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def istft(Zxx, fs=1.0, window='hann', nperseg=None, noverlap=None, nfft=None, input_onesided=True, boundary=True, time_axis=-1, freq_axis=-2, scaling='spectrum')
Perform the inverse Short Time Fourier transform (legacy function).
.. legacy:: function
`ShortTimeFFT` is a newer STFT / ISTFT implementation with more
features. A :ref:`comparison <tutorial_stft_legacy_stft>` between the
implementations can be found in the :ref:`tutorial_stft` section of the
:ref:`user_guide`.
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``).
scaling: {'spectrum', 'psd'}
The default 'spectrum' scaling allows each frequency line of `Zxx` to
be interpreted as a magnitude spectrum. The 'psd' option scales each
line to a power spectral density - it allows to calculate the signal's
energy by numerically integrating over ``abs(Zxx)**2``.
Returns
-------
t : ndarray
Array of output data times.
x : ndarray
iSTFT of `Zxx`.
See Also
--------
stft: Short Time Fourier Transform
ShortTimeFFT: Newer STFT/ISTFT implementation providing more features.
check_COLA: Check whether the Constant OverLap Add (COLA) constraint
is met
check_NOLA: Check whether the Nonzero Overlap Add (NOLA) 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 "nonzero
overlap add" (NOLA):
.. math:: \sum_{t}w^{2}[n-tH] \ne 0
This ensures that the normalization factors that appear in the denominator
of the overlap-add reconstruction equation
.. math:: x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[n-tH]}
are not zero. The NOLA constraint can be checked with the `check_NOLA`
function.
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 estimation
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. Lim "Signal Estimation from
Modified Short-Time Fourier Transform", IEEE 1984,
10.1109/TASSP.1984.1164317
Examples
--------
>>> import numpy as np
>>> from scipy import signal
>>> import matplotlib.pyplot as plt
>>> rng = np.random.default_rng()
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 = rng.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, shading='gouraud')
>>> 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()
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