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"""Tools for spectral analysis. """
'spectrogram', 'stft', 'istft', 'check_COLA']
y, freqs, precenter=False, normalize=False): """ lombscargle(x, y, freqs)
Computes the Lomb-Scargle periodogram.
The Lomb-Scargle periodogram was developed by Lomb [1]_ and further extended by Scargle [2]_ to find, and test the significance of weak periodic signals with uneven temporal sampling.
When *normalize* is False (default) the computed periodogram is unnormalized, it takes the value ``(A**2) * N/4`` for a harmonic signal with amplitude A for sufficiently large N.
When *normalize* is True the computed periodogram is is normalized by the residuals of the data around a constant reference model (at zero).
Input arrays should be one-dimensional and will be cast to float64.
Parameters ---------- x : array_like Sample times. y : array_like Measurement values. freqs : array_like Angular frequencies for output periodogram. precenter : bool, optional Pre-center amplitudes by subtracting the mean. normalize : bool, optional Compute normalized periodogram.
Returns ------- pgram : array_like Lomb-Scargle periodogram.
Raises ------ ValueError If the input arrays `x` and `y` do not have the same shape.
Notes ----- This subroutine calculates the periodogram using a slightly modified algorithm due to Townsend [3]_ which allows the periodogram to be calculated using only a single pass through the input arrays for each frequency.
The algorithm running time scales roughly as O(x * freqs) or O(N^2) for a large number of samples and frequencies.
References ---------- .. [1] N.R. Lomb "Least-squares frequency analysis of unequally spaced data", Astrophysics and Space Science, vol 39, pp. 447-462, 1976
.. [2] J.D. Scargle "Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data", The Astrophysical Journal, vol 263, pp. 835-853, 1982
.. [3] R.H.D. Townsend, "Fast calculation of the Lomb-Scargle periodogram using graphics processing units.", The Astrophysical Journal Supplement Series, vol 191, pp. 247-253, 2010
Examples -------- >>> import scipy.signal >>> import matplotlib.pyplot as plt
First define some input parameters for the signal:
>>> A = 2. >>> w = 1. >>> phi = 0.5 * np.pi >>> nin = 1000 >>> nout = 100000 >>> frac_points = 0.9 # Fraction of points to select
Randomly select a fraction of an array with timesteps:
>>> r = np.random.rand(nin) >>> x = np.linspace(0.01, 10*np.pi, nin) >>> x = x[r >= frac_points]
Plot a sine wave for the selected times:
>>> y = A * np.sin(w*x+phi)
Define the array of frequencies for which to compute the periodogram:
>>> f = np.linspace(0.01, 10, nout)
Calculate Lomb-Scargle periodogram:
>>> import scipy.signal as signal >>> pgram = signal.lombscargle(x, y, f, normalize=True)
Now make a plot of the input data:
>>> plt.subplot(2, 1, 1) >>> plt.plot(x, y, 'b+')
Then plot the normalized periodogram:
>>> plt.subplot(2, 1, 2) >>> plt.plot(f, pgram) >>> plt.show()
"""
x = np.asarray(x, dtype=np.float64) y = np.asarray(y, dtype=np.float64) freqs = np.asarray(freqs, dtype=np.float64)
assert x.ndim == 1 assert y.ndim == 1 assert freqs.ndim == 1
if precenter: pgram = _lombscargle(x, y - y.mean(), freqs) else: pgram = _lombscargle(x, y, freqs)
if normalize: pgram *= 2 / np.dot(y, y)
return pgram
return_onesided=True, scaling='density', axis=-1): """ Estimate power spectral density using a periodogram.
Parameters ---------- x : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to 'boxcar'. nfft : int, optional Length of the FFT used. If `None` the length of `x` will be used. detrend : str or function or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. 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 and computing the power spectrum ('spectrum') where `Pxx` has units of V**2, if `x` is measured in V and `fs` is measured in Hz. 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`.
Notes ----- .. versionadded:: 0.12.0
See Also -------- welch: Estimate power spectral density using Welch's method lombscargle: Lomb-Scargle periodogram for unevenly sampled data
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt >>> np.random.seed(1234)
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.periodogram(x, fs) >>> plt.semilogy(f, Pxx_den) >>> plt.ylim([1e-7, 1e2]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('PSD [V**2/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[25000:]) 0.00099728892368242854
Now compute and plot the power spectrum.
>>> f, Pxx_spec = signal.periodogram(x, fs, 'flattop', scaling='spectrum') >>> plt.figure() >>> plt.semilogy(f, np.sqrt(Pxx_spec)) >>> plt.ylim([1e-4, 1e1]) >>> 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 window is None: window = 'boxcar'
if nfft is None: nperseg = x.shape[axis] elif nfft == x.shape[axis]: nperseg = nfft elif nfft > x.shape[axis]: nperseg = x.shape[axis] elif nfft < x.shape[axis]: s = [np.s_[:]]*len(x.shape) s[axis] = np.s_[:nfft] x = x[s] nperseg = nfft nfft = None
return welch(x, fs, window, nperseg, 0, nfft, detrend, return_onesided, scaling, axis)
detrend='constant', return_onesided=True, scaling='density', axis=-1): r""" 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. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. 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 and computing the power spectrum ('spectrum') where `Pxx` has units of V**2, if `x` is measured in V and `fs` is measured in Hz. 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 Hann 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 >>> np.random.seed(1234)
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, nperseg=1024) >>> plt.semilogy(f, Pxx_den) >>> plt.ylim([0.5e-3, 1]) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('PSD [V**2/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
"""
freqs, Pxx = csd(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis)
return freqs, Pxx.real
detrend='constant', return_onesided=True, scaling='density', axis=-1): r""" Estimate the cross power spectral density, Pxy, using Welch's method.
