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Cleanup
author Daniele Nicolodi <nicolodi@science.unitn.it>
date Mon, 05 Dec 2011 18:04:34 +0100
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1 <p>
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2 A digital filter is an operation that associates an input time series x[n] into an output one, y[n]. Methods developed in the LTPDA Toolbox deal with linear digital filters, i.e. those which fulfill that a linear combination of inputs results in a linear combination of outputs with the same coefficients (provided that these are not time dependent). In these conditions, the filter can be expressed as
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3 </p>
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4 <div align="center">
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5 <IMG src="images/sigproc_1.png" width="173" height="55" align="center" border="0">
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6 </div>
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7 <p>
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8 described in these terms, the filter is completely described by the impulse response h[k], and can then be subdivided into the following classes:
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9 </p>
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11 <ul>
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12 <li> Causal: if there is no output before input is fed in.
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13 <div align="center">
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14 <IMG src="images/sigproc_2.png" width="103" height="28" align="center" border="0">
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15 </div>
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16 </li>
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17 <li> Stable: if finite input results in finite output.
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18 <div align="center">
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19 <IMG src="images/sigproc_3.png" width="105" height="55" align="center" border="0">
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20 </div>
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21 </li>
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22 <li> Shift invariant: if time shift in the input results in a time shift in the output by the same amount.
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23 <div align="center">
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24 <IMG src="images/sigproc_4.png" width="84" height="28" align="center" border="0">
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25 </div>
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26 </li>
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27 </ul>
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28 <br>
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29 <h2><a name="ARMA">Digital filters classification</a></h2>
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30 <p>
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31 Digital filters can be described as difference equations. If we consider an input time series x and an output y, three specific cases can then be distinguished:
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32 </p>
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33 <ul>
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34 <li> Autoregressive (AR) process: the difference equation in this case is given by:
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35 <div align="center">
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36 <br>
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37 <IMG src="images/sigproc_12.png" width="154" height="56" align="center" border="0">
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38 </div>
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39 <br>
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40 AR processes can be also classified as <a href="sigproc_iir.html"> IIR Filters</a>.
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41 <br>
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42 <br></li>
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43 <li> Moving Averrage (MA) process:the difference equation in this case is given by: <br>
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44 <div align="center">
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45 <br>
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46 <IMG src="images/sigproc_11.png" width="156" height="56" align="center" border="0">
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47 </div>
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48 <br>
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49 MA processes can be also classified as <a href="sigproc_fir.html"> FIR Filters</a>.
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50 <br>
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51 <br></li>
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52 <li>Autoregressive Moving Average (ARMA) process: the difference equation in this case contains both an AR and a MA process:
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53 <div align="center">
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54 <br>
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55 <IMG src="images/sigproc_7.png" width="283" height="56" align="center" border="0">
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56 </div>
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57 </li>
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