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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <p>
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
3 </p>
4 <div align="center">
5 <IMG src="images/sigproc_1.png" width="173" height="55" align="center" border="0">
6 </div>
7 <p>
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:
9 </p>
10
11 <ul>
12 <li> Causal: if there is no output before input is fed in.
13 <div align="center">
14 <IMG src="images/sigproc_2.png" width="103" height="28" align="center" border="0">
15 </div>
16 </li>
17 <li> Stable: if finite input results in finite output.
18 <div align="center">
19 <IMG src="images/sigproc_3.png" width="105" height="55" align="center" border="0">
20 </div>
21 </li>
22 <li> Shift invariant: if time shift in the input results in a time shift in the output by the same amount.
23 <div align="center">
24 <IMG src="images/sigproc_4.png" width="84" height="28" align="center" border="0">
25 </div>
26 </li>
27 </ul>
28 <br>
29 <h2><a name="ARMA">Digital filters classification</a></h2>
30 <p>
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:
32 </p>
33 <ul>
34 <li> Autoregressive (AR) process: the difference equation in this case is given by:
35 <div align="center">
36 <br>
37 <IMG src="images/sigproc_12.png" width="154" height="56" align="center" border="0">
38 </div>
39 <br>
40 AR processes can be also classified as <a href="sigproc_iir.html"> IIR Filters</a>.
41 <br>
42 <br></li>
43 <li> Moving Averrage (MA) process:the difference equation in this case is given by: <br>
44 <div align="center">
45 <br>
46 <IMG src="images/sigproc_11.png" width="156" height="56" align="center" border="0">
47 </div>
48 <br>
49 MA processes can be also classified as <a href="sigproc_fir.html"> FIR Filters</a>.
50 <br>
51 <br></li>
52 <li>Autoregressive Moving Average (ARMA) process: the difference equation in this case contains both an AR and a MA process:
53 <div align="center">
54 <br>
55 <IMG src="images/sigproc_7.png" width="283" height="56" align="center" border="0">
56 </div>
57 </li>
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59