Parameters ---------- x : array_like Time series of measurement values y : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` and `y` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. 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 cross spectral density ('density') where `Pxy` has units of V**2/Hz and computing the cross spectrum ('spectrum') where `Pxy` has units of V**2, if `x` and `y` are measured in V and `fs` is measured in Hz. Defaults to 'density' axis : int, optional Axis along which the CSD is computed for both inputs; the default is over the last axis (i.e. ``axis=-1``).
Returns ------- f : ndarray Array of sample frequencies. Pxy : ndarray Cross spectral density or cross power spectrum of x,y.
See Also -------- periodogram: Simple, optionally modified periodogram lombscargle: Lomb-Scargle periodogram for unevenly sampled data welch: Power spectral density by Welch's method. [Equivalent to csd(x,x)] coherence: Magnitude squared coherence by Welch's method.
Notes -------- By convention, Pxy is computed with the conjugate FFT of X multiplied by the FFT of Y.
If the input series differ in length, the shorter series will be zero-padded to match.
An appropriate amount of overlap will depend on the choice of window and on your requirements. For the default Hann 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.
.. versionadded:: 0.16.0
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] Rabiner, Lawrence R., and B. Gold. "Theory and Application of Digital Signal Processing" Prentice-Hall, pp. 414-419, 1975
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt
Generate two test signals with some common features.
>>> fs = 10e3 >>> N = 1e5 >>> amp = 20 >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> b, a = signal.butter(2, 0.25, 'low') >>> x = np.random.normal(scale=np.sqrt(noise_power), size=time.shape) >>> y = signal.lfilter(b, a, x) >>> x += amp*np.sin(2*np.pi*freq*time) >>> y += np.random.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
Compute and plot the magnitude of the cross spectral density.
>>> f, Pxy = signal.csd(x, y, fs, nperseg=1024) >>> plt.semilogy(f, np.abs(Pxy)) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('CSD [V**2/Hz]') >>> plt.show() """
freqs, _, Pxy = _spectral_helper(x, y, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='psd')
# Average over windows. if len(Pxy.shape) >= 2 and Pxy.size > 0: if Pxy.shape[-1] > 1: Pxy = Pxy.mean(axis=-1) else: Pxy = np.reshape(Pxy, Pxy.shape[:-1])
return freqs, Pxy
nfft=None, detrend='constant', return_onesided=True, scaling='density', axis=-1, mode='psd'): """ Compute a spectrogram with consecutive Fourier transforms.
Spectrograms can be used as a way of visualizing the change of a nonstationary signal's frequency content over time.
Parameters ---------- x : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Tukey window with shape parameter of 0.25. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. noverlap : int, optional Number of points to overlap between segments. If `None`, ``noverlap = nperseg // 8``. 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. 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 `Sxx` has units of V**2/Hz and computing the power spectrum ('spectrum') where `Sxx` has units of V**2, if `x` is measured in V and `fs` is measured in Hz. Defaults to 'density'. axis : int, optional Axis along which the spectrogram is computed; the default is over the last axis (i.e. ``axis=-1``). mode : str, optional Defines what kind of return values are expected. Options are ['psd', 'complex', 'magnitude', 'angle', 'phase']. 'complex' is equivalent to the output of `stft` with no padding or boundary extension. 'magnitude' returns the absolute magnitude of the STFT. 'angle' and 'phase' return the complex angle of the STFT, with and without unwrapping, respectively.
Returns ------- f : ndarray Array of sample frequencies. t : ndarray Array of segment times. Sxx : ndarray Spectrogram of x. By default, the last axis of Sxx corresponds to the segment times.
See Also -------- periodogram: Simple, optionally modified periodogram lombscargle: Lomb-Scargle periodogram for unevenly sampled data welch: Power spectral density by Welch's method. csd: Cross spectral density by Welch's method.
Notes ----- An appropriate amount of overlap will depend on the choice of window and on your requirements. In contrast to welch's method, where the entire data stream is averaged over, one may wish to use a smaller overlap (or perhaps none at all) when computing a spectrogram, to maintain some statistical independence between individual segments. It is for this reason that the default window is a Tukey window with 1/8th of a window's length overlap at each end.
.. versionadded:: 0.16.0
References ---------- .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time Signal Processing", Prentice Hall, 1999.
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt
Generate a test signal, a 2 Vrms sine wave whose frequency is slowly modulated around 3kHz, corrupted by white noise of exponentially decreasing magnitude sampled at 10 kHz.
