digital systems</strong>
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<ul>
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<li>fast.</li>
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</ul>
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<table border="1px">
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<tr>
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<td  width="50%" align="center" valign="top">
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<strong>Continuous time world</strong>
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<ul>
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<li>most natural signals live here</li>
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<li>things are easy to write, understand, conceptualize</li>
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</ul>
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</td>
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<td width="50%" align="center" valign="top">
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<strong>Digital World</strong>
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<ul>
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<li>digital media signals live here along with computers, MATLAB, digital circuits</li>
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</ul>
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</td>
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</tr>
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</table>
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<p>These world are brought together using sampling & quantization, as well as reconstruction</p>
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== Signal Characteristics ==
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<ul>
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  <li>Deterministic vs. random
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    <ul>
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      <li>x(t) well defined , s.a. <math>x(t) =  e^{j\pi t}</math></li>
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      <li>x(n) well defined , s.a. <math>x(n) = j^{n}</math> <br/>ex: Lena's image</li>
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    </ul>
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  </li>
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  <li>Random
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    <ul>
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      <li>x(t) drawn according to some distribution</li>
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      <li>example: x(t) white noise<br/>x = rand(10) (almost) random</li>
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    </ul>
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  </li>
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</ul>
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<ul>
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  <li>Periodic vs. non-periodic
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  <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>
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  </li>
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</ul>
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== ECE438 Course Notes January 16, 2009 ==
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<p><strong>Todays Goals</strong>
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  <ul>
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    <li>Signal Characteristics</li>
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    <li>Signal Transformations</li>
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    <li>Special Signals</li>
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    <li>Singularity Functions</li>
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  </ul>
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</p>
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<p><strong>right sided signal: </strong><br/>
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<math>\exists t_{min} (n_{min})</math> such that <math>x(t) = 0</math> when <math>t < t_{min}</math>
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</p>
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<p><strong>left sided signal: </strong><br/>
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<math>\exists t_{max} (n_{max})</math> such that <math>x(t) = 0</math> when <math>t > t_{max}</math><br/>
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if <math>t_{max} \leq 0</math> we say the signal is <u>anticausal</u>
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</p>
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<p><strong>two sided (mixed causal):</strong><br/>
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neither left sided nor right sided
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</p>
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<p><strong>Finite Duration Signal: </strong><br/>
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both right and left sided, <math>\exists t_{min},t_{max}</math> such that <math>x(t) = 0</math> for <math>t > t_{max}</math> and <math>t < t_{min}</math>
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</p>
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<h3>Signal Metrics</h3>
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<br/>
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<ul style="list-style:none;">
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  <li><strong>Signal Energy</strong>
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    <ul style="list-style:none;">
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      <li>
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        <p><math>E_x = \int_{-\infty}^{\infty} |x(t)|^2\,dt</math> for ct (continuous time)</p>
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        <p><math>E_x = \sum_{n=-\infty}^{\infty} |x(n)|^2</math> for dt (discrete time)</p>
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      </li>
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    </ul>
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  </li>
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  <li><strong>Signal Power</strong>
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    <ul style="list-style:none;">
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      <li>
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        <p><math>P_x = \lim_{T \Rightarrow \infty}\frac{1}{2T}\int_{-T}^{T} |x(t)|^2\,dt</math> for ct (continuous time)</p>
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        <p><math>P_x = \lim_{N \Rightarrow \infty}\sum_{n=-N}^{N} |x(n)|^2</math> for ct (continuous time)</p>
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        <p>note: for periodic signals <br/>
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        <math>P_x = \frac{1}{N}\sum_{n=0}^{N-1}|x(n)|^2</math>
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        </p>
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      </li>
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    </ul>
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  </li>
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  <li><strong>Signal RMS (root-mean-square)</strong>
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    <ul style="list-style:none;">
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      <li>
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      <math>X_{rms} = \sqrt{P_x}</math>
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      </li>
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    </ul>
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  </li>
