scipy.signal.

hilbert#

scipy.signal.hilbert(x, N=None, axis=-1)[source]#

FFT-based computation of the analytic signal.

The analytic signal is calculated by filtering 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.

Parameters:

xarray_like: Signal data. Must be real.
Nint, optional: Number of Fourier components. Default: x.shape[axis]
axisint, optional: Axis along which to do the transformation. Default: -1.

Returns:

xandarray: 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)).

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="Phase in rad", ylim=(0, 120))
>>> ax1.plot(t[1:], instantaneous_frequency, 'C2-', label='Instantaneous Phase')
>>> ax1.legend()
>>> plt.show()

../../_images/scipy-signal-hilbert-1.png