>>> fs = 10e3 >>> N = 1e5 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = np.arange(N) / float(fs) >>> mod = 500*np.cos(2*np.pi*0.25*time) >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) >>> noise = np.random.normal(scale=np.sqrt(noise_power), size=time.shape) >>> noise *= np.exp(-time/5) >>> x = carrier + noise
Compute and plot the spectrogram.
>>> f, t, Sxx = signal.spectrogram(x, fs) >>> plt.pcolormesh(t, f, Sxx) >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() """ modelist = ['psd', 'complex', 'magnitude', 'angle', 'phase'] if mode not in modelist: raise ValueError('unknown value for mode {}, must be one of {}' .format(mode, modelist))
# need to set default for nperseg before setting default for noverlap below window, nperseg = _triage_segments(window, nperseg, input_length=x.shape[axis])
# Less overlap than welch, so samples are more statisically independent if noverlap is None: noverlap = nperseg // 8
if mode == 'psd': freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='psd')
else: freqs, time, Sxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling, axis, mode='stft')
if mode == 'magnitude': Sxx = np.abs(Sxx) elif mode in ['angle', 'phase']: Sxx = np.angle(Sxx) if mode == 'phase': # Sxx has one additional dimension for time strides if axis < 0: axis -= 1 Sxx = np.unwrap(Sxx, axis=axis)
# mode =='complex' is same as `stft`, doesn't need modification
return freqs, time, Sxx
r""" Check whether the Constant OverLap Add (COLA) constraint is met
Parameters ---------- window : str or tuple or array_like Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. nperseg : int Length of each segment. noverlap : int Number of points to overlap between segments. tol : float, optional The allowed variance of a bin's weighted sum from the median bin sum.
Returns ------- verdict : bool `True` if chosen combination satisfies COLA within `tol`, `False` otherwise
See Also -------- stft: Short Time Fourier Transform istft: Inverse Short Time Fourier Transform
Notes ----- In order to enable inversion of an STFT via the inverse STFT in `istft`, the signal windowing must obey the constraint of "Constant OverLap Add" (COLA). This ensures that every point in the input data is equally weighted, thereby avoiding aliasing and allowing full reconstruction.
Some examples of windows that satisfy COLA: - Rectangular window at overlap of 0, 1/2, 2/3, 3/4, ... - Bartlett window at overlap of 1/2, 3/4, 5/6, ... - Hann window at 1/2, 2/3, 3/4, ... - Any Blackman family window at 2/3 overlap - Any window with ``noverlap = nperseg-1``
A very comprehensive list of other windows may be found in [2]_, wherein the COLA condition is satisfied when the "Amplitude Flatness" is unity.
.. versionadded:: 0.19.0
References ---------- .. [1] Julius O. Smith III, "Spectral Audio Signal Processing", W3K Publishing, 2011,ISBN 978-0-9745607-3-1. .. [2] G. Heinzel, A. Ruediger and R. Schilling, "Spectrum and spectral density estimation by the Discrete Fourier transform (DFT), including a comprehensive list of window functions and some new at-top windows", 2002, http://hdl.handle.net/11858/00-001M-0000-0013-557A-5
Examples -------- >>> from scipy import signal
Confirm COLA condition for rectangular window of 75% (3/4) overlap:
>>> signal.check_COLA(signal.boxcar(100), 100, 75) True
COLA is not true for 25% (1/4) overlap, though:
>>> signal.check_COLA(signal.boxcar(100), 100, 25) False
"Symmetrical" Hann window (for filter design) is not COLA:
>>> signal.check_COLA(signal.hann(120, sym=True), 120, 60) False
"Periodic" or "DFT-even" Hann window (for FFT analysis) is COLA for overlap of 1/2, 2/3, 3/4, etc.:
>>> signal.check_COLA(signal.hann(120, sym=False), 120, 60) True
>>> signal.check_COLA(signal.hann(120, sym=False), 120, 80) True
>>> signal.check_COLA(signal.hann(120, sym=False), 120, 90) True
"""
nperseg = int(nperseg)
if nperseg < 1: raise ValueError('nperseg must be a positive integer')
if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg.') noverlap = int(noverlap)
if isinstance(window, string_types) 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] != nperseg: raise ValueError('window must have length of nperseg')
step = nperseg - noverlap binsums = sum(win[ii*step:(ii+1)*step] for ii in range(nperseg//step))
if nperseg % step != 0: binsums[:nperseg % step] += win[-(nperseg % step):]
deviation = binsums - np.median(binsums) return np.max(np.abs(deviation)) < tol
detrend=False, return_onesided=True, boundary='zeros', padded=True, axis=-1): r""" Compute the Short Time Fourier Transform (STFT).
STFTs can be used as a way of quantifying the change of a nonstationary signal's frequency and phase content over time.
