# -*- coding: utf-8 -*-
#
# This file is part of the pyFDA project hosted at https://github.com/chipmuenk/pyfda
#
# Copyright © pyFDA Project Contributors
# Licensed under the terms of the MIT License
# (see file LICENSE in root directory for details)
import numpy as np
import scipy.signal as sig
import scipy
# import logging
# logger = logging.getLogger(__name__)
"""
Reference dictionary with available FFT windows, their function names and
their properties.
When the function name `fn_name` is just a string, it is taken from
`scipy.signal.windows`, otherwise it has to be fully qualified name.
"""
bartlett_info =\
'''<span>
The Bartlett and triangular windows are similar, except that the end
point(s) of the Bartlett window are at zero. Its side lobes fall off with
12 dB/oct. or 40 dB/dec., the min. side lobe suppression is 26 dB.<br /><br />
It can be constructed as the convolution of two rectangular windows,
hence, its Fourier transform is the product of two (periodic) sinc
functions, decaying twice as fast as the spectrum of a rectangular window.
</span>'''
rectangular_info =\
'''<span>
Boxcar or Rectangular window, best suited for analyzing <br />
a) <b>coherent signals</b>, i.e. where the window length is an integer number of the
signal period.<br />
b) <b>impulses</b> where the signal length is shorter than the window
length (e.g. for the impulse response of a FIR filter).<br />
c) <b>noisy sinusoids with low SNR</b> because this window has the best equivalent
noise bandwidth.<br />
It should not be used for other signals as it has the worst sidelobe suppression
(13 dB) of all windows. Sidelobes decay with only 6 dB/oct. or 20 dB / dec.
<br /><br />
When used for FIR filter design, a filter with the least square error
is returned, created by truncating the sinc-law frequency response after
<i>N</i> terms and transforming back to the time domain. It has the sharpest
transition of all windowed FIR filters but the worst stop band attenuation
and a large ripple in the passband. Attenuation in the stop band grows with
only 6 dB/oct. or 20 dB / dec.
</span>'''
all_wins_dict_ref = {
'boxcar': {
'app': ['fir', 'spec', 'stft', 'all'],
'disp_name': 'Boxcar',
'fn_name': 'boxcar',
'id': 'boxcar',
'info': rectangular_info,
'par': [],
'par_val': []
},
'rectangular': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Rectangular',
'fn_name': 'boxcar',
'id': 'rectangular',
'info': rectangular_info,
'par': [],
'par_val': []
},
'barthann': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Barthann',
'fn_name': 'barthann',
'id': 'barthann',
'info':
'''<span>
The modified Bartlett-Hann window is a weighted combination of Bartlett
(triangular) and Hann window and has a similar mainlobe width as those two.
Sidelobes are asymptotically decaying, near sidelobes have a lower level
than Bartlett and Hann windows, far sidelobes have lower levels than
Bartlett and Hamming windows.
</span>''',
'par': [],
'par_val': []
},
'bartlett': {
'app': ['fir', 'spec', 'stft', 'all'],
'disp_name': 'Bartlett',
'fn_name': 'bartlett',
'id': 'bartlett',
'info': bartlett_info,
'par': [],
'par_val': []
},
'blackman': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Blackman',
'fn_name': 'blackman',
'id': 'blackman',
'info':
'''<span>
The Blackman window is used for both FIR filter design and spectral analysis.
Compared to Hann and Hamming window, it has a wider main lobe (less sharp
transition between pass and stop band / worse frequency resolution)
and lower sidelobe levels (improved stopband rejection / less leakage of
high-frequency interferers).<br /><br />
The Blackman window is a three term cosine window with coefficients of
a_0 = 0.42, a1 = 0.5, a2 = 0.08. The maximum sidelobe level is -57 dB,
sidelobes have a fall-off rate of -18 dB/dec.
Its main lobe width is 12 π / <i>N</i>.
</span>''',
'par': [],
'par_val': []
},
'blackmanharris': {
'app': ['fir', 'spec', 'all'],
'disp_name':'Blackman-Harris',
'fn_name': 'pyfda.libs.pyfda_fft_windows_lib.blackmanharris',
'id': 'blackmanharris',
'info':
'''<span>
The minimum 4-term Blackman-Harris window gives an excellent
constant side-lobe suppression of more than 90 dB while keeping a
reasonably narrow main lobe.<br /><br />
5-, 7- and 9-term Blackman-Harris windows achieve side-lobe suppressions
of up to 125, 180 and 230 dB.
