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"""The suite of window functions."""
import numpy as np
from scipy import special, linalg
from scipy.fftpack import fft
__all__ = ['boxcar', 'triang', 'parzen', 'bohman', 'blackman', 'nuttall',
'blackmanharris', 'flattop', 'bartlett', 'hanning', 'barthann',
'hamming', 'kaiser', 'gaussian', 'general_gaussian', 'chebwin',
'slepian', 'hann', 'get_window']
def boxcar(M, sym=True):
"""The M-point boxcar window.
"""
return np.ones(M, float)
def triang(M, sym=True):
"""The M-point triangular window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(1, int((M + 1) / 2) + 1)
if M % 2 == 0:
w = (2 * n - 1.0) / M
w = np.r_[w, w[::-1]]
else:
w = 2 * n / (M + 1.0)
w = np.r_[w, w[-2::-1]]
if not sym and not odd:
w = w[:-1]
return w
def parzen(M, sym=True):
"""The M-point Parzen window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(-(M - 1) / 2.0, (M - 1) / 2.0 + 0.5, 1.0)
na = np.extract(n < -(M - 1) / 4.0, n)
nb = np.extract(abs(n) <= (M - 1) / 4.0, n)
wa = 2 * (1 - np.abs(na) / (M / 2.0)) ** 3.0
wb = (1 - 6 * (np.abs(nb) / (M / 2.0)) ** 2.0 +
6 * (np.abs(nb) / (M / 2.0)) ** 3.0)
w = np.r_[wa, wb, wa[::-1]]
if not sym and not odd:
w = w[:-1]
return w
def bohman(M, sym=True):
"""The M-point Bohman window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
fac = np.abs(np.linspace(-1, 1, M)[1:-1])
w = (1 - fac) * np.cos(np.pi * fac) + 1.0 / np.pi * np.sin(np.pi * fac)
w = np.r_[0, w, 0]
if not sym and not odd:
w = w[:-1]
return w
def blackman(M, sym=True):
"""The M-point Blackman window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
w = (0.42 - 0.5 * np.cos(2.0 * np.pi * n / (M - 1)) +
0.08 * np.cos(4.0 * np.pi * n / (M - 1)))
if not sym and not odd:
w = w[:-1]
return w
def nuttall(M, sym=True):
"""A minimum 4-term Blackman-Harris window according to Nuttall.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
a = [0.3635819, 0.4891775, 0.1365995, 0.0106411]
n = np.arange(0, M)
fac = n * 2 * np.pi / (M - 1.0)
w = (a[0] - a[1] * np.cos(fac) +
a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac))
if not sym and not odd:
w = w[:-1]
return w
def blackmanharris(M, sym=True):
"""The M-point minimum 4-term Blackman-Harris window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
a = [0.35875, 0.48829, 0.14128, 0.01168]
n = np.arange(0, M)
fac = n * 2 * np.pi / (M - 1.0)
w = (a[0] - a[1] * np.cos(fac) +
a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac))
if not sym and not odd:
w = w[:-1]
return w
def flattop(M, sym=True):
"""The M-point Flat top window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
a = [0.2156, 0.4160, 0.2781, 0.0836, 0.0069]
n = np.arange(0, M)
fac = n * 2 * np.pi / (M - 1.0)
w = (a[0] - a[1] * np.cos(fac) +
a[2] * np.cos(2 * fac) - a[3] * np.cos(3 * fac) +
a[4] * np.cos(4 * fac))
if not sym and not odd:
w = w[:-1]
return w
def bartlett(M, sym=True):
"""The M-point Bartlett window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
w = np.where(np.less_equal(n, (M - 1) / 2.0),
2.0 * n / (M - 1), 2.0 - 2.0 * n / (M - 1))
if not sym and not odd:
w = w[:-1]
return w
def hanning(M, sym=True):
"""The M-point Hanning window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
w = 0.5 - 0.5 * np.cos(2.0 * np.pi * n / (M - 1))
if not sym and not odd:
w = w[:-1]
return w
hann = hanning