Parameters ---------- x : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. 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`. When specified, the COLA constraint must be met (see Notes below). 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. Defaults to `False`. 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. Defaults to `True`. boundary : str or None, optional Specifies whether the input signal is extended at both ends, and how to generate the new values, in order to center the first windowed segment on the first input point. This has the benefit of enabling reconstruction of the first input point when the employed window function starts at zero. Valid options are ``['even', 'odd', 'constant', 'zeros', None]``. Defaults to 'zeros', for zero padding extension. I.e. ``[1, 2, 3, 4]`` is extended to ``[0, 1, 2, 3, 4, 0]`` for ``nperseg=3``. padded : bool, optional Specifies whether the input signal is zero-padded at the end to make the signal fit exactly into an integer number of window segments, so that all of the signal is included in the output. Defaults to `True`. Padding occurs after boundary extension, if `boundary` is not `None`, and `padded` is `True`, as is the default. axis : int, optional Axis along which the STFT is computed; the default is over the last axis (i.e. ``axis=-1``).
Returns ------- f : ndarray Array of sample frequencies. t : ndarray Array of segment times. Zxx : ndarray STFT of `x`. By default, the last axis of `Zxx` corresponds to the segment times.
See Also -------- istft: Inverse Short Time Fourier Transform check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met welch: Power spectral density by Welch's method. spectrogram: Spectrogram by Welch's method. csd: Cross spectral density by Welch's method. lombscargle: Lomb-Scargle periodogram for unevenly sampled data
Notes ----- In order to enable inversion of an STFT via the inverse STFT in `istft`, the signal windowing must obey the constraint of "Constant OverLap Add" (COLA), and the input signal must have complete windowing coverage (i.e. ``(x.shape[axis] - nperseg) % (nperseg-noverlap) == 0``). The `padded` argument may be used to accomplish this.
The COLA constraint ensures that every point in the input data is equally weighted, thereby avoiding aliasing and allowing full reconstruction. Whether a choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can be tested with `check_COLA`.
.. versionadded:: 0.19.0
References ---------- .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time Signal Processing", Prentice Hall, 1999. .. [2] Daniel W. Griffin, Jae S. Limdt "Signal Estimation from Modified Short Fourier Transform", IEEE 1984, 10.1109/TASSP.1984.1164317
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt
Generate a test signal, a 2 Vrms sine wave whose frequency is slowly modulated around 3kHz, corrupted by white noise of exponentially decreasing magnitude sampled at 10 kHz.
>>> fs = 10e3 >>> N = 1e5 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.01 * fs / 2 >>> time = np.arange(N) / float(fs) >>> mod = 500*np.cos(2*np.pi*0.25*time) >>> carrier = amp * np.sin(2*np.pi*3e3*time + mod) >>> noise = np.random.normal(scale=np.sqrt(noise_power), ... size=time.shape) >>> noise *= np.exp(-time/5) >>> x = carrier + noise
Compute and plot the STFT's magnitude.
>>> f, t, Zxx = signal.stft(x, fs, nperseg=1000) >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp) >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.show() """
freqs, time, Zxx = _spectral_helper(x, x, fs, window, nperseg, noverlap, nfft, detrend, return_onesided, scaling='spectrum', axis=axis, mode='stft', boundary=boundary, padded=padded)
return freqs, time, Zxx
input_onesided=True, boundary=True, time_axis=-1, freq_axis=-2): r""" Perform the inverse Short Time Fourier transform (iSTFT).
Parameters ---------- Zxx : array_like STFT of the signal to be reconstructed. If a purely real array is passed, it will be cast to a complex data type. fs : float, optional Sampling frequency of the time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. Must match the window used to generate the STFT for faithful inversion. nperseg : int, optional Number of data points corresponding to each STFT segment. This parameter must be specified if the number of data points per segment is odd, or if the STFT was padded via ``nfft > nperseg``. If `None`, the value depends on the shape of `Zxx` and `input_onesided`. If `input_onesided` is True, ``nperseg=2*(Zxx.shape[freq_axis] - 1)``. Otherwise, ``nperseg=Zxx.shape[freq_axis]``. Defaults to `None`. noverlap : int, optional Number of points to overlap between segments. If `None`, half of the segment length. Defaults to `None`. When specified, the COLA constraint must be met (see Notes below), and should match the parameter used to generate the STFT. Defaults to `None`. nfft : int, optional Number of FFT points corresponding to each STFT segment. This parameter must be specified if the STFT was padded via ``nfft > nperseg``. If `None`, the default values are the same as for `nperseg`, detailed above, with one exception: if `input_onesided` is True and ``nperseg==2*Zxx.shape[freq_axis] - 1``, `nfft` also takes on that value. This case allows the proper inversion of an odd-length unpadded STFT using ``nfft=None``. Defaults to `None`. input_onesided : bool, optional If `True`, interpret the input array as one-sided FFTs, such as is returned by `stft` with ``return_onesided=True`` and `numpy.fft.rfft`. If `False`, interpret the input as a a two-sided FFT. Defaults to `True`. boundary : bool, optional Specifies whether the input signal was extended at its boundaries by supplying a non-`None` ``boundary`` argument to `stft`. Defaults to `True`. time_axis : int, optional Where the time segments of the STFT is located; the default is the last axis (i.e. ``axis=-1``). freq_axis : int, optional Where the frequency axis of the STFT is located; the default is the penultimate axis (i.e. ``axis=-2``).