</span>''',
'par': [{
'name': 'L', 'name_tex': r'$L$', 'list': ['4', '5', '7', '9'],
'tooltip': '<span>Number of cosine terms</span>'}],
'par_val': ['4']
},
'bohman': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Bohman',
'fn_name': 'bohman',
'id': 'bohman',
'info':
'''<span>Sidelobes of the Bohman window drop with 24 dB/oct.
</span>''',
'par': [],
'par_val': []
},
'cosine': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Cosine',
'fn_name': 'cosine',
'id': 'cosine',
'info':
'''<span>
The window is half a cosine period, shifted by π/2.
For that reason it is also known as "half-cosine" or "sine" window.
</span>''',
'par': [],
'par_val': []
},
'chebwin': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Dolph-Chebyshev',
'fn_name': 'chebwin',
'id': 'chebwin',
'info':
'''<span>
This one-parameter family of window allows choosing the (constant) sidelobe
attenuation <i>a</i>. It gives the narrowest main lobe width for
a given order <i>M</i> and sidelobe equiripple attenuation <i>a</i>,
using Chebyshev polynomials.
</span>''',
'par': [{
'name': 'a', 'name_tex': r'$a$', 'min': 45, 'max': 300,
'tooltip': '<span>Side lobe attenuation in dB.</span>'}],
'par_val': [80]
},
'dpss': {
'app': ['spec', 'stft', 'fir', 'all'],
'disp_name': 'DPSS',
'fn_name': 'dpss',
'id': 'dpss',
'info':
'''<span>
Digital Prolate Spheroidal Sequences (DPSS) (or Slepian
sequences) are often used in multitaper power spectral density
estimation. The first window in the sequence can be used to maximize
the energy concentration in the main lobe, and is also called the
Slepian window. <br /><br />
The Kaiser window is an easier to calculate approximation for the
Slepian window with β = π <i>NW</i> .
</span>''',
'par': [{
'name': 'NW', 'name_tex': r'$NW$', 'min': 0, 'max': 100,
'tooltip':
'<span>Standardized half bandwidth, <i>NW = BW · N</i> / 2</span>'
}],
'par_val': [3]
},
#
'flattop': {
'app': ['spec'],
'disp_name': 'Flattop',
'fn_name': 'flattop',
'id': 'flattop',
'info':
'''<span>
Flattop windows are a family of windows with very low amplitude error /
passband ripple used frequently in spectrum analyzers and other measurement
equipment. Their 3dB bandwidth is approx. 2.5 times larger than that of a
Hann window.
They are 2nd ... 4th-order (3 ... 5 term) cosine windows.
</span>''',
'par': [],
'par_val': []
},
'general_gaussian': {
'app': ['spec', 'all'],
'disp_name': 'General Gaussian',
'fn_name': 'general_gaussian',
'id': 'general_gaussian',
'info':
'''<span>
General Gaussian window, <i>p</i> = 1 yields a Gaussian window,
<i>p</i> = 0.5 yields the shape of a Laplace distribution.
</span>''',
'par': [{
'name': 'p', 'name_tex': r'$p$', 'min': 0, 'max': 20,
'tooltip': '<span>Shape parameter p</span>'
},
{
'name': 'σ', 'name_tex': r'$\sigma$', 'min': 0,
'max': 100, 'tooltip': '<span>Standard deviation σ</span>'
}],
'par_val': [1.5, 5]
},
'gaussian': {
'app': ['spec', 'all'],
'disp_name': 'Gauss',
'fn_name': 'gaussian',
'id': 'gaussian',
'info':
'''<span>
The Gaussian function provides the minimum time-bandwidth product and
hence a great compromise between localization in the time and frequency
domain.
A higher standard deviation creates a wider Gaussian distribution in the
time domain, yielding a narrower main lobe and higher sidelobe levels in
the frequency domain and vice versa.
When used as a window, it has to be truncated and no longer
provides the minimum time-bandwidth product. For σ > 3, the
truncation error is sufficiently small for most applications.
σ = (N -1)/(2a)
</span>''',
'par': [{
'name': 'σ', 'name_tex': r'$\sigma$', 'min': 0,
'max': 100, 'tooltip': '<span>Standard deviation σ</span>'}],
'par_val': [5]
},
'hamming': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Hamming',
'fn_name': 'hamming',
'id': 'hamming',
'info':
'''<span>
The Hamming Window is used for both FIR filter design and spectral analysis.
