def hilbert_transform(X, rate, filters=None, phase=None, X_fft_h=None):
"""
Apply bandpass filtering with Hilbert transform using
a prespecified set of filters.
Parameters
----------
X : ndarray (n_channels, n_time)
Input data, dimensions
rate : float
Number of samples per second.
filters : filter or list of filters (optional)
One or more bandpass filters
Returns
-------
Xh : ndarray, complex
Bandpassed analytic signal
"""
if not isinstance(filters, list):
filters = [filters]
time = X.shape[-1]
freq = fftfreq(time, 1. / rate)
Xh = np.zeros((len(filters),) + X.shape, dtype=np.complex)
if X_fft_h is None:
# Heavyside filter
h = np.zeros(len(freq))
h[freq > 0] = 2.
h[0] = 1.
h = h[np.newaxis, :]
X_fft_h = fft(X) * h
if phase is not None:
X_fft_h *= phase
for ii, f in enumerate(filters):
if f is None:
Xh[ii] = ifft(X_fft_h)
else:
f = f / np.linalg.norm(f)
Xh[ii] = ifft(X_fft_h * f)
if Xh.shape[0] == 1:
return Xh[0], X_fft_h
return Xh, X_fft_h
def hilbert_transform(X, rate, filters=None):
"""
Apply bandpass filtering with Hilbert transform using
a prespecified set of filters.
Parameters
----------
X : ndarray (n_channels, n_time)
Input data, dimensions
rate : float
Number of samples per second.
filters : filter or list of filters (optional)
One or more bandpass filters
Returns
-------
Xc : array
Bandpassed analytical signal (dtype: complex)
"""
if not isinstance(filters, list):
filters = [filters]
n_channels, time = X.shape
freq = fftfreq(time, 1. / rate)
# Heavyside filter
h = np.zeros(len(freq))
h[freq > 0] = 2.
h[0] = 1.
h = h[np.newaxis, :]
Xh = np.zeros((len(filters), ) + X.shape, dtype=np.complex)
X_fft_h = fft(X) * h
for ii, f in enumerate(filters):
if f is None:
Xh[ii] = ifft(X_fft_h)
else:
Xh[ii] = ifft(X_fft_h * f)
if Xh.shape[0] == 1:
return Xh[0]
return Xh
def spec(x, dt, axis=-1):
"""
Returns the normalized power spectrum (1 / sqrt(n) forward, 1 / sqrt(n)
backwards). This is so that Perseval's rule may be used easily.
Usage:
[f, g] = spec(x, dt)
INPUTS:
x - Array of data to perform the FFT on.
dt - Time element between array elements. (Scalar). Used to generate the
frequency array.
OUTPUTS:
f - Frequency Array
g - FFT array.
"""
n = x.shape[axis]
f0 = 1/abs(n * dt)
P = fp.fft(x, axis=axis)/sqrt(n)
#Generate an array 1:N, subtract (N + 1) /2 if odd, (N+2)/2 if even.
f = (linspace(1, n, n) - (n + (1 + (n % 2 ==0 )))/2) * f0
P = spfp.fftshift(P, axes=axis)
return f, P
def wavelet(data, freqs, srate, wave_num, demean=True):
'''
@title Wavelet Transformation With Phase
@param data - vector of time series to be decomposed
@param freqs - vector of center frequencies for decomposition
@param srate - sample rate (in Hz)
@param wave_num - desired number of cycles in wavelet (typically 3-20 for frequencies 2-200).
@param demean - whether to de-mean data first?
@details
Wavelet
Function for decomposing time series data into time-frequency
representation (spectral decomposition) using wavelet transform. Employs
Morlet wavelet method (gaussian taper sine wave) to obtain the analytic
signal for specified frequencies (via convolution).
Inputs:
data - vector of time series to be decomposed
freqs - vector of center frequencies for decomposition
srate - sample rate (in Hz)
wave_num - desired number of cycles in wavelet (typically 5-10).
Translated from Matlab script written by Brett Foster,
Stanford Memory Lab, Feb. 2015
'''
srate = round(srate);
# wave_num can be an array equals to length of freqs or just two
if not len(wave_num) == len(freqs):
ratio = (np.log(np.max(wave_num)) - np.log(np.min(wave_num))) / (np.log(np.max(freqs)) - np.log(np.min(freqs)))
wavelet_cycles = np.exp((np.log(freqs) - np.log(np.min(freqs))) * ratio + np.log(np.min(wave_num)))
else:
wavelet_cycles = np.array(wave_num)
f_l = len(freqs)
d_l = data.shape[-1]
# normalize data, and fft
if demean:
fft_data = fft(data - np.mean(data))
else:
fft_data = fft(data)
# wavelet window calc - each columns of final wave is a wavelet kernel (after fft)
# sts = wavelet_cycles / (2 * pi * freqs)
# wavelet_wins = cbind(-3 * sts, 3 * sts)
# calculate and fft wavelet kernels
fft_waves = np.ndarray(shape = [f_l, d_l], dtype = np.complex128)
for ii in range(f_l):
fq = freqs[ii]
cycles = wavelet_cycles[ii]
# standard error
st = cycles / (2 * np.pi * fq)
# calculate window size
wavelet_win = np.arange(-3 * st, 3 * st, 1/srate)
# half of window length
w_l_half = (len(wavelet_win) - 1) / 2
# wavelet 1: calc sinus in complex domain
tmp_sine = np.exp((1j) * 2 * np.pi * fq / srate * np.arange(-w_l_half,w_l_half+1))
# Gaussian normalization part
A = 1/np.sqrt(st*np.sqrt(np.pi))
# wavelet 2: calc gaussian wrappers
tmp_gaus_win = A * np.exp(-np.power(wavelet_win, 2)/(2 * np.power(cycles/(2 * np.pi * fq), 2)))
# wave kernel
tmp_wavelet = tmp_sine * tmp_gaus_win
# padding
w_l = len(tmp_wavelet)
n_pre = int(np.ceil(d_l / 2) - np.floor(w_l/2))
n_post = int(d_l - n_pre - w_l)
wvi = np.pad(tmp_wavelet, [n_pre, n_post], 'constant', constant_values = 0)
fft_wave = np.conj(fft(wvi))
fft_waves[ii, :] = fft_wave
# Convolution Notice that since we don't pad zeros to data
# d_l is nrows of wave_spectrum. However, if wave_spectrum is changed
# we should not use d_l anymore. instead, use nrow(fft_waves)
wave_len = fft_waves.shape[-1]
wave_spectrum = ifft(fft_data * fft_waves)
# use numpy fancy index
cut_off = np.ceil(wave_len / 2)
ind = np.concatenate([np.arange(cut_off, wave_len, dtype = np.int), np.arange(cut_off, dtype = np.int)])
wave_spectrum = wave_spectrum[:, ind] / np.sqrt(srate / 2)
# returns amplitude and phase data
phase = np.angle(wave_spectrum)
power = np.power(np.real(wave_spectrum), 2) + np.power(np.imag(wave_spectrum), 2)
coef = np.sqrt(power)
return {
'power' : power,
'phase' : phase,
'coef' : coef
}
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.
As noted, `resample` uses FFT transformations, which can be very
slow if the number of input samples is large and prime, see
`scipy.fftpack.fft`.
"""
x = np.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, np.ndarray):
if window.shape != (Nx, ):
raise ValueError('window must have the same length as data')
W = window
else:
W = ifftshift(get_window(window, Nx))
newshape = [1] * x.ndim
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 = np.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 = np.arange(0, num) * (t[1] - t[0]) * Nx / float(num) + t[0]
return y, new_t