def real_cepstrum(x, n=None, axis=-1):
r"""
Calculates the real cepstrum of an input sequence x where the cepstrum is
defined as the inverse Fourier transform of the log magnitude DFT
(spectrum) of a signal. It's primarily used for source/speaker separation
in speech signal processing
Parameters
----------
x : ndarray
Input sequence, if x is a matrix, return cepstrum in direction of axis
n : int
Size of Fourier Transform; If none, will use length of input array
axis: int
Direction for cepstrum calculation
Returns
-------
ceps : ndarray
Complex cepstrum result
"""
x = cp.asarray(x)
spectrum = fft.fft(x, n=n, axis=axis)
ceps = fft.ifft(cp.log(cp.abs(spectrum)), n=n, axis=axis).real
return ceps
def cceps(x, n=None, axis=-1):
r"""
Calculates the complex cepstrum of a real valued input sequence x
where the cepstrum is defined as the inverse Fourier transform
of the log magnitude DFT (spectrum) of a signal. It's primarily
used for source/speaker separation in speech signal processing.
The input is altered to have zero-phase at pi radians (180 degrees)
Parameters
----------
x : ndarray
Input sequence, if x is a matrix, return cepstrum in direction of axis
n : int
Size of Fourier Transform; If none, will use length of input array
axis: int
Direction for cepstrum calculation
Returns
-------
ceps : ndarray
Complex cepstrum result
"""
x = cp.asarray(x)
h = fft.fft(x, n=n, axis=axis)
ah = cceps_unwrap(cp.angle(h))
logh = cp.log(cp.abs(h)) + 1j * ah
cceps = fft.ifft(logh, n=n, axis=axis).real
return cceps
def complex_cepstrum(x, n=None, axis=-1):
r"""
Calculates the complex cepstrum of a real valued input sequence x
where the cepstrum is defined as the inverse Fourier transform
of the log magnitude DFT (spectrum) of a signal. It's primarily
used for source/speaker separation in speech signal processing.
The input is altered to have zero-phase at pi radians (180 degrees)
Parameters
----------
x : ndarray
Input sequence, if x is a matrix, return cepstrum in direction of axis
n : int
Size of Fourier Transform; If none, will use length of input array
axis: int
Direction for cepstrum calculation
Returns
-------
ceps : ndarray
Complex cepstrum result
"""
x = cp.asarray(x)
def _unwrap(x):
r"""
Unwrap phase for complex cepstrum calculation; helper function
"""
samples = len(x)
unwrapped = cp.unwrap(x)
center = math.floor((samples + 1) / 2)
ndelay = cp.round_(unwrapped[center] / cp.pi)
unwrapped -= cp.pi * ndelay * cp.arange(samples) / center
return unwrapped, ndelay
spectrum = fft.fft(x, n=n, axis=axis)
unwrapped_phase, ndelay = _unwrap(cp.angle(spectrum))
log_spectrum = cp.log(cp.abs(spectrum)) + 1j * unwrapped_phase
ceps = fft.ifft(log_spectrum, n=n, axis=axis).real
return ceps, ndelay
def resample(x, num, t=None, axis=0, window=None, domain="time"):
"""
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.
domain : string, optional
A string indicating the domain of the input `x`:
``time``
Consider the input `x` as time-domain. (Default)
``freq``
Consider the input `x` as frequency-domain.
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.
See Also
--------
decimate : Downsample the signal after applying an FIR or IIR filter.
resample_poly : Resample using polyphase filtering and an FIR filter.
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 `cusignal.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 or output samples is large and prime;
see `scipy.fftpack.fft`.
Examples
--------
Note that the end of the resampled data rises to meet the first
sample of the next cycle:
>>> import cusignal
>>> import cupy as cp
>>> x = cp.linspace(0, 10, 20, endpoint=False)
>>> y = cp.cos(-x**2/6.0)
>>> f = cusignal.resample(y, 100)
>>> xnew = cp.linspace(0, 10, 100, endpoint=False)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'go-', xnew, f, '.-', 10, y[0], 'ro')
>>> plt.legend(['data', 'resampled'], loc='best')
>>> plt.show()
"""
x = asarray(x)
Nx = x.shape[axis]
if domain == "time":
X = fftpack.fft(x, axis=axis)
elif domain == "freq":
X = x
else:
raise NotImplementedError("domain should be 'time' or 'freq'")
if window is not None:
if callable(window):
W = window(fftpack.fftfreq(Nx))
elif isinstance(window, 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)] * x.ndim
newshape = list(x.shape)
newshape[axis] = num
N = int(cp.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 = fftpack.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 hilbert(x, N=None, axis=-1):
"""
Compute the analytic signal, using the Hilbert transform.
