def resample(x, num, t=None, axis=0, window=None):
"""
Resample `x` to `num` samples using Fourier method along the given axis.
The resampled signal starts at the same value as `x` but is sampled
with a spacing of ``len(x) / num * (spacing of x)``. Because a
Fourier method is used, the signal is assumed to be periodic.
Parameters
----------
x : array_like
The data to be resampled.
num : int
The number of samples in the resampled signal.
t : array_like, optional
If `t` is given, it is assumed to be the sample positions
associated with the signal data in `x`.
axis : int, optional
The axis of `x` that is resampled. Default is 0.
window : array_like, callable, string, float, or tuple, optional
Specifies the window applied to the signal in the Fourier
domain. See below for details.
Returns
-------
resampled_x or (resampled_x, resampled_t)
Either the resampled array, or, if `t` was given, a tuple
containing the resampled array and the corresponding resampled
positions.
Notes
-----
The argument `window` controls a Fourier-domain window that tapers
the Fourier spectrum before zero-padding to alleviate ringing in
the resampled values for sampled signals you didn't intend to be
interpreted as band-limited.
If `window` is a function, then it is called with a vector of inputs
indicating the frequency bins (i.e. fftfreq(x.shape[axis]) ).
If `window` is an array of the same length as `x.shape[axis]` it is
assumed to be the window to be applied directly in the Fourier
domain (with dc and low-frequency first).
For any other type of `window`, the function `scipy.signal.get_window`
is called to generate the window.
The first sample of the returned vector is the same as the first
sample of the input vector. The spacing between samples is changed
from dx to:
dx * len(x) / num
If `t` is not None, then it represents the old sample positions,
and the new sample positions will be returned as well as the new
samples.
"""
x = asarray(x)
X = fft(x, axis=axis)
Nx = x.shape[axis]
if window is not None:
if callable(window):
W = window(fftfreq(Nx))
elif isinstance(window, ndarray) and window.shape == (Nx, ):
W = window
else:
W = ifftshift(get_window(window, Nx))
newshape = ones(len(x.shape))
newshape[axis] = len(W)
W.shape = newshape
X = X * W
sl = [slice(None)] * len(x.shape)
newshape = list(x.shape)
newshape[axis] = num
N = int(np.minimum(num, Nx))
Y = zeros(newshape, 'D')
sl[axis] = slice(0, (N + 1) / 2)
Y[sl] = X[sl]
sl[axis] = slice(-(N - 1) / 2, None)
Y[sl] = X[sl]
y = ifft(Y, axis=axis) * (float(num) / float(Nx))
if x.dtype.char not in ['F', 'D']:
y = y.real
if t is None:
return y
else:
new_t = arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0]
return y, new_t
def resample(x, num, t=None, axis=0, window=None):
"""
Resample `x` to `num` samples using Fourier method along the given axis.
The resampled signal starts at the same value as `x` but is sampled
with a spacing of ``len(x) / num * (spacing of x)``. Because a
Fourier method is used, the signal is assumed to be periodic.
Parameters
----------
x : array_like
The data to be resampled.
num : int
The number of samples in the resampled signal.
t : array_like, optional
If `t` is given, it is assumed to be the sample positions
associated with the signal data in `x`.
axis : int, optional
The axis of `x` that is resampled. Default is 0.
window : array_like, callable, string, float, or tuple, optional
Specifies the window applied to the signal in the Fourier
domain. See below for details.
Returns
-------
resampled_x or (resampled_x, resampled_t)
Either the resampled array, or, if `t` was given, a tuple
containing the resampled array and the corresponding resampled
positions.
Notes
-----
The argument `window` controls a Fourier-domain window that tapers
the Fourier spectrum before zero-padding to alleviate ringing in
the resampled values for sampled signals you didn't intend to be
interpreted as band-limited.
If `window` is a function, then it is called with a vector of inputs
indicating the frequency bins (i.e. fftfreq(x.shape[axis]) ).