def barthann(M, sym=True):
"""Return the M-point modified Bartlett-Hann window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
fac = np.abs(n / (M - 1.0) - 0.5)
w = 0.62 - 0.48 * fac + 0.38 * np.cos(2 * np.pi * fac)
if not sym and not odd:
w = w[:-1]
return w
def hamming(M, sym=True):
"""The M-point Hamming window.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
w = 0.54 - 0.46 * np.cos(2.0 * np.pi * n / (M - 1))
if not sym and not odd:
w = w[:-1]
return w
def kaiser(M, beta, sym=True):
"""Return a Kaiser window of length M with shape parameter beta.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M)
alpha = (M - 1) / 2.0
w = (special.i0(beta * np.sqrt(1 - ((n - alpha) / alpha) ** 2.0)) /
special.i0(beta))
if not sym and not odd:
w = w[:-1]
return w
def gaussian(M, std, sym=True):
"""Return a Gaussian window of length M with standard-deviation std.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M) - (M - 1.0) / 2.0
sig2 = 2 * std * std
w = np.exp(-n ** 2 / sig2)
if not sym and not odd:
w = w[:-1]
return w
def general_gaussian(M, p, sig, sym=True):
"""Return a window with a generalized Gaussian shape.
The Gaussian shape is defined as ``exp(-0.5*(x/sig)**(2*p))``, the
half-power point is at ``(2*log(2)))**(1/(2*p)) * sig``.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
n = np.arange(0, M) - (M - 1.0) / 2.0
w = np.exp(-0.5 * (n / sig) ** (2 * p))
if not sym and not odd:
w = w[:-1]
return w
# `chebwin` contributed by Kumar Appaiah.
def chebwin(M, at, sym=True):
"""Dolph-Chebyshev window.
Parameters
----------
M : int
Window size.
at : float
Attenuation (in dB).
sym : bool
Generates symmetric window if True.
"""
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
# compute the parameter beta
order = M - 1.0
beta = np.cosh(1.0 / order * np.arccosh(10 ** (np.abs(at) / 20.)))
k = np.r_[0:M] * 1.0
x = beta * np.cos(np.pi * k / M)
# Find the window's DFT coefficients
# Use analytic definition of Chebyshev polynomial instead of expansion
# from scipy.special. Using the expansion in scipy.special leads to errors.
p = np.zeros(x.shape)
p[x > 1] = np.cosh(order * np.arccosh(x[x > 1]))
p[x < -1] = (1 - 2 * (order % 2)) * np.cosh(order * np.arccosh(-x[x < -1]))
p[np.abs(x) <= 1] = np.cos(order * np.arccos(x[np.abs(x) <= 1]))
# Appropriate IDFT and filling up
# depending on even/odd M
if M % 2:
w = np.real(fft(p))
n = (M + 1) / 2
w = w[:n] / w[0]
w = np.concatenate((w[n - 1:0:-1], w))
else:
p = p * np.exp(1.j * np.pi / M * np.r_[0:M])
w = np.real(fft(p))
n = M / 2 + 1
w = w / w[1]
w = np.concatenate((w[n - 1:0:-1], w[1:n]))
if not sym and not odd:
w = w[:-1]
return w
def slepian(M, width, sym=True):
"""Return the M-point slepian window.
"""
if (M * width > 27.38):
raise ValueError("Cannot reliably obtain slepian sequences for"
" M*width > 27.38.")
if M < 1:
return np.array([])
if M == 1:
return np.ones(1, 'd')
odd = M % 2
if not sym and not odd:
M = M + 1
twoF = width / 2.0
alpha = (M - 1) / 2.0
m = np.arange(0, M) - alpha
n = m[:, np.newaxis]
k = m[np.newaxis, :]
AF = twoF * special.sinc(twoF * (n - k))
[lam, vec] = linalg.eig(AF)
ind = np.argmax(abs(lam), axis=-1)
w = np.abs(vec[:, ind])
w = w / max(w)
if not sym and not odd:
w = w[:-1]
return w
def get_window(window, Nx, fftbins=True):
"""
Return a window of length `Nx` and type `window`.
