# scipy.signal.stft

scipy.signal.stft(x, fs=1.0, window='hann', nperseg=256, noverlap=None, nfft=None, detrend=False, return_onesided=True, boundary='zeros', padded=True, axis=-1, scaling='spectrum')

  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. Defaults to True, but for complex data, a two-sided spectrum is always returned.

    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`).

    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`.

      New in version 1.9.0.

  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

  `check_NOLA`

    Check whether the Nonzero Overlap Add (NOLA) 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 “Nonzero OverLap Add” (NOLA), 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.

  Given a time-domain signal \(x[n]\), a window \(w[n]\), and a hop size \(H\) = nperseg - noverlap, the windowed frame at time index \(t\) is given by

  \[x_{t}[n]=x[n]w[n-tH]\]

  The overlap-add (OLA) reconstruction equation is given by

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

  The NOLA constraint ensures that every normalization term that appears in the denomimator of the OLA reconstruction equation is nonzero. Whether a choice of window, nperseg, and noverlap satisfy this constraint can be tested with `check_NOLA`.

  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 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 = rng.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, shading='gouraud')
      >>> plt.title('STFT Magnitude')
      >>> plt.ylabel('Frequency [Hz]')
      >>> plt.xlabel('Time [sec]')
      >>> plt.show()

  Compare the energy of the signal x with the energy of its STFT:

      >>> E_x = sum(x**2) / fs  # Energy of x
      >>> # Calculate a two-sided STFT with PSD scaling:
      >>> f, t, Zxx = signal.stft(x, fs, nperseg=1000, return_onesided=False,
      ...                         scaling='psd')
      >>> # Integrate numerically over abs(Zxx)**2:
      >>> df, dt = f[1] - f[0], t[1] - t[0]
      >>> E_Zxx = sum(np.sum(Zxx.real**2 + Zxx.imag**2, axis=0) * df) * dt
      >>> # The energy is the same, but the numerical errors are quite large:
      >>> np.isclose(E_x, E_Zxx, rtol=1e-2)
      True