If `window` is an array of the same length as `x.shape[axis]` it is
assumed to be the window to be applied directly in the Fourier
domain (with dc and low-frequency first).
For any other type of `window`, the function `scipy.signal.get_window`
is called to generate the window.
The first sample of the returned vector is the same as the first
sample of the input vector. The spacing between samples is changed
from dx to:
dx * len(x) / num
If `t` is not None, then it represents the old sample positions,
and the new sample positions will be returned as well as the new
samples.
"""
x = asarray(x)
X = fft(x, axis=axis)
Nx = x.shape[axis]
if window is not None:
if callable(window):
W = window(fftfreq(Nx))
elif isinstance(window, ndarray) and window.shape == (Nx,):
W = window
else:
W = ifftshift(get_window(window, Nx))
newshape = ones(len(x.shape))
newshape[axis] = len(W)
W.shape = newshape
X = X * W
sl = [slice(None)] * len(x.shape)
newshape = list(x.shape)
newshape[axis] = num
N = int(np.minimum(num, Nx))
Y = zeros(newshape, 'D')
sl[axis] = slice(0, (N + 1) / 2)
Y[sl] = X[sl]
sl[axis] = slice(-(N - 1) / 2, None)
Y[sl] = X[sl]
y = ifft(Y, axis=axis) * (float(num) / float(Nx))
if x.dtype.char not in ['F', 'D']:
y = y.real
if t is None:
return y
else:
new_t = arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0]
return y, new_t
def resample(x, num, t=None, axis=0, window=None):
"""Resample to num samples using Fourier method along the given axis.
The resampled signal starts at the same value of x but is sampled
with a spacing of len(x) / num * (spacing of x). Because a
Fourier method is used, the signal is assumed periodic.
Window controls a Fourier-domain window that tapers the Fourier
spectrum before zero-padding to alleviate ringing in the resampled
values for sampled signals you didn't intend to be interpreted as
band-limited.
If window is a function, then it is called with a vector of inputs
indicating the frequency bins (i.e. fftfreq(x.shape[axis]) )
If window is an array of the same length as x.shape[axis] it is
assumed to be the window to be applied directly in the Fourier
domain (with dc and low-frequency first).
If window is a string then use the named window. If window is a
float, then it represents a value of beta for a kaiser window. If
window is a tuple, then the first component is a string
representing the window, and the next arguments are parameters for
that window.
Possible windows are:
'flattop' -- 'flat', 'flt'
'boxcar' -- 'ones', 'box'
'triang' -- 'traing', 'tri'
'parzen' -- 'parz', 'par'
'bohman' -- 'bman', 'bmn'
'blackmanharris' -- 'blackharr', 'bkh'
'nuttall', -- 'nutl', 'nut'
'barthann' -- 'brthan', 'bth'
'blackman' -- 'black', 'blk'
'hamming' -- 'hamm', 'ham'
'bartlett' -- 'bart', 'brt'
'hanning' -- 'hann', 'han'
('kaiser', beta) -- 'ksr'
('gaussian', std) -- 'gauss', 'gss'
('general gauss', power, width) -- 'general', 'ggs'
('slepian', width) -- 'slep', 'optimal', 'dss'
The first sample of the returned vector is the same as the first
sample of the input vector, the spacing between samples is changed
from dx to
dx * len(x) / num
If t is not None, then it represents the old sample positions, and the new
sample positions will be returned as well as the new samples.
"""
x = asarray(x)
X = fft(x, axis=axis)
Nx = x.shape[axis]
if window is not None:
if callable(window):
W = window(fftfreq(Nx))
elif isinstance(window, ndarray) and window.shape == (Nx, ):
W = window
else:
W = ifftshift(get_window(window, Nx))
newshape = ones(len(x.shape))
newshape[axis] = len(W)
W.shape = newshape
X = X * W
sl = [slice(None)] * len(x.shape)
newshape = list(x.shape)
newshape[axis] = num
N = int(np.minimum(num, Nx))
Y = zeros(newshape, 'D')
sl[axis] = slice(0, (N + 1) / 2)
Y[sl] = X[sl]
sl[axis] = slice(-(N - 1) / 2, None)
Y[sl] = X[sl]
y = ifft(Y, axis=axis) * (float(num) / float(Nx))
if x.dtype.char not in ['F', 'D']:
y = y.real
if t is None:
return y
else:
new_t = arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0]
return y, new_t
def resample(x,num,t=None,axis=0,window=None):
"""Resample to num samples using Fourier method along the given axis.
