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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Digital Filtering</h1>
  <hr>
  
  <p>
	<p>
  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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<IMG src="images/sigproc_1.png" width="173" height="55" align="center" border="0">
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<p>
described in these terms, the filter is completely described by the impulse response h[k], and can then be subdivided into the following classes: 
</p>

<ul>
<li> Causal: if there is no output before input is fed in.
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<IMG src="images/sigproc_2.png" width="103" height="28" align="center" border="0">
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  </li>
<li> Stable: if finite input results in finite output.  
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<IMG src="images/sigproc_3.png" width="105" height="55" align="center" border="0">
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</li>
<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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<IMG src="images/sigproc_4.png" width="84" height="28" align="center" border="0">
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</li>
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<br>
<h2><a name="ARMA">Digital filters classification</a></h2>
<p>
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:
</p>
<ul>
<li> Autoregressive (AR) process: the difference equation in this case is given by:  
<div align="center">
<br>
<IMG src="images/sigproc_12.png" width="154" height="56" align="center" border="0">
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<br>
AR processes can be also classified as <a href="sigproc_iir.html"> IIR Filters</a>.
    <br>
    <br></li>
<li> Moving Averrage (MA) process:the difference equation in this case is given by: <br>  
<div align="center">
<br>
<IMG src="images/sigproc_11.png" width="156" height="56" align="center" border="0">
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<br>
MA processes can be also classified as <a href="sigproc_fir.html"> FIR Filters</a>.
    <br>
    <br></li>
<li>Autoregressive Moving Average (ARMA) process: the difference equation in this case contains both an AR and a MA process:   
<div align="center">
<br>
<IMG src="images/sigproc_7.png" width="283" height="56" align="center" border="0">
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