Hamming etc. windows are wrong

[Jerry]

I’ve noticed that the Hamming window (and I suppose other windows) provided by 
Octave, SciPy****, and NumPy***** is wrong, returning a window that is 
symmetric rather than one that is “DFT symmetric.” This is easily spotted by 
looking at the first and last points of the window; if they are the same 
values, then the window is incorrect.

These windows are normally applied to data that are equally spaced, thus 
implying that their DFTs are periodic. Let’s assume that we are windowing N 
data points, indexed from 0 to N-1 as is normal in signal processing. Then the 
DFT is also N points long, also indexed from 0 to N-1. The periodic extension 
principal implies that the point indexed by N in the DFT domain has the same 
value as the point indexed by 0. The window can also be imagined to have the 
same kind of periodic extension, so that what would be the N-th point of the 
window matches the N-point of the data with the same weight as was applied to 
the 0-th point of the data but without duplicating that point at N-1.

This mistake is widespread, infecting many software packages. (I am sending 
this same note to the NumPy, SciPy, and Octave lists.) It is certainly wrong in 
most textbooks, including the classic, Oppenheim & Schafer, 1975. However, the 
definitive and widely referenced paper on windows by Fredric J. Harris, “On the 
Use of Windows for Harmonic Analysis with the Discrete Fourier Transform,” 
Proceedings of the IEEE, vol. 66, NO. 1 , January, 1978 * discusses this 
problem, both its roots and its correction. For example, quoting from the paper:

“Since the DFT essentially considers sequences to be periodic, we can consider 
the missing end point to be the beginning of the next period of the periodic 
extension of this sequence. In fact, under the periodic extension, the next 
sample (at 16 s in Fig. 1.) is indistinguishable from the sample at zero 
seconds.

“This apparent lack of symmetry due to the missing (but implied) end point is a 
source of confusion in sampled window design. This can be traced to the early 
work related to convergence factors for the partial sums of the Fourier series. 
The partial sums (or the finite Fourier transform) always include an odd number 
of points and exhibit even symmetry about the origin. Hence much of the 
literature and many software libraries incorporate windows designed with true 
even symmetry rather than the implied symmetry with the missing end point!”

...and later...

“We will now catalog some well-known (and some not well-known windows. For each 
window we will comment on the justification for its use and identify its 
significant parameters. All the windows will be presented as even (about the 
origin) sequences with an odd number of points. To convert the window to DFT 
even, the right endpoint will be discarded and the sequence will be shifted so 
that the left end point coincides with the origin.”

Note that he limits himself therefore to even-numbered "DFT even" windows, but 
the rule remains--honor the implied periodicity even for windows of odd length.

Lest one consider this a trivial difference, consider the following 
Octave-Matlab code (corresponding NumPy and SciPy code would yield the same 
results).

x = ones(8, 1); # An even sequence; its DFT should be real.
hWrong = hamming(8)
h9 = hamming(9);
hRight = h9(1:8)
xWrong = x .* hWrong;   # Multiplication by "ones"
xRight = x .* hRight;   # shown for clarity.
XWrong = fft(xWrong)    # Contains nonzero imaginary part.
XRight = fft(xRight)    # Real, as expected from evenness of x.
XWrongMag = abs(XWrong) # The magnitudes are
XRightMag = abs(XRight) # also different.

This causes the following output:

hWrong =

  0.080000
  0.253195
  0.642360
  0.954446
  0.954446
  0.642360
  0.253195
  0.080000

hRight =

  0.080000
  0.214731
  0.540000
  0.865269
  1.000000
  0.865269
  0.540000
  0.214731

XWrong =

  3.86000 + 0.00000i
 -1.76795 - 0.73231i
  0.13889 + 0.13889i
  0.01906 + 0.04602i
  0.00000 + 0.00000i
  0.01906 - 0.04602i
  0.13889 - 0.13889i
 -1.76795 + 0.73231i

XRight =

  4.32000 + 0.00000i
 -1.84000 + 0.00000i
  0.00000 + 0.00000i
  0.00000 + 0.00000i
  0.00000 + 0.00000i
  0.00000 - 0.00000i
  0.00000 - 0.00000i
 -1.84000 - 0.00000i

XWrongMag =

  3.86000
  1.91362
  0.19642
  0.04981
  0.00000
  0.04981
  0.19642
  1.91362

XRightMag =

  4.32000
  1.84000
  0.00000
  0.00000
  0.00000
  0.00000
  0.00000
  1.84000

This news will likely upset many and I will be interested to see what arguments 
will flow in favor of keeping the status quo--I’m sure one will be, “we’ve 
always done it this way” and others will be more substantial, possibly 
expressing decent arguments why the current situation is useful for some 
applications. In any case, I refer again to the Harris paper.

So rather than host an argument on this list, this is what I propose: Do what 
Matlab** does. Acknowledge both uses by making a flag to handle the “periodic” 
or the “symmetric” cases. The Matlab default is “symmetric” which is of course 
unfortunate but at least such inclusion in Octave, NumPy, and SciPy would 
retain compatibility with the existing usage. Then it’s up to the user whether 
to shoot him/herself in the foot, assuming that such a decision is guided by 
actually referring to the documentation for the package being used and not 
blindly using the default.

Jerry Bauck


* 
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1455106&url=http%3A%2F%2Fieeexplore.ieee.org%2Fxpls%2Fabs_all.jsp%3Farnumber%3D1455106

** http://www.mathworks.com/help/signal/ref/hamming.html

*** 
http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.signal.hamming.html

**** I am just learning Python-NumPy-SciPy but it appears as though the SciPy 
situation is that the documentation page above *** mentions the flag but the 
flag has not been implemented into SciPy itself. I would be glad to stand 
corrected.

***** http://docs.scipy.org/doc/numpy/reference/generated/numpy.hamming.html