Returns ------- t : ndarray Array of output data times. x : ndarray iSTFT of `Zxx`.
See Also -------- stft: Short Time Fourier Transform check_COLA: Check whether the Constant OverLap Add (COLA) constraint is met
Notes ----- In order to enable inversion of an STFT via the inverse STFT with `istft`, the signal windowing must obey the constraint of "Constant OverLap Add" (COLA). This ensures that every point in the input data is equally weighted, thereby avoiding aliasing and allowing full reconstruction. Whether a choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can be tested with `check_COLA`, by using ``nperseg = Zxx.shape[freq_axis]``.
An STFT which has been modified (via masking or otherwise) is not guaranteed to correspond to a exactly realizible signal. This function implements the iSTFT via the least-squares esimation algorithm detailed in [2]_, which produces a signal that minimizes the mean squared error between the STFT of the returned signal and the modified STFT.
.. versionadded:: 0.19.0
References ---------- .. [1] Oppenheim, Alan V., Ronald W. Schafer, John R. Buck "Discrete-Time Signal Processing", Prentice Hall, 1999. .. [2] Daniel W. Griffin, Jae S. Limdt "Signal Estimation from Modified Short Fourier Transform", IEEE 1984, 10.1109/TASSP.1984.1164317
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt
Generate a test signal, a 2 Vrms sine wave at 50Hz corrupted by 0.001 V**2/Hz of white noise sampled at 1024 Hz.
>>> fs = 1024 >>> N = 10*fs >>> nperseg = 512 >>> amp = 2 * np.sqrt(2) >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / float(fs) >>> carrier = amp * np.sin(2*np.pi*50*time) >>> noise = np.random.normal(scale=np.sqrt(noise_power), ... size=time.shape) >>> x = carrier + noise
Compute the STFT, and plot its magnitude
>>> f, t, Zxx = signal.stft(x, fs=fs, nperseg=nperseg) >>> plt.figure() >>> plt.pcolormesh(t, f, np.abs(Zxx), vmin=0, vmax=amp) >>> plt.ylim([f[1], f[-1]]) >>> plt.title('STFT Magnitude') >>> plt.ylabel('Frequency [Hz]') >>> plt.xlabel('Time [sec]') >>> plt.yscale('log') >>> plt.show()
Zero the components that are 10% or less of the carrier magnitude, then convert back to a time series via inverse STFT
>>> Zxx = np.where(np.abs(Zxx) >= amp/10, Zxx, 0) >>> _, xrec = signal.istft(Zxx, fs)
Compare the cleaned signal with the original and true carrier signals.
>>> plt.figure() >>> plt.plot(time, x, time, xrec, time, carrier) >>> plt.xlim([2, 2.1]) >>> plt.xlabel('Time [sec]') >>> plt.ylabel('Signal') >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier']) >>> plt.show()
Note that the cleaned signal does not start as abruptly as the original, since some of the coefficients of the transient were also removed:
>>> plt.figure() >>> plt.plot(time, x, time, xrec, time, carrier) >>> plt.xlim([0, 0.1]) >>> plt.xlabel('Time [sec]') >>> plt.ylabel('Signal') >>> plt.legend(['Carrier + Noise', 'Filtered via STFT', 'True Carrier']) >>> plt.show()
"""
# Make sure input is an ndarray of appropriate complex dtype Zxx = np.asarray(Zxx) + 0j freq_axis = int(freq_axis) time_axis = int(time_axis)
if Zxx.ndim < 2: raise ValueError('Input stft must be at least 2d!')
if freq_axis == time_axis: raise ValueError('Must specify differing time and frequency axes!')
nseg = Zxx.shape[time_axis]
if input_onesided: # Assume even segment length n_default = 2*(Zxx.shape[freq_axis] - 1) else: n_default = Zxx.shape[freq_axis]
# Check windowing parameters if nperseg is None: nperseg = n_default else: nperseg = int(nperseg) if nperseg < 1: raise ValueError('nperseg must be a positive integer')
if nfft is None: if (input_onesided) and (nperseg == n_default + 1): # Odd nperseg, no FFT padding nfft = nperseg else: nfft = n_default elif nfft < nperseg: raise ValueError('nfft must be greater than or equal to nperseg.') else: nfft = int(nfft)
if noverlap is None: noverlap = nperseg//2 else: noverlap = int(noverlap) if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg.') nstep = nperseg - noverlap
if not check_COLA(window, nperseg, noverlap): raise ValueError('Window, STFT shape and noverlap do not satisfy the ' 'COLA constraint.')