The transformation is done along the last axis by default.
Parameters
----------
x : array_like
Signal data. Must be real.
N : int, optional
Number of Fourier components. Default: ``x.shape[axis]``
axis : int, optional
Axis along which to do the transformation. Default: -1.
Returns
-------
xa : ndarray
Analytic signal of `x`, of each 1-D array along `axis`
Notes
-----
The analytic signal ``x_a(t)`` of signal ``x(t)`` is:
.. math:: x_a = F^{-1}(F(x) 2U) = x + i y
where `F` is the Fourier transform, `U` the unit step function,
and `y` the Hilbert transform of `x`. [1]_
In other words, the negative half of the frequency spectrum is zeroed
out, turning the real-valued signal into a complex signal. The Hilbert
transformed signal can be obtained from ``cp.imag(hilbert(x))``, and the
original signal from ``cp.real(hilbert(x))``.
Examples
---------
In this example we use the Hilbert transform to determine the amplitude
envelope and instantaneous frequency of an amplitude-modulated signal.
>>> import cupy as cp
>>> import matplotlib.pyplot as plt
>>> from cusignal import hilbert, chirp
>>> duration = 1.0
>>> fs = 400.0
>>> samples = int(fs*duration)
>>> t = cp.arange(samples) / fs
We create a chirp of which the frequency increases from 20 Hz to 100 Hz and
apply an amplitude modulation.
>>> signal = chirp(t, 20.0, t[-1], 100.0)
>>> signal *= (1.0 + 0.5 * cp.sin(2.0*cp.pi*3.0*t) )
The amplitude envelope is given by magnitude of the analytic signal. The
instantaneous frequency can be obtained by differentiating the
instantaneous phase in respect to time. The instantaneous phase corresponds
to the phase angle of the analytic signal.
>>> analytic_signal = hilbert(signal)
>>> amplitude_envelope = cp.abs(analytic_signal)
>>> instantaneous_phase = cp.unwrap(cp.angle(analytic_signal))
>>> instantaneous_frequency = (cp.diff(instantaneous_phase) /
... (2.0*cp.pi) * fs)
>>> fig = plt.figure()
>>> ax0 = fig.add_subplot(211)
>>> ax0.plot(cp.asnumpy(t), cp.asnumpy(signal), label='signal')
>>> ax0.plot(cp.asnumpy(t), cp.asnumpy(amplitude_envelope), \
label='envelope')
>>> ax0.set_xlabel("time in seconds")
>>> ax0.legend()
>>> ax1 = fig.add_subplot(212)
>>> ax1.plot(t[1:], instantaneous_frequency)
>>> ax1.set_xlabel("time in seconds")
>>> ax1.set_ylim(0.0, 120.0)
References
----------
.. [1] Wikipedia, "Analytic signal".
https://en.wikipedia.org/wiki/Analytic_signal
.. [2] Leon Cohen, "Time-Frequency Analysis", 1995. Chapter 2.
.. [3] Alan V. Oppenheim, Ronald W. Schafer. Discrete-Time Signal
Processing, Third Edition, 2009. Chapter 12.
ISBN 13: 978-1292-02572-8
"""
x = asarray(x)
if iscomplexobj(x):
raise ValueError("x must be real.")
if N is None:
N = x.shape[axis]
if N <= 0:
raise ValueError("N must be positive.")
Xf = fftpack.fft(x, N, axis=axis)
h = zeros(N)
if N % 2 == 0:
h[0] = h[N // 2] = 1
h[1 : N // 2] = 2
else:
h[0] = 1
h[1 : (N + 1) // 2] = 2
if x.ndim > 1:
ind = [newaxis] * x.ndim
ind[axis] = slice(None)
h = h[tuple(ind)]
x = fftpack.ifft(Xf * h, axis=axis)
return x
def ifft(x):
global plan1D
if plan1D == None and Plan:
plan1D = fftpack.get_fft_plan(x, axes=(-1))
return fftpack.ifft(x, overwrite_x=owrite, plan=plan1D)