It has been optimized for suppression of
the first side lobe. Compared to the Hann window, this comes at
the cost of a worse (constant) level of higher side lobes.<br /><br />
Mathematically, it is a two-term raised cosine
window with non-zero endpoints (DC-offset).
</span>''',
'par': [],
'par_val': []
},
'hann': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Hann',
'fn_name': 'hann',
'id': 'hann',
'info':
'''<span>
The Hann (or, falsely, "Hanning") window is smooth at the
edges. In the frequency domain this corresponds to side-lobes falling
off with a rate of 18 dB/oct or 30 dB/dec. The first sidelobe is quite
high (-32 dB). It is a good compromise for many applications, especially
when higher frequency components need to be suppressed.
<br /><br />
Mathematically, it is the most simple two-term raised cosine
or squared sine window.
</span>''',
'par': [],
'par_val': []
},
'kaiser': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Kaiser',
'fn_name': 'kaiser',
'id': 'kaiser',
'info':
'''<span>
The Kaiser or Kaiser-Bessel window is a family of one parameter windows
where the β parameter provides an easy control over the trade-off
between main-lobe width and side-lobe level. High values give low
side-lobe levels but a wide main-lobe width and vice versa.<br><br>
It is a very good approximation to the
Digital Prolate Spheroidal Sequence (DPSS), or Slepian window,
which maximizes the energy in the main lobe of the window relative
to the total energy but is easier to compute.
</span>''',
'par': [{
'name': 'β', 'name_tex': r'$\beta$', 'min': 0, 'max': 30,
'tooltip':
'<span>Shape parameter; lower values reduce main lobe width, '
'higher values reduce side lobe level, typ. in the range '
'5 ... 20.</span>'}],
'par_val': [10]
},
'nuttall': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Nuttall',
'fn_name': 'nuttall',
'id': 'nuttall',
'info':
'''<span>
A nuttall window is very similar to a four-term Blackman-Harris window.
It gives an excellent constant side-lobe suppression of more
than 90 dB while keeping a reasonably narrow main lobe.<br /><br /></span>''',
'par': [],
'par_val': []
},
'parzen': {
'app': ['fir', 'spec', 'all'],
'disp_name': 'Parzen',
'fn_name': 'parzen',
'id': 'parzen',
'info':
'''<span>
The Parzen window is a 4th order B-spline window whose sidelobes
fall off with -24 dB/oct.
<br/ ><br />
It can be constructed by convolving a rectangular window four times
(or multiplying its frequency response four times).
<br /><br />
See also: Boxcar and Triangular / Bartlett windows.
</span>''',
'par': [],
'par_val': []
},
'triang': {
'app': ['fir', 'spec', 'stft', 'all'],
'disp_name': 'Triangular',
'fn_name': 'triang',
'id': 'triang',
'info': bartlett_info,
'par': [],
'par_val': []
},
'tukey': {
'app': ['spec', 'stft'],
'disp_name': 'Tukey',
'fn_name': 'tukey',
'id': 'tukey',
'info':
'''<span>
Also known as "tapered cosine window", this window is constructed from a
rectangular window whose edges are tapered with cosine functions. The shape
factor α defines the fraction of the window inside the cosine tapered
region. Hence, α = 0 returns a rectangular window, α = 1 a
Hann window.
<br /><br />
Tukey windows are used a.o. for analyzing transient data containing short
bursts. It is the default window for scipy.signal.spectrogram with
α = 0.25). Amplitudes of transient events are less likely to be
altered by this window than e.g. by a Hann window.
</span>''',
'par': [{
'name': 'α', 'name_tex': r'$\alpha$', 'min': 0, 'max': 1,
'tooltip': '<span>Shape parameter (see window tool tipp)</span>'}],
'par_val': [0.25]
},
'ultraspherical': {
# for some reason, this window crashes pyfda
'app': [],
'disp_name': 'Ultraspherical',
'fn_name': 'pyfda.libs.pyfda_fft_windows_lib.ultraspherical',
'id': 'ultraspherical',
'info':'''<span>
Ultraspherical or Gegenbauer window, <i>μ</i> = 1 yields a Gaussian
window, <i>μ</i> = 0.5 yields the shape of a Laplace distribution.
This is a three-parameter window (<i>N</i>, μ,x_0).