# Rearrange axes if necessary if time_axis != Zxx.ndim-1 or freq_axis != Zxx.ndim-2: # Turn negative indices to positive for the call to transpose if freq_axis < 0: freq_axis = Zxx.ndim + freq_axis if time_axis < 0: time_axis = Zxx.ndim + time_axis zouter = list(range(Zxx.ndim)) for ax in sorted([time_axis, freq_axis], reverse=True): zouter.pop(ax) Zxx = np.transpose(Zxx, zouter+[freq_axis, time_axis])
# Get window as array if isinstance(window, string_types) 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] != nperseg: raise ValueError('window must have length of {0}'.format(nperseg))
if input_onesided: ifunc = np.fft.irfft else: ifunc = fftpack.ifft
xsubs = ifunc(Zxx, axis=-2, n=nfft)[..., :nperseg, :]
# Initialize output and normalization arrays outputlength = nperseg + (nseg-1)*nstep x = np.zeros(list(Zxx.shape[:-2])+[outputlength], dtype=xsubs.dtype) norm = np.zeros(outputlength, dtype=xsubs.dtype)
if np.result_type(win, xsubs) != xsubs.dtype: win = win.astype(xsubs.dtype)
xsubs *= win.sum() # This takes care of the 'spectrum' scaling
# Construct the output from the ifft segments # This loop could perhaps be vectorized/strided somehow... for ii in range(nseg): # Window the ifft x[..., ii*nstep:ii*nstep+nperseg] += xsubs[..., ii] * win norm[..., ii*nstep:ii*nstep+nperseg] += win**2
# Divide out normalization where non-tiny x /= np.where(norm > 1e-10, norm, 1.0)
# Remove extension points if boundary: x = x[..., nperseg//2:-(nperseg//2)]
if input_onesided: x = x.real
# Put axes back if x.ndim > 1: if time_axis != Zxx.ndim-1: if freq_axis < time_axis: time_axis -= 1 x = np.rollaxis(x, -1, time_axis)
time = np.arange(x.shape[0])/float(fs) return time, x
nfft=None, detrend='constant', axis=-1): r""" Estimate the magnitude squared coherence estimate, Cxy, of discrete-time signals X and Y using Welch's method.
``Cxy = abs(Pxy)**2/(Pxx*Pyy)``, where `Pxx` and `Pyy` are power spectral density estimates of X and Y, and `Pxy` is the cross spectral density estimate of X and Y.
Parameters ---------- x : array_like Time series of measurement values y : array_like Time series of measurement values fs : float, optional Sampling frequency of the `x` and `y` time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. Defaults to 'constant'. axis : int, optional Axis along which the coherence is computed for both inputs; the default is over the last axis (i.e. ``axis=-1``).
Returns ------- f : ndarray Array of sample frequencies. Cxy : ndarray Magnitude squared coherence of x and y.
See Also -------- periodogram: Simple, optionally modified periodogram lombscargle: Lomb-Scargle periodogram for unevenly sampled data welch: Power spectral density by Welch's method. csd: Cross spectral density by Welch's method.
Notes -------- An appropriate amount of overlap will depend on the choice of window and on your requirements. For the default Hann 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.
.. versionadded:: 0.16.0
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] Stoica, Petre, and Randolph Moses, "Spectral Analysis of Signals" Prentice Hall, 2005
Examples -------- >>> from scipy import signal >>> import matplotlib.pyplot as plt
Generate two test signals with some common features.
>>> fs = 10e3 >>> N = 1e5 >>> amp = 20 >>> freq = 1234.0 >>> noise_power = 0.001 * fs / 2 >>> time = np.arange(N) / fs >>> b, a = signal.butter(2, 0.25, 'low') >>> x = np.random.normal(scale=np.sqrt(noise_power), size=time.shape) >>> y = signal.lfilter(b, a, x) >>> x += amp*np.sin(2*np.pi*freq*time) >>> y += np.random.normal(scale=0.1*np.sqrt(noise_power), size=time.shape)
Compute and plot the coherence.
>>> f, Cxy = signal.coherence(x, y, fs, nperseg=1024) >>> plt.semilogy(f, Cxy) >>> plt.xlabel('frequency [Hz]') >>> plt.ylabel('Coherence') >>> plt.show() """
freqs, Pxx = welch(x, fs, window, nperseg, noverlap, nfft, detrend, axis=axis) _, Pyy = welch(y, fs, window, nperseg, noverlap, nfft, detrend, axis=axis) _, Pxy = csd(x, y, fs, window, nperseg, noverlap, nfft, detrend, axis=axis)
Cxy = np.abs(Pxy)**2 / Pxx / Pyy
return freqs, Cxy
nfft=None, detrend='constant', return_onesided=True, scaling='spectrum', axis=-1, mode='psd', boundary=None, padded=False): """ Calculate various forms of windowed FFTs for PSD, CSD, etc.
This is a helper function that implements the commonality between the stft, psd, csd, and spectrogram functions. It is not designed to be called externally. The windows are not averaged over; the result from each window is returned.
