sform (legacy function). STFTs can be used as a way of quantifying the change of a nonstationary signal's frequency and phase content over time. .. legacy:: function `ShortTimeFFT` is a newer STFT / ISTFT implementation with more features. A :ref:`comparison ` between the implementations can be found in the :ref:`tutorial_stft` section of the :ref:`user_guide`. 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``. .. versionadded:: 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 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 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 :math:`x[n]`, a window :math:`w[n]`, and a hop size :math:`H` = `nperseg - noverlap`, the windowed frame at time index :math:`t` is given by .. math:: x_{t}[n]=x[n]w[n-tH] The overlap-add (OLA) reconstruction equation is given by .. math:: 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 denominator of the OLA reconstruction equation is nonzero. Whether a choice of `window`, `nperseg`, and `noverlap` satisfy this constraint can be tested with `check_NOLA`. .. 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 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 rk