The resampled signal starts at the same value of x but is sampled
with a spacing of len(x) / num * (spacing of x). Because a
Fourier method is used, the signal is assumed periodic.
Window controls a Fourier-domain window that tapers the Fourier
spectrum before zero-padding to alleviate ringing in the resampled
values for sampled signals you didn't intend to be interpreted as
band-limited.
If window is a function, then it is called with a vector of inputs
indicating the frequency bins (i.e. fftfreq(x.shape[axis]) )
If window is an array of the same length as x.shape[axis] it is
assumed to be the window to be applied directly in the Fourier
domain (with dc and low-frequency first).
If window is a string then use the named window. If window is a
float, then it represents a value of beta for a kaiser window. If
window is a tuple, then the first component is a string
representing the window, and the next arguments are parameters for
that window.
Possible windows are:
'flattop' -- 'flat', 'flt'
'boxcar' -- 'ones', 'box'
'triang' -- 'traing', 'tri'
'parzen' -- 'parz', 'par'
'bohman' -- 'bman', 'bmn'
'blackmanharris' -- 'blackharr', 'bkh'
'nuttall', -- 'nutl', 'nut'
'barthann' -- 'brthan', 'bth'
'blackman' -- 'black', 'blk'
'hamming' -- 'hamm', 'ham'
'bartlett' -- 'bart', 'brt'
'hanning' -- 'hann', 'han'
('kaiser', beta) -- 'ksr'
('gaussian', std) -- 'gauss', 'gss'
('general gauss', power, width) -- 'general', 'ggs'
('slepian', width) -- 'slep', 'optimal', 'dss'
The first sample of the returned vector is the same as the first
sample of the input vector, the spacing between samples is changed
from dx to
dx * len(x) / num
If t is not None, then it represents the old sample positions, and the new
sample positions will be returned as well as the new samples.
"""
x = asarray(x)
X = fft(x,axis=axis)
Nx = x.shape[axis]
if window is not None:
if callable(window):
W = window(fftfreq(Nx))
elif isinstance(window, ndarray) and window.shape == (Nx,):
W = window
else:
W = ifftshift(get_window(window,Nx))
newshape = ones(len(x.shape))
newshape[axis] = len(W)
W.shape = newshape
X = X*W
sl = [slice(None)]*len(x.shape)
newshape = list(x.shape)
newshape[axis] = num
N = int(np.minimum(num,Nx))
Y = zeros(newshape,'D')
sl[axis] = slice(0,(N+1)/2)
Y[sl] = X[sl]
sl[axis] = slice(-(N-1)/2,None)
Y[sl] = X[sl]
y = ifft(Y,axis=axis)*(float(num)/float(Nx))
if x.dtype.char not in ['F','D']:
y = y.real
if t is None:
return y
else:
new_t = arange(0,num)*(t[1]-t[0])* Nx / float(num) + t[0]
return y, new_t
def welch(x, fs=1.0, window='hanning', nperseg=256, noverlap=None, nfft=None,
detrend='constant', return_onesided=True, scaling='density', axis=-1):
"""
Estimate power spectral density using Welch's method.
Welch's method [1]_ computes an estimate of the power spectral density
by dividing the data into overlapping segments, computing a modified
periodogram for each segment and averaging the periodograms.
Parameters
----------
x : array_like
Time series of measurement values
fs : float, optional
Sampling frequency of the `x` time series in units of Hz. Defaults
to 1.0.
window : str or tuple or array_like, optional
Desired window to use. See `get_window` for a list of windows and
required parameters. If `window` is array_like it will be used
directly as the window and its length will be used for nperseg.