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  <li><strong>Signal Magnitude</strong>
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    <ul style="list-style:none;">
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      <li>
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        <p><math>m_x = max|x(t)|</math>, for CT</p>
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        <p><math>m_x = max|x(n)|</math>, for DT</p>
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        <p> if <math>m_x < \infty</math>, we say signal is bounded</p>
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      </li>
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    </ul>
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  </li>
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  <li><strong>Scaling (<math>y(t) = x(\frac{t}{a})</math>)</strong>
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    <ul style="list-style:none;">
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      <li>
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        <p>note: y(0) = x(0), fixed point at t=0<br/>
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        if a > 1, graph will narrow<br/>
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        if a < 1, graph will expand<br/><br/>
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        <font style="color:red;">if a>1 will not work for digital signals</font>
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        </p><br/>
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        <p><strong>Down Sampler:</strong><br/>
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          <math>y(n) = x(Dn)</math>, D = integer > 1<br/>
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          <math>x(n) \Rightarrow D\Downarrow \Rightarrow y(n)</math>
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        </p>
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        <p><strong>Up Sampler:</strong> <math>x(n) \Rightarrow D\Uparrow \Rightarrow y(n)</math><br/>
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          <math>y(n) = x(\frac{n}{D})</math>, if n/D is an integer
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        </p>
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        <p><strong>Scaling and Shifting</strong> <math>y(t) = x(\frac{t}{a}-t_0)</math><br/>
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          note: <math>y(0) = x(-t_0)</math>
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        </p>
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      </li>
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    </ul>
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  </li>
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</ul>
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== ECE438 Course Notes January 21, 2009 ==
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<p><h3><u>Delta Functions</u></h3><br/>
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<strong>Continuous-time:</strong> (a.k.a. Dirac delta function)<br/>
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<math>\delta(t) = \lim_{\triangle \Rightarrow 0} \frac{1}{\triangle}rect(\frac{t}{\triangle})</math><br/>
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<u>Properties</u><br/>
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<ul>
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  <li><math>\int_{-\infty}^{\infty} \delta(t)\,dt = 1</math>(unit area)</li>
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  <li><math>\int_{-\infty}^{\infty} x(t) \delta(t-t_0)\,dt = x(t_0)</math>(sifting property)</li>
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</ul>
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</p>
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<p><strong>Discrete-time:</strong> (a.k.a. Kronecher delta fn.)<br/>
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<math>\delta[n] = 1 | n=0, 0 | 0 > n < 0</math>
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<ul>
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  <li>Sifting Property: <math>\sum_{n=-\infty}^{\infty} x[n] \delta[n-n_0] = x[n_0]</math></li>
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</ul>
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</p>
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<h3>Comb &amp; Rep operators (for CT signals)</h3>
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<p>Comb operator multiplies a signal by an "impulse train".
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<ul>
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<li><math>\sum_{k=-\infty}^{\infty} \delta(t-kT)</math></li>
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<li><math>Comb_T{{x(t)}} = x(t)\sum_{k=-\infty}^{\infty} \delta(t-kT) = \sum_{k=-\infty}^{\infty} x(kT)\delta(t-kT)</math></li>
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</ul>
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</p>
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<p><strong>Rep operator</strong> simply replicates a signal every T units:<br/>
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<math>rep_T{{x(t)}} = \sum_{k=-\infty}^{\infty} x(t-kT)</math>
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</p>
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<h3>Systems</h3>
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<p>A system maps an input signal x(t) to a unique output signal, y(t).
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<math>x(t) \Rightarrow \mbox {System} \Rightarrow y(t)</math><br/>
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<math>y(t) = S[x(t)]</math>
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</p>
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<p><Strong>Examples:</strong><br/>
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<math>y(n) = \frac{1}{3}[x(n) + x(n-1) + x(n-2)]</math> (moving averaging function, seen in Lab2)
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</p>
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<p><u>System Properties:</u><br/>
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<ul>
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  <li>Linearity
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      <li>Definition: A system S is linear if for any two input signals <math>x_1(t)</math> and <math>x_2(t)</math>, and any (complex) constant, a, it satisfies the following two properties:</li?
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      <li>Superposition: <math>S[x_1(t) + x_2(t)] = S[x_1(t)] + S[x_2(t)]</math></li>
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      <li>Homogeneity: <math>S[ax_1(t)] = aS[x_1(t)]</math></li>
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    </ul>
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  </li>
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  <li>Time-Invariance
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    <ul>
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      <li>Definition: A system S is time-invariant(TI) if delaying the input signal results only in an identical delayin the output signal.</li>
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      <li>If <math>y_1(t) = S[x_1(t)]</math> and <math> y_2(t) = S[x_1(t-t_0)]</math> than <math>y_2(t) = y_1(t-t_0)</math></li>
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    </ul>
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  </li>
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</ul>
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</p>
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Latest revision as of 05:37, 16 September 2013


Lecture Notes for ECE438 Spring 2009, Prof. Boutin

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