Window functions precision (they are not exactly symmetric)

[kimstik]

I noted that window functions is perfectly bit-to-bit symmetric in case of float32, but in float64 they are not.
Error is about 2 mantissa bit in case of Blackman window.
Numpy issue was solved in #17195 .

[kimstik]

Copy-paste of original report:

I noted that window functions is perfectly bit-to-bit symmetric in case of float32, but in float64 they are not.
Error is about 2 mantissa bit in case of Blackman window.

Reproducing code example:

import numpy as np
import scipy.signal.windows


wnd = (	np.blackman,
		np.bartlett,
		np.hamming,
		np.hanning,
		scipy.signal.windows.flattop            ,
		scipy.signal.windows.barthann           ,
		scipy.signal.windows.bartlett           ,
		scipy.signal.windows.blackman           ,
		scipy.signal.windows.blackmanharris     ,
		scipy.signal.windows.bohman             ,
		scipy.signal.windows.boxcar             ,
		scipy.signal.windows.cosine             ,
		scipy.signal.windows.flattop            ,
		scipy.signal.windows.hamming            ,
		scipy.signal.windows.hann               ,
		scipy.signal.windows.nuttall            ,
		scipy.signal.windows.parzen             ,
		scipy.signal.windows.triang				,
		)

for t in (np.float32, np.float64):	#, np.longdouble):
	print (t, np.finfo(t))
	for w in wnd:
		x = w(4096).astype(t)
		err = np.max(np.abs( x[:2048] - x[4095:2047:-1] ))
		print (w, err, np.log2(err))

Results:

<class 'numpy.float32'> Machine parameters for float32
---------------------------------------------------------------
precision =   6   resolution = 1.0000000e-06
machep =    -23   eps =        1.1920929e-07
negep =     -24   epsneg =     5.9604645e-08
minexp =   -126   tiny =       1.1754944e-38
maxexp =    128   max =        3.4028235e+38
nexp =        8   min =        -max
---------------------------------------------------------------

<function blackman at 0x000001B4D49289D0> 0.0 -inf
<function bartlett at 0x000001B4D4928A60> 0.0 -inf
<function hamming at 0x000001B4D4928B80> 0.0 -inf
<function hanning at 0x000001B4D4928AF0> 0.0 -inf
<function flattop at 0x000001B4E3591E50> 0.0 -inf
<function barthann at 0x000001B4E35971F0> 0.0 -inf
<function bartlett at 0x000001B4E3591EE0> 0.0 -inf
<function blackman at 0x000001B4E3591CA0> 0.0 -inf
<function blackmanharris at 0x000001B4E3591DC0> 0.0 -inf
<function bohman at 0x000001B4E3591C10> 0.0 -inf
<function boxcar at 0x000001B4E3591A60> 0.0 -inf
<function cosine at 0x000001B4E3597670> 0.0 -inf
<function flattop at 0x000001B4E3591E50> 0.0 -inf
<function hamming at 0x000001B4E3597310> 0.0 -inf
<function hann at 0x000001B4E3591F70> 0.0 -inf
<function nuttall at 0x000001B4E3591D30> 0.0 -inf
<function parzen at 0x000001B4E3591B80> 0.0 -inf
<function triang at 0x000001B4E3591AF0> 0.0 -inf


<class 'numpy.float64'> Machine parameters for float64
---------------------------------------------------------------
precision =  15   resolution = 1.0000000000000001e-15
machep =    -52   eps =        2.2204460492503131e-16
negep =     -53   epsneg =     1.1102230246251565e-16
minexp =  -1022   tiny =       2.2250738585072014e-308
maxexp =   1024   max =        1.7976931348623157e+308
nexp =       11   min =        -max
---------------------------------------------------------------

<function blackman at 0x000001B4D49289D0> 7.771561172376096e-16 -50.192645077942394
<function bartlett at 0x000001B4D4928A60> 1.1102230246251565e-16 -53.0
<function hamming at 0x000001B4D4928B80> 5.551115123125783e-16 -50.67807190511264
<function hanning at 0x000001B4D4928AF0> 5.551115123125783e-16 -50.67807190511264
<function flattop at 0x000001B4E3591E50> 5.551115123125783e-16 -50.67807190511264
<function barthann at 0x000001B4E35971F0> 3.3306690738754696e-16 -51.415037499278846
<function bartlett at 0x000001B4E3591EE0> 1.1102230246251565e-16 -53.0
<function blackman at 0x000001B4E3591CA0> 3.3306690738754696e-16 -51.415037499278846
<function blackmanharris at 0x000001B4E3591DC0> 4.440892098500626e-16 -51.0
<function bohman at 0x000001B4E3591C10> 4.440892098500626e-16 -51.0
<function boxcar at 0x000001B4E3591A60> 0.0 -inf
<function cosine at 0x000001B4E3597670> 3.469446951953614e-16 -51.35614381022528
<function flattop at 0x000001B4E3591E50> 5.551115123125783e-16 -50.67807190511264
<function hamming at 0x000001B4E3597310> 2.220446049250313e-16 -52.0
<function hann at 0x000001B4E3591F70> 2.220446049250313e-16 -52.0
<function nuttall at 0x000001B4E3591D30> 4.440892098500626e-16 -51.0
<function parzen at 0x000001B4E3591B80> 0.0 -inf
<function triang at 0x000001B4E3591AF0> 0.0 -inf

Numpy/Python version information:

1.19.0 3.8.4 (tags/v3.8.4:dfa645a, Jul 13 2020, 16:46:45) [MSC v.1924 64 bit (AMD64)]

[kimstik]

May this issue be classified as issue?

[kimstik]

What is more that "scipy.signal.windows.dpss" is worst window function in terms of symmetricity.

import numpy as np
import scipy.signal.windows

for w in (	scipy.signal.windows.dpss	(4096,5)      ,
		scipy.signal.windows.kaiser	(4096,5*np.pi),	):
	err = np.max(np.abs( w-np.flip(w) ))
	exp = np.abs(np.log2(err))
	print( err, exp )

shows that Kaiser is perfect, but DPSS have error of 9.44e-12

[kimstik]

scipy 1.9.0rc2 python 3.10.3
a lot of useful windows are still have limited precision (non-symmetric):

func                 	error   	exp bits
dpss                 	9.45e-12	36.6
general_cosine       	3.11e-15	48.2
flattop              	5.55e-16	50.7
nuttall              	4.44e-16	51.0
...