code: 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",
  ):
      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
          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):
  
      .. 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.
  
      See [1]_, [2]_ for more information.
  
      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 cupy
      >>> from cupyx.scipy.signal import stft, istft
      >>> 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 = cupy.arange(N) / float(fs)
      >>> carrier = amp * cupy.sin(2*cupy.pi*50*time)
      >>> noise = cupy.random.normal(scale=cupy.sqrt(noise_power),
      ...                          size=time.shape)
      >>> x = carrier + noise
  
      Compute the STFT, and plot its magnitude
  
      >>> f, t, Zxx = cusignal.stft(x, fs=fs, nperseg=nperseg)
      >>> f = cupy.asnumpy(f)
      >>> t = cupy.asnumpy(t)
      >>> Zxx = cupy.asnumpy(Zxx)
      >>> 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 = cupy.where(cupy.abs(Zxx) >= amp/10, Zxx, 0)
      >>> _, xrec = cusignal.istft(Zxx, fs)
      >>> xrec = cupy.asnumpy(xrec)
      >>> x = cupy.asnumpy(x)
      >>> time = cupy.asnumpy(time)
      >>> carrier = cupy.asnumpy(carrier)
  
      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 = cupy.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
  
      # 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 = cupy.transpose(Zxx, zouter + [freq_axis, time_axis])
  
      # Get window as array
      if isinstance(window, str) or type(window) is tuple:
          win = get_window(window, nperseg)
      else:
          win = cupy.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)
              )
  
      ifunc = cupy.fft.irfft if input_onesided else cupy.fft.ifft
      xsubs = ifunc(Zxx, axis=-2, n=nfft)[..., :nperseg, :]
  
      # Initialize output and normalization arrays
      outputlength = nperseg + (nseg - 1) * nstep
      x = cupy.zeros(
          list(Zxx.shape[:-2]) + [outputlength], dtype=xsubs.dtype
      )
      norm = cupy.zeros(outputlength, dtype=xsubs.dtype)
  
      if cupy.result_type(win, xsubs) != xsubs.dtype:
          win = win.astype(xsubs.dtype)
  
      if scaling == "spectrum":
          xsubs *= win.sum()
      elif scaling == "psd":
          xsubs *= cupy.sqrt(fs * cupy.sum(win**2))
      else:
          raise ValueError(
              f"Parameter {scaling=} not in ['spectrum', 'psd']!"
          )
  
      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
  
      # Remove extension points
      if boundary:
          x = x[..., nperseg // 2 : -(nperseg // 2)]
          norm = norm[..., nperseg // 2 : -(nperseg // 2)]
  
      # Divide out normalization where non-tiny
      if cupy.sum(norm > 1e-10) != len(norm):
          warnings.warn(
              "NOLA condition failed, STFT may not be invertible"
          )
      x /= cupy.where(norm > 1e-10, norm, 1.0)
  
      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 = cupy.rollaxis(x, -1, time_axis)
  
      time = cupy.arange(x.shape[0]) / float(fs)
      return time, x
  
file_path: cupy/cupyx/scipy/signal/_spectral.py
imports: import warnings
  import cupy
  from cupyx.scipy.signal.windows._windows import get_window
  from cupyx.scipy.signal._spectral_impl import _lombscargle, _spectral_helper, _median_bias, _triage_segments
label_desc: Use image node_id 8 for calling a global function with example usage: istft(Zxx, fs, window, nperseg, noverlap, nfft, input_onesided, boundary, time_axis, freq_axis, scaling) and returns: time,  x