</span>''',
'par': [{
'name': 'μ', 'name_tex': r'$\mu$', 'min': -0.5, 'max': 10,
'tooltip': '<span>Shape parameter μ or α</span>'
},
{
'name': 'x0', 'name_tex': r'$x_0$', 'min': -10, 'max': 10,
'tooltip': '<span>Amplitude</span>'}
],
'par_val': [0.5, 1]
}
}
# -------------------------------------------------------------------------------------
[docs]
def blackmanharris(N, L, sym):
if L == '4':
return sig.windows.blackmanharris(N, sym)
elif L == '5':
""" 5 Term Cosine, 125.427 dB, NBW 2.21535 bins, 9.81016 dB gain """
a = [3.232153788877343e-001,
-4.714921439576260e-001,
1.755341299601972e-001,
-2.849699010614994e-002,
1.261357088292677e-003]
elif L == '7':
""" 7 Term Cosine, 180.468 dB, NBW 2.63025 bins, 11.33355 dB gain"""
a = [2.712203605850388e-001,
-4.334446123274422e-001,
2.180041228929303e-001,
-6.578534329560609e-002,
1.076186730534183e-002,
-7.700127105808265e-004,
1.368088305992921e-005]
elif L == '9':
""" 9 Term Cosine, 234.734 dB, NBW 2.98588 bins, 12.45267 dB gain"""
a = [2.384331152777942e-001,
-4.005545348643820e-001,
2.358242530472107e-001,
-9.527918858383112e-002,
2.537395516617152e-002,
-4.152432907505835e-003,
3.685604163298180e-004,
-1.384355593917030e-005,
1.161808358932861e-007]
else:
raise Exception(
"Undefined parameter '{0}' for order L!"
.format(L))
return
return calc_cosine_window(N, sym, a)
[docs]
def calc_cosine_window(N, sym, a):
"""
Return window based on cosine functions with amplitudes specified
by the list `a`.
"""
if sym:
L = N-1
else:
L = N
x = np.arange(N) * 2 * np.pi / L
win = a[0]
for k in range(1, len(a)):
win += a[k] * np.cos(k*x)
return win
[docs]
def ultraspherical(N, alpha=0.5, x_0=1, sym=True):
""" The window does not work yet!
More info: https://www.recordingblogs.com/wiki/ultraspherical-window
and https://www.ece.uvic.ca/~andreas/RLectures/UltraSpherWinJASP.pdf
"""
if sym:
L = N-1
else:
L = N
# x = np.arange(N) * np.pi / (N)
geg_ev = scipy.special.eval_gegenbauer
w0 = geg_ev(N, alpha, x_0)
w = np.zeros(N)
# a = 2
# for n in range(5 + 1):
# x = np.linspace(-1.1, 1.1, 5001)
# y = eval_gegenbauer(n, a, x)
# plt.plot(x, y, label=r'$C_{%i}^{(2)}$' % n, zorder=-n)
# plt.ylim((-10,10))
for n in range(0, N):
w[n] = w0
for k in range(1, N//2+1):
w[n] += geg_ev(N, alpha, x_0 * np.cos(k*np.pi/(N+1)))\
* np.cos(2*n*np.pi*k/(N+1))
# rtn += np.cos(x*k)
# w = geg_ev(N-1, alpha, x_0 * np.cos(x))
# logger.error(W[0].dtype, len(W))
# W = np.abs(fft.ifft(w))
# logger.error(type(w[0].dtype), len(w))
return w
[docs]
class UserWindows(object):
def __init__(self, parent):
super(UserWindows, self).__init__(parent)
# =======
# see also:
# https://www.electronicdesign.com/technologies/analog/article/21798689/choose-the-right-fft-window-function-when-evaluating-precision-adcs
# https://github.com/capitanov/blackman_harris_win
# https://en.m.wikipedia.org/wiki/Window_function
# https://www.dsprelated.com/freebooks/sasp/Blackman_Harris_Window_Family.html
# https://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/46092/versions/3/previews/coswin.m/index.html
# Refs:
# "A Family of Cosine-Sum Windows for High-Resolution Measurements"
# Hans-Helge Albrecht
# Physikalisch-Technische Bundesanstalt
# Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP '01).
# 2001 IEEE International Conference on (Volume:5 )
# pgs. 3081-3084
# "Tailoring of Minimum Sidelobe Cosine-Sum Windows for High-Resolution Measurements"
# Hans-Helge Albrecht
# Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany
# The Open Signal Processing Journal, 2010, 3, pp. 20-29
# Heinzel G. et al.,
# "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT),
# including a comprehensive list of window functions and some new flat-top windows",
# February 15, 2002
# https://holometer.fnal.gov/GH_FFT.pdf