Parameters --------- x : array_like Array or sequence containing the data to be analyzed. y : array_like Array or sequence containing the data to be analyzed. If this is the same object in memory as `x` (i.e. ``_spectral_helper(x, x, ...)``), the extra computations are spared. fs : float, optional Sampling frequency of the time series. Defaults to 1.0. window : str or tuple or array_like, optional Desired window to use. If `window` is a string or tuple, it is passed to `get_window` to generate the window values, which are DFT-even by default. 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 must be nperseg. Defaults to a Hann window. nperseg : int, optional Length of each segment. Defaults to None, but if window is str or tuple, is set to 256, and if window is array_like, is set to the length of the window. 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 or `False`, optional Specifies how to detrend each segment. If `detrend` is a string, it is passed as the `type` argument to the `detrend` function. If it is a function, it takes a segment and returns a detrended segment. If `detrend` is `False`, no detrending is done. 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 cross spectral density ('density') where `Pxy` has units of V**2/Hz and computing the cross spectrum ('spectrum') where `Pxy` has units of V**2, if `x` and `y` are measured in V and `fs` is measured in Hz. Defaults to 'density' axis : int, optional Axis along which the FFTs are computed; the default is over the last axis (i.e. ``axis=-1``). mode: str {'psd', 'stft'}, optional Defines what kind of return values are expected. Defaults to 'psd'. boundary : str or None, optional Specifies whether the input signal is extended at both ends, and how to generate the new values, in order to center the first windowed segment on the first input point. This has the benefit of enabling reconstruction of the first input point when the employed window function starts at zero. Valid options are ``['even', 'odd', 'constant', 'zeros', None]``. Defaults to `None`. padded : bool, optional Specifies whether the input signal is zero-padded at the end to make the signal fit exactly into an integer number of window segments, so that all of the signal is included in the output. Defaults to `False`. Padding occurs after boundary extension, if `boundary` is not `None`, and `padded` is `True`. Returns ------- freqs : ndarray Array of sample frequencies. t : ndarray Array of times corresponding to each data segment result : ndarray Array of output data, contents dependent on *mode* kwarg.
Notes ----- Adapted from matplotlib.mlab
.. versionadded:: 0.16.0 """ if mode not in ['psd', 'stft']: raise ValueError("Unknown value for mode %s, must be one of: " "{'psd', 'stft'}" % mode)
boundary_funcs = {'even': even_ext, 'odd': odd_ext, 'constant': const_ext, 'zeros': zero_ext, None: None}
if boundary not in boundary_funcs: raise ValueError("Unknown boundary option '{0}', must be one of: {1}" .format(boundary, list(boundary_funcs.keys())))
# If x and y are the same object we can save ourselves some computation. same_data = y is x
if not same_data and mode != 'psd': raise ValueError("x and y must be equal if mode is 'stft'")
axis = int(axis)
# Ensure we have np.arrays, get outdtype x = np.asarray(x) if not same_data: y = np.asarray(y) outdtype = np.result_type(x, y, np.complex64) else: outdtype = np.result_type(x, np.complex64)
if not same_data: # Check if we can broadcast the outer axes together xouter = list(x.shape) youter = list(y.shape) xouter.pop(axis) youter.pop(axis) try: outershape = np.broadcast(np.empty(xouter), np.empty(youter)).shape except ValueError: raise ValueError('x and y cannot be broadcast together.')
if same_data: if x.size == 0: return np.empty(x.shape), np.empty(x.shape), np.empty(x.shape) else: if x.size == 0 or y.size == 0: outshape = outershape + (min([x.shape[axis], y.shape[axis]]),) emptyout = np.rollaxis(np.empty(outshape), -1, axis) return emptyout, emptyout, emptyout
if x.ndim > 1: if axis != -1: x = np.rollaxis(x, axis, len(x.shape)) if not same_data and y.ndim > 1: y = np.rollaxis(y, axis, len(y.shape))
# Check if x and y are the same length, zero-pad if necessary if not same_data: if x.shape[-1] != y.shape[-1]: if x.shape[-1] < y.shape[-1]: pad_shape = list(x.shape) pad_shape[-1] = y.shape[-1] - x.shape[-1] x = np.concatenate((x, np.zeros(pad_shape)), -1) else: pad_shape = list(y.shape) pad_shape[-1] = x.shape[-1] - y.shape[-1] y = np.concatenate((y, np.zeros(pad_shape)), -1)
if nperseg is not None: # if specified by user nperseg = int(nperseg) if nperseg < 1: raise ValueError('nperseg must be a positive integer')
# parse window; if array like, then set nperseg = win.shape win, nperseg = _triage_segments(window, nperseg,input_length=x.shape[-1])
if nfft is None: nfft = nperseg elif nfft < nperseg: raise ValueError('nfft must be greater than or equal to nperseg.') else: nfft = int(nfft)
if noverlap is None: noverlap = nperseg//2 else: noverlap = int(noverlap) if noverlap >= nperseg: raise ValueError('noverlap must be less than nperseg.') nstep = nperseg - noverlap
# Padding occurs after boundary extension, so that the extended signal ends # in zeros, instead of introducing an impulse at the end. # I.e. if x = [..., 3, 2] # extend then pad -> [..., 3, 2, 2, 3, 0, 0, 0] # pad then extend -> [..., 3, 2, 0, 0, 0, 2, 3]
if boundary is not None: ext_func = boundary_funcs[boundary] x = ext_func(x, nperseg//2, axis=-1) if not same_data: y = ext_func(y, nperseg//2, axis=-1)
