The transformation is done along the last axis by default.

The analytic signal x_a(t) of signal x(t) is:

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

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

Compute the analytic signal, using the Hilbert transform.

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import hilbert, chirp

duration = 1.0
fs = 400.0
samples = int(fs*duration)
t = np.arange(samples) / fs

signal = chirp(t, 20.0, t[-1], 100.0)
signal *= (1.0 + 0.5 * np.sin(2.0*np.pi*3.0*t) )

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)
ax0.plot(t, signal, label='signal')
ax0.plot(t, amplitude_envelope, label='envelope')
ax0.set_xlabel("time in seconds")
ax0.legend()
ax1.plot(t[1:], instantaneous_frequency)
ax1.set_xlabel("time in seconds")
ax1.set_ylim(0.0, 120.0)
fig.tight_layout()