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<TITLE>Convolution of an odd number of points</TITLE>
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<P><font size="+3" color="green"><B>Convolution of an odd number of points</B></font></P>
<p>
Suppose the blurring vector <CODE>b</CODE>, the second argument,
contains an odd number of points. Let <i>N</i> be the length of <CODE>
b</CODE>, and let <i>M</i> be the length of the data vector,
<CODE>y</CODE>. The convolution of <CODE>y</CODE>
with <CODE>b</CODE> is:</p>
<p><center>
<img src="convolute1.png"></center></p>
<p>
for <img src="convolute2.png">. References to subscripts out of range
of <CODE>y</CODE> are not summed. The blurring vector is
normalized to <code>1</code> to insure that the integrals of the the original data and the
convolution result are identical. This normalization is internal, the blurring vector is
returned unchanged.</p>
<p>
To ensure proper convolution, <CODE>y</CODE> should be padded at
it's upper and lower ends with zeros so it's length is at least the minimum of the non-zero
length of <CODE>b</CODE> and one-half the length of <CODE>b</CODE>.</p>
<p>
Note that the lengths of <CODE>b</CODE> and <CODE>y</CODE> can differ. To avoid centroid
shifts in the output, center the blurring vector properly. For example, suppose that
<CODE>b</CODE> has <code>2N-1</code> elements containing a Gaussian, then its peak should
be at <code>N</code>.</p>
<P>
<a href="convolute.htm"><img align="top" border="0" src="../../../shadow_left.gif">
<font size="+1" color="olive">CONVOLUTE function</font></a><br />
<a href="even.htm"><img align="top" border="0" src="../../../shadow_right.gif">
<font size="+1" color="olive">Convolution of an even number of points</font></a>
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