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scipy.signal.

# hilbert

scipy.signal.hilbert(x, N=None, axis=-1)

  FFT-based computation of the analytic signal.

  The analytic signal is calculated by zeroing out the negative frequencies and doubling the amplitudes of the positive frequencies in the FFT domain. The imaginary part of the result is the hilbert transform of the real-valued input signal.

  The transformation is done along the last axis by default.

  For numpy arrays, `scipy.fft.set_workers` can be used to change the number of workers used for the FFTs.

  Parameters:
    x array_like

      Signal data. Must be real.

    N int, optional

      Number of output samples. x is initially cropped or zero-padded to length N along axis. Default: `x.shape[axis]`

    axis int, optional

      Axis along which to do the transformation. Default: -1.

  Returns:
    xa ndarray

      Analytic signal of x, of each 1-D array along axis

  See also

  `envelope`

    Compute envelope of a real- or complex-valued signal.

  Notes

  The analytic signal `x_a(t)` of a real-valued signal `x(t)` can be expressed as [1]

  \[x_a = F^{-1}(F(x) 2U) = x + i y\ ,\]

  where F is the Fourier transform, U the unit step function, and y the Hilbert transform of x. [2]

  In other words, the negative half of the frequency spectrum is zeroed out, turning the real-valued signal into a complex-valued signal. The Hilbert transformed signal can be obtained from `np.imag(hilbert(x))`, and the original signal from `np.real(hilbert(x))`.

  Array API Standard Support

  `hilbert` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ⛔ | ⛔
  Dask | ✅ | n/a

  > See Support for the array API standard for more information.

  References

  [1]

  Wikipedia, “Analytic signal”. https://en.wikipedia.org/wiki/Analytic_signal

  [2]

  Wikipedia, “Hilbert Transform”. https://en.wikipedia.org/wiki/Hilbert_transform

  [3]

  Leon Cohen, “Time-Frequency Analysis”, 1995. Chapter 2.

  [4]

  Alan V. Oppenheim, Ronald W. Schafer. Discrete-Time Signal Processing, Third Edition, 2009. Chapter 12. ISBN 13: 978-1292-02572-8

  Examples

  In this example we use the Hilbert transform to determine the amplitude envelope and instantaneous frequency of an amplitude-modulated signal.

  Let’s create a chirp of which the frequency increases from 20 Hz to 100 Hz and apply an amplitude modulation:

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy.signal import hilbert, chirp
      ...
      >>> duration, fs = 1, 400  # 1 s signal with sampling frequency of 400 Hz
      >>> t = np.arange(int(fs*duration)) / fs  # timestamps of samples
      >>> signal = chirp(t, 20.0, t[-1], 100.0)
      >>> signal *= (1.0 + 0.5 * np.sin(2.0*np.pi*3.0*t) )

  The amplitude envelope is given by the magnitude of the analytic signal. The instantaneous frequency can be obtained by differentiating the instantaneous phase in respect to time. The instantaneous phase corresponds to the phase angle of the analytic signal.

      >>> analytic_signal = hilbert(signal)
      >>> amplitude_envelope = np.abs(analytic_signal)
      >>> instantaneous_phase = np.unwrap(np.angle(analytic_signal))
      >>> instantaneous_frequency = np.diff(instantaneous_phase) / (2.0*np.pi) * fs
      ...
      >>> fig, (ax0, ax1) = plt.subplots(nrows=2, sharex='all', tight_layout=True)
      >>> ax0.set_title("Amplitude-modulated Chirp Signal")
      >>> ax0.set_ylabel("Amplitude")
      >>> ax0.plot(t, signal, label='Signal')
      >>> ax0.plot(t, amplitude_envelope, label='Envelope')
      >>> ax0.legend()
      >>> ax1.set(xlabel="Time in seconds", ylabel="Frequency in Hz", ylim=(0, 120))
      >>> ax1.plot(t[1:], instantaneous_frequency, 'C2-',
      ...          label='Instantaneous Frequency')
      >>> ax1.legend()
      >>> plt.show()