ncies 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
----------
x : array_like
    Signal data.  Must be real.
N : int, optional
    Number of Fourier components.  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`

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

.. math:: 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

See Also
--------
envelope: Compute envelope of a real- or complex-valued signal.

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()

úx must be real.Nr