if padded: # Pad to integer number of windowed segments # I.e make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg nadd = (-(x.shape[-1]-nperseg) % nstep) % nperseg zeros_shape = list(x.shape[:-1]) + [nadd] x = np.concatenate((x, np.zeros(zeros_shape)), axis=-1) if not same_data: zeros_shape = list(y.shape[:-1]) + [nadd] y = np.concatenate((y, np.zeros(zeros_shape)), axis=-1)
# Handle detrending and window functions if not detrend: def detrend_func(d): return d elif not hasattr(detrend, '__call__'): def detrend_func(d): return signaltools.detrend(d, type=detrend, axis=-1) elif axis != -1: # Wrap this function so that it receives a shape that it could # reasonably expect to receive. def detrend_func(d): d = np.rollaxis(d, -1, axis) d = detrend(d) return np.rollaxis(d, axis, len(d.shape)) else: detrend_func = detrend
if np.result_type(win,np.complex64) != outdtype: win = win.astype(outdtype)
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 mode == 'stft': scale = np.sqrt(scale)
if return_onesided: if np.iscomplexobj(x): sides = 'twosided' warnings.warn('Input data is complex, switching to ' 'return_onesided=False') else: sides = 'onesided' if not same_data: if np.iscomplexobj(y): sides = 'twosided' warnings.warn('Input data is complex, switching to ' 'return_onesided=False') else: sides = 'twosided'
if sides == 'twosided': freqs = fftpack.fftfreq(nfft, 1/fs) elif sides == 'onesided': freqs = np.fft.rfftfreq(nfft, 1/fs)
# Perform the windowed FFTs result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides)
if not same_data: # All the same operations on the y data result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft, sides) result = np.conjugate(result) * result_y elif mode == 'psd': result = np.conjugate(result) * result
result *= scale if sides == 'onesided' and mode == 'psd': if nfft % 2: result[..., 1:] *= 2 else: # Last point is unpaired Nyquist freq point, don't double result[..., 1:-1] *= 2
time = np.arange(nperseg/2, x.shape[-1] - nperseg/2 + 1, nperseg - noverlap)/float(fs) if boundary is not None: time -= (nperseg/2) / fs
result = result.astype(outdtype)
# All imaginary parts are zero anyways if same_data and mode != 'stft': result = result.real
# Output is going to have new last axis for time/window index, so a # negative axis index shifts down one if axis < 0: axis -= 1
# Roll frequency axis back to axis where the data came from result = np.rollaxis(result, -1, axis)
return freqs, time, result
""" Calculate windowed FFT, for internal use by scipy.signal._spectral_helper
This is a helper function that does the main FFT calculation for `_spectral helper`. All input validation is performed there, and the data axis is assumed to be the last axis of x. It is not designed to be called externally. The windows are not averaged over; the result from each window is returned.
Returns ------- result : ndarray Array of FFT data
Notes ----- Adapted from matplotlib.mlab
.. versionadded:: 0.16.0 """ # Created strided array of data segments if nperseg == 1 and noverlap == 0: result = x[..., np.newaxis] else: # http://stackoverflow.com/a/5568169 step = nperseg - noverlap shape = x.shape[:-1]+((x.shape[-1]-noverlap)//step, nperseg) strides = x.strides[:-1]+(step*x.strides[-1], x.strides[-1]) result = np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)
# Detrend each data segment individually result = detrend_func(result)
# Apply window by multiplication result = win * result
# Perform the fft. Acts on last axis by default. Zero-pads automatically if sides == 'twosided': func = fftpack.fft else: result = result.real func = np.fft.rfft result = func(result, n=nfft)
return result
""" Parses window and nperseg arguments for spectrogram and _spectral_helper. This is a helper function, not meant to be called externally.
Parameters --------- window : string, tuple, or ndarray If window is specified by a string or tuple and nperseg is not specified, nperseg is set to the default of 256 and returns a window of that length. If instead the window is array_like and nperseg is not specified, then nperseg is set to the length of the window. A ValueError is raised if the user supplies both an array_like window and a value for nperseg but nperseg does not equal the length of the window.
nperseg : int Length of each segment
input_length: int Length of input signal, i.e. x.shape[-1]. Used to test for errors.
Returns ------- win : ndarray window. If function was called with string or tuple than this will hold the actual array used as a window.
nperseg : int Length of each segment. If window is str or tuple, nperseg is set to 256. If window is array_like, nperseg is set to the length of the 6 window. """
#parse window; if array like, then set nperseg = win.shape if isinstance(window, string_types) or isinstance(window, tuple): # if nperseg not specified if nperseg is None: nperseg = 256 # then change to default if nperseg > input_length: warnings.warn('nperseg = {0:d} is greater than input length ' ' = {1:d}, using nperseg = {1:d}' .format(nperseg, input_length)) nperseg = input_length win = get_window(window, nperseg) else: win = np.asarray(window) if len(win.shape) != 1: raise ValueError('window must be 1-D') if input_length < win.shape[-1]: raise ValueError('window is longer than input signal') if nperseg is None: nperseg = win.shape[0] elif nperseg is not None: if nperseg != win.shape[0]: raise ValueError("value specified for nperseg is different from" " length of window") return win, nperseg
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