Conditions for which the Hilbert transform returns a correct phase

Question

I'm quite new to signal analysis, and I'm currently trying to understand under which conditions a Hilbert transform can be used to compute the correct instantaneous phase and enveloppe of a given signal.

Say I start from the example in Python given here (from the scipy website):

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 = plt.figure()
ax0 = fig.add_subplot(211)
ax0.plot(t, signal, label='signal')
ax0.plot(t, amplitude_envelope, label='envelope')
ax0.set_xlabel("time in seconds")
ax0.legend()
ax1 = fig.add_subplot(212)
ax1.plot(t[1:], instantaneous_frequency)
ax1.set_xlabel("time in seconds")
ax1.set_ylim(0.0, 120.0)

I get a nice result showing me the enveloppe and desired frequency:

However, if for instance I decide to add a little bit of noise:

signal+=random.rand(len(signal))*0.2

Then the inferred enveloppe and frequencies get not that good (some of the noise goes into the enveloppe, some into the frequencies):

This was quite expectable, and I know I just have to find a way to smooth my signal to solve that. My real problem is, say you add a trend to your signal:

signal+=np.linspace(-1,1,400)

Then the results get completely messed up: *can't add any more links since I'm new to the forum*

And the results are also messed up even for just a vertical shift... So, basically, my question is: what are the conditions needed for the Hilbert transform to return the correct phase and amplitude ?

For now, this is what I suppose:

• the signal must not be noisy
• the signal must be centered around zero
• the signal must not have any trend
• amplitude and frequency can vary

Am I right? Thank you.

Answer 1

A single instantaneous phase estimate may or may not make any sense if there is more than one frequency peak in the signal's local spectrum. So, to get a better single frequency and phase estimate, you may want to first high-pass filter your signal to remove the spectrum of any D.C. bias or slow trend, and also band-pass filter the signal to remove any outside spectral peaks due to noise or interfering signals.

So my off-hand guess for a condition is that the local spectrum has a single clear frequency peak with a monotonicly decreasing spectral magnitude away from that one peak. All the way to zero at DC for real signals.