Defaults to 'hanning'.
nperseg : int, optional
Length of each segment. Defaults to 256.
noverlap: int, optional
Number of points to overlap between segments. If None,
``noverlap = nperseg / 2``. Defaults to None.
nfft : int, optional
Length of the FFT used, if a zero padded FFT is desired. If None,
the FFT length is `nperseg`. Defaults to None.
detrend : str or function, optional
Specifies how to detrend each segment. If `detrend` is a string,
it is passed as the ``type`` argument to `detrend`. If it is a
function, it takes a segment and returns a detrended segment.
Defaults to 'constant'.
return_onesided : bool, optional
If True, return a one-sided spectrum for real data. If False return
a two-sided spectrum. Note that for complex data, a two-sided
spectrum is always returned.
scaling : { 'density', 'spectrum' }, optional
Selects between computing the power spectral density ('density')
where Pxx has units of V**2/Hz if x is measured in V and computing
the power spectrum ('spectrum') where Pxx has units of V**2 if x is
measured in V. Defaults to 'density'.
axis : int, optional
Axis along which the periodogram is computed; the default is over
the last axis (i.e. ``axis=-1``).
Returns
-------
f : ndarray
Array of sample frequencies.
Pxx : ndarray
Power spectral density or power spectrum of x.
See Also
--------
periodogram: Simple, optionally modified periodogram
lombscargle: Lomb-Scargle periodogram for unevenly sampled data
Notes
-----
An appropriate amount of overlap will depend on the choice of window
and on your requirements. For the default 'hanning' window an
overlap of 50% is a reasonable trade off between accurately estimating
the signal power, while not over counting any of the data. Narrower
windows may require a larger overlap.
If `noverlap` is 0, this method is equivalent to Bartlett's method [2]_.
.. versionadded:: 0.12.0
References
----------
.. [1] P. Welch, "The use of the fast Fourier transform for the
estimation of power spectra: A method based on time averaging
over short, modified periodograms", IEEE Trans. Audio
Electroacoust. vol. 15, pp. 70-73, 1967.
.. [2] M.S. Bartlett, "Periodogram Analysis and Continuous Spectra",
Biometrika, vol. 37, pp. 1-16, 1950.
Examples
--------
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
Generate a test signal, a 2 Vrms sine wave at 1234 Hz, corrupted by
0.001 V**2/Hz of white noise sampled at 10 kHz.
>>> fs = 10e3
>>> N = 1e5
>>> amp = 2*np.sqrt(2)
>>> freq = 1234.0
>>> noise_power = 0.001 * fs / 2
>>> time = np.arange(N) / fs
>>> x = amp*np.sin(2*np.pi*freq*time)
>>> x += np.random.normal(scale=np.sqrt(noise_power), size=time.shape)
Compute and plot the power spectral density.
>>> f, Pxx_den = signal.welch(x, fs, 1024)
>>> plt.semilogy(f, Pxx_den)
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('PSD [V/sqrt(Hz)]')
>>> plt.show()
If we average the last half of the spectral density, to exclude the
peak, we can recover the noise power on the signal.
>>> np.mean(Pxx_den[256:])
0.0009924865443739191
Now compute and plot the power spectrum.
>>> f, Pxx_spec = signal.welch(x, fs, 'flattop', 1024, scaling='spectrum')
>>> plt.figure()
>>> plt.semilogy(f, np.sqrt(Pxx_spec))
>>> plt.xlabel('frequency [Hz]')
>>> plt.ylabel('Linear spectrum [V RMS]')
>>> plt.show()
The peak height in the power spectrum is an estimate of the RMS amplitude.
