bug in IDL's hanning() window-generating function

[bennetsc(at)ucs.orst.edu (Scott Bennett)]

     Perhaps this has been discussed or reported before, but if so,
RSI has not yet fixed it.
     I just noticed that IDL's hanning() has a bug described by Harris
(1978) as common in software packages.  The bug is easy to look for.
To check, generate a window of even length.  Then look at its
coefficients.  The array should *not* be exactly symmetrical.  It
should have only one highest coefficient next to, and to the right of,
the "center" of the array, noting that the center of the time
(or space) period covered by the discrete window actually falls at the
largest window coefficient, which is immediately to the right of the
"center" of the array, because of the even number of coefficients.
IDL's hanning() produces a symmetric window with two identical
coefficients at the center and zeros at the ends, which is incorrect.
A Hann window of even length should only be zero at the left (i.e.
starting) end, whereas the other end should be equal to the first
coefficient in from the zero.  (A Hann window of odd length should
also be zero at the start, but not at the end.  See further below.)
     For example, "print, hanning(6)" gives

0.00000     0.345492     0.904509     0.904508     0.345491      0.00000

whereas the correct sequence is

0.00000     0.250000     0.750000      1.00000     0.750000     0.250000

     Next, generate a window of odd length.  An odd-length window
should have its largest coefficient at its central element, but the
next coefficient should be identical to it, followed by decreasing
values out to the end of the window that are the mirror image of values
going the other direction from the central element.  This results in
the last coefficient of the window still matching the second
coefficient of the window.  The duplication of the highest coefficient
in the middle of the window, with the duplicated value being less than
1, seems to me to account for the time (or space) shift in sampling
a given periodic function that one finds when using 2n + 1 samples
instead of 2n samples.  In other words, the center of the time (or
space) period covered by the discrete window falls halfway between
the two largest data window array elements.  Note that this means
that the single highest coefficient of a typical window of even length
will be 1, but the two highest coefficients of a window of odd length
will be equal and somewhat less than 1; i.e. the value 1 would fall
halfway between the two coefficients actually present at the discrete
sampling interval.
     For example, "print, hanning(5)" gives

0.00000     0.500000      1.00000     0.500000      0.00000

but the correct sequence should be

0.00000     0.345492     0.904509     0.904508     0.345491

Note that the third and fourth coefficients above are "identical"
within the limits of precision and the default format used in the
print statement, as are the second and fifth.  Also, note that the
same kind of error is produced regardless of the value of ALPHA coded
(default is .5, which yields a Hann window) in the call to hanning(),
excluding ALPHA=1., of course.
     In any case, I think the point Harris made is that a discrete
sampling of a window function should not taper to the same value at
the end that it has at the beginning because to do so would include
the first sample of the *next* period (windowed segment.)  So IDL's
hanning() gets it wrong for both even- and odd-length windows. :-(  

Ref.:
Fredric J. Harris, On the Use of Windows for Harmonic Analysis
	with the Discrete Fourier Transform, Proc. IEEE, vol. 66
	no. 1, January 1978


                                  Scott Bennett, Comm. ASMELG, CFIAG
                                  College of Oceanic and Atmospheric
					Sciences
                                  Oregon State University
                                  Corvallis, Oregon 97331
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