# scipy.signal.istft

scipy.signal.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 (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`).

    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

  `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):

  \[\sum_{t}w^{2}[n-tH] \ne 0\]

  This ensures that the normalization factors that appear in the denominator of the overlap-add reconstruction equation

  \[x[n]=\frac{\sum_{t}x_{t}[n]w[n-tH]}{\sum_{t}w^{2}[n-tH]}\]

  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.

  New in version 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()