>>> np.sqrt(Pxx_spec.max())
2.0077340678640727
"""
x = np.asarray(x)
if x.size == 0:
return np.empty(x.shape), np.empty(x.shape)
if axis != -1:
x = np.rollaxis(x, axis, len(x.shape))
if x.shape[-1] < nperseg:
warnings.warn('nperseg = %d, is greater than x.shape[%d] = %d, using '
'nperseg = x.shape[%d]'
% (nperseg, axis, x.shape[axis], axis))
nperseg = x.shape[-1]
if isinstance(window, basestring) or type(window) is tuple:
win = get_window(window, nperseg)
else:
win = np.asarray(window)
if len(win.shape) != 1:
raise ValueError('window must be 1-D')
if win.shape[0] > x.shape[-1]:
raise ValueError('window is longer than x.')
nperseg = win.shape[0]
if scaling == 'density':
scale = 1.0 / (fs * (win*win).sum())
elif scaling == 'spectrum':
scale = 1.0 / win.sum()**2
else:
raise ValueError('Unknown scaling: %r' % scaling)
if noverlap is None:
noverlap = nperseg // 2
elif noverlap >= nperseg:
raise ValueError('noverlap must be less than nperseg.')
if nfft is None:
nfft = nperseg
elif nfft < nperseg:
raise ValueError('nfft must be greater than or equal to nperseg.')
if not hasattr(detrend, '__call__'):
detrend_func = lambda seg: signaltools.detrend(seg, type=detrend)
elif axis != -1:
# Wrap this function so that it receives a shape that it could
# reasonably expect to receive.
def detrend_func(seg):
seg = np.rollaxis(seg, -1, axis)
seg = detrend(seg)
return np.rollaxis(seg, axis, len(seg.shape))
else:
detrend_func = detrend
step = nperseg - noverlap
indices = np.arange(0, x.shape[-1]-nperseg+1, step)
if np.isrealobj(x) and return_onesided:
outshape = list(x.shape)
if nfft % 2 == 0: # even
outshape[-1] = nfft // 2 + 1
Pxx = np.empty(outshape, x.dtype)
for k, ind in enumerate(indices):
x_dt = detrend_func(x[..., ind:ind+nperseg])
xft = fftpack.rfft(x_dt*win, nfft)
# fftpack.rfft returns the positive frequency part of the fft
# as real values, packed r r i r i r i ...
# this indexing is to extract the matching real and imaginary
# parts, while also handling the pure real zero and nyquist
# frequencies.
if k == 0:
Pxx[..., (0,-1)] = xft[..., (0,-1)]**2
Pxx[..., 1:-1] = xft[..., 1:-1:2]**2 + xft[..., 2::2]**2
else:
Pxx *= k/(k+1.0)
Pxx[..., (0,-1)] += xft[..., (0,-1)]**2 / (k+1.0)
Pxx[..., 1:-1] += (xft[..., 1:-1:2]**2 + xft[..., 2::2]**2) \
/ (k+1.0)
else: # odd
outshape[-1] = (nfft+1) // 2
Pxx = np.empty(outshape, x.dtype)
for k, ind in enumerate(indices):
x_dt = detrend_func(x[..., ind:ind+nperseg])
xft = fftpack.rfft(x_dt*win, nfft)
if k == 0:
Pxx[..., 0] = xft[..., 0]**2
Pxx[..., 1:] = xft[..., 1::2]**2 + xft[..., 2::2]**2
else:
Pxx *= k/(k+1.0)
Pxx[..., 0] += xft[..., 0]**2 / (k+1)
Pxx[..., 1:] += (xft[..., 1::2]**2 + xft[..., 2::2]**2) \
/ (k+1.0)
Pxx[..., 1:-1] *= 2*scale
Pxx[..., (0,-1)] *= scale
f = np.arange(Pxx.shape[-1]) * (fs/nfft)
else:
for k, ind in enumerate(indices):
x_dt = detrend_func(x[..., ind:ind+nperseg])
xft = fftpack.fft(x_dt*win, nfft)
if k == 0:
Pxx = (xft * xft.conj()).real
else:
Pxx *= k/(k+1.0)
Pxx += (xft * xft.conj()).real / (k+1.0)
Pxx *= scale
f = fftpack.fftfreq(nfft, 1.0/fs)
if axis != -1:
Pxx = np.rollaxis(Pxx, -1, axis)
return f, Pxx