Parameters
----------
window : string, float, or tuple
The type of window to create. See below for more details.
Nx : int
The number of samples in the window.
fftbins : bool, optional
If True, create a "periodic" window ready to use with ifftshift
and be multiplied by the result of an fft (SEE ALSO fftfreq).
Notes
-----
Window types:
boxcar, triang, blackman, hamming, hanning, bartlett,
parzen, bohman, blackmanharris, nuttall, barthann,
kaiser (needs beta), gaussian (needs std),
general_gaussian (needs power, width),
slepian (needs width), chebwin (needs attenuation)
If the window requires no parameters, then `window` can be a string.
If the window requires parameters, then `window` must be a tuple
with the first argument the string name of the window, and the next
arguments the needed parameters.
If `window` is a floating point number, it is interpreted as the beta
parameter of the kaiser window.
Each of the window types listed above is also the name of
a function that can be called directly to create a window of
that type.
Examples
--------
>>> get_window('triang', 7)
array([ 0.25, 0.5 , 0.75, 1. , 0.75, 0.5 , 0.25])
>>> get_window(('kaiser', 4.0), 9)
array([ 0.08848053, 0.32578323, 0.63343178, 0.89640418, 1. ,
0.89640418, 0.63343178, 0.32578323, 0.08848053])
>>> get_window(4.0, 9)
array([ 0.08848053, 0.32578323, 0.63343178, 0.89640418, 1. ,
0.89640418, 0.63343178, 0.32578323, 0.08848053])
"""
sym = not fftbins
try:
beta = float(window)
except (TypeError, ValueError):
args = ()
if isinstance(window, tuple):
winstr = window[0]
if len(window) > 1:
args = window[1:]
elif isinstance(window, str):
if window in ['kaiser', 'ksr', 'gaussian', 'gauss', 'gss',
'general gaussian', 'general_gaussian',
'general gauss', 'general_gauss', 'ggs',
'slepian', 'optimal', 'slep', 'dss',
'chebwin', 'cheb']:
raise ValueError("The '" + window + "' window needs one or "
"more parameters -- pass a tuple.")
else:
winstr = window
if winstr in ['blackman', 'black', 'blk']:
winfunc = blackman
elif winstr in ['triangle', 'triang', 'tri']:
winfunc = triang
elif winstr in ['hamming', 'hamm', 'ham']:
winfunc = hamming
elif winstr in ['bartlett', 'bart', 'brt']:
winfunc = bartlett
elif winstr in ['hanning', 'hann', 'han']:
winfunc = hanning
elif winstr in ['blackmanharris', 'blackharr', 'bkh']:
winfunc = blackmanharris
elif winstr in ['parzen', 'parz', 'par']:
winfunc = parzen
elif winstr in ['bohman', 'bman', 'bmn']:
winfunc = bohman
elif winstr in ['nuttall', 'nutl', 'nut']:
winfunc = nuttall
elif winstr in ['barthann', 'brthan', 'bth']:
winfunc = barthann
elif winstr in ['flattop', 'flat', 'flt']:
winfunc = flattop
elif winstr in ['kaiser', 'ksr']:
winfunc = kaiser
elif winstr in ['gaussian', 'gauss', 'gss']:
winfunc = gaussian
elif winstr in ['general gaussian', 'general_gaussian',
'general gauss', 'general_gauss', 'ggs']:
winfunc = general_gaussian
elif winstr in ['boxcar', 'box', 'ones']:
winfunc = boxcar
elif winstr in ['slepian', 'slep', 'optimal', 'dss']:
winfunc = slepian
elif winstr in ['chebwin', 'cheb']:
winfunc = chebwin
else:
raise ValueError("Unknown window type.")
params = (Nx,) + args + (sym,)
else:
winfunc = kaiser
params = (Nx, beta, sym)
return winfunc(*params)
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