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# scipy.signal.ShortTimeFFT.from_dual

classmethod ShortTimeFFT.from_dual(dual_win, hop, fs, *, fft_mode='onesided', mfft=None, scale_to=None, phase_shift=0)

  Instantiate a `ShortTimeFFT` by only providing a dual window.

  If an STFT is invertible, it is possible to calculate the window `win` from a given dual window `dual_win`. All other parameters have the same meaning as in the initializer of `ShortTimeFFT`.

  As explained in the Short-Time Fourier Transform section of the SciPy User Guide, an invertible STFT can be interpreted as series expansion of time-shifted and frequency modulated dual windows. E.g., the series coefficient S[q,p] belongs to the term, which shifted `dual_win` by p * `delta_t` and multiplied it by exp( 2 * j * pi * t * q * `delta_f`).

  See also

  `from_window`

    Create instance by wrapping `get_window`.

  `ShortTimeFFT`

    Create instance using standard initializer.

  Examples

  The following example discusses decomposing a signal into time- and frequency-shifted Gaussians. A Gaussian with standard deviation of one made up of 51 samples will be used:

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy.signal import ShortTimeFFT
      >>> from scipy.signal.windows import gaussian
      ...
      >>> T, N = 0.1, 51
      >>> d_win = gaussian(N, std=1/T, sym=True)  # symmetric Gaussian window
      >>> t = T * (np.arange(N) - N//2)
      ...
      >>> fg1, ax1 = plt.subplots()
      >>> ax1.set_title(r"Dual Window: Gaussian with $\sigma_t=1$")
      >>> ax1.set(xlabel=f"Time $t$ in seconds ({N} samples, $T={T}$ s)",
      ...        xlim=(t[0], t[-1]), ylim=(0, 1.1*max(d_win)))
      >>> ax1.plot(t, d_win, 'C0-')

  The following plot with the overlap of 41, 11 and 2 samples show how the `hop` interval affects the shape of the window `win`:

      >>> fig2, axx = plt.subplots(3, 1, sharex='all')
      ...
      >>> axx[0].set_title(r"Windows for hop$\in\{10, 40, 49\}$")
      >>> for c_, h_ in enumerate([10, 40, 49]):
      ...     SFT = ShortTimeFFT.from_dual(d_win, h_, 1/T)
      ...     axx[c_].plot(t + h_ * T, SFT.win, 'k--', alpha=.3, label=None)
      ...     axx[c_].plot(t - h_ * T, SFT.win, 'k:', alpha=.3, label=None)
      ...     axx[c_].plot(t, SFT.win, f'C{c_+1}',
      ...                     label=r"$\Delta t=%0.1f\,$s" % SFT.delta_t)
      ...     axx[c_].set_ylim(0, 1.1*max(SFT.win))
      ...     axx[c_].legend(loc='center')
      >>> axx[-1].set(xlabel=f"Time $t$ in seconds ({N} samples, $T={T}$ s)",
      ...             xlim=(t[0], t[-1]))
      >>> plt.show()

  Beside the window `win` centered at t = 0 the previous (t = -`delta_t`) and following window (t = `delta_t`) are depicted. It can be seen that for small `hop` intervals, the window is compact and smooth, having a good time-frequency concentration in the STFT. For the large `hop` interval of 4.9 s, the window has small values around t = 0, which are not covered by the overlap of the adjacent windows, which could lead to numeric inaccuracies. Furthermore, the peaky shape at the beginning and the end of the window points to a higher bandwidth, resulting in a poorer time-frequency resolution of the STFT. Hence, the choice of the `hop` interval will be a compromise between a time-frequency resolution and memory requirements demanded by small `hop` sizes.