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<TITLE>Method</TITLE>
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<P><font size="+3" color="green"><B>Method</B></font></P>
<p>
A nonrecursive filter is defined by the convolution formula:</p>
<p><center><img align="top" src="savgolI01.png"></center></p>
<p>
The <img align="bottom" src="savgolI02.png"> 's are the coefficients of the filter, the
<img align="bottom" src="savgolI03.png"> 's are the input data, and the
<img align="bottom" src="savgolI04.png"> 's are the
outputs. The set of points <img align="bottom" src="savgolI05.png"> to
<img align="bottom" src="savgolI06.png"> define the moving window of the filter, of width
<img align="bottom" src="savgolI07.png">, where
<img align="bottom" src="savgolI08.png"> and
<img align="bottom" src="savgolI09.png">. The
Savitzky-Golay smoothing method finds filter coefficients
that preserve the zeroth, the first, the second, and higher moments. The idea is to approximate the
underlying function within a moving filter window by a polynomial of order <code>2</code>, or
of order <code>4</code>. For each data point, <img align="bottom" src="savgolI10.png">,
least-squares fit a polynomial to all <img align="bottom" src="savgolI11.png"> points
within the filter window, and then set <img align="bottom" src="savgolI12.png"> to be the
value of that polynomial at position <img align="bottom" src="savgolI13.png">.
No use is made of the value of that polynomial at any other point. For the next point,
<img align="bottom" src="savgolI14.png">, a whole new least-squares fit is done using the
shifted window.</p>
<p>
Since the process of least-squares fitting involves only a linear matrix inversion, the
coefficients of a fitted polynomial are themselves linear in the values of the data. All the
least-squares fitting can be done in advance, for fictitious data consisting of all zeros except
for a single <code>1</code>, and then the fits on the real data are done by taking linear
combinations. Thus, there are particular sets of filter coefficients,
<img align="bottom" src="savgolI13.png">, for which the
convolution formula automatically accomplishes the process of polynomial least-squares fitting
inside a moving window.</p>
<p>
The Savitzky-Golay filters provide smoothing without loss of resolution when the underlying
function can be locally well fitted by a polynomial. This is true for smooth line profiles
not too much narrower than the filter width. When this is not true, these filters have no
advantage over other smoothing methods.</p>
<p>
If the data is irregularly sampled, that is, the <img align="bottom" src="savgolI14.png">
values are not equally spaced, one can
pretend that the data points are equally spaced. This amounts to virtually shifting, within
each moving window, the data points to equally spaced positions. Such a shift introduces
the equivalent of an additional source of noise into the function values. In those cases where
smoothing is useful, this noise will often be much smaller than the noise already present.
Specifically, if the location of the points is approximately random within the window, then
a rough criterion is this: If the change in <img align="bottom" src="savgolI14.png">
across the full width of the window, <img align="bottom" src="savgolI11.png">, is less
than <img align="bottom" src="savgolI15.png"> times the measurement noise on a single
point, then this implementation of the Savitzky-Golay filter can be used.</p>
<P>
<a href="savgol.htm"><img align="top" border="0" src="../shadow_left.gif">
<font size="+1" color="olive">SAVGOL function</font></a></P>
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