Numerically computing envelope of Gibbs oscillation

Question

If I numerically compute the envelope of $\sin(\pi t)$ using a Hilbert transform, I obtain exactly what I expect:

If I do the same for $\mathrm{sinc}(t)$, still I obtain an envelope which agrees with my intuitions:

However, if I try to find the envelope of a Gibbs oscillation using the Fourier series for the Heaviside $\theta$ function, I obtain the following very wild figure which does not agree with my intuitions about what an envelope should look like:

Are there techniques for computing the envelope of Gibbs oscillations? I have tried a Kaiser window on the times series before computing the Hilbert transform, but no joy.

Code to reproduce:

#!/usr/bin/env python3
from math import pi as π
import random
import scipy
import numpy

import matplotlib
import matplotlib.pyplot as plt

def sine_envelope():
    times = numpy.linspace(-1, 1, 512)
    values = numpy.sin(π*times)
    analytic_signal = scipy.signal.hilbert(values)
    hilbert_transform = numpy.imag(analytic_signal)
    plt.plot(times, values, color="orange", label="Signal")
    plt.plot(times, hilbert_transform, color="green", label="Hilbert transform")
    plt.plot(times, numpy.abs(analytic_signal), color="steelblue", label="Envelope")
    plt.legend()
    plt.tight_layout()
    plt.show()

def sinc_envelope():
    times = numpy.linspace(-1, 1, 512)
    values = numpy.sinc(π*times)
    analytic_signal = scipy.signal.hilbert(values)
    hilbert_transform = numpy.imag(analytic_signal)
    plt.plot(times, values, color="orange", label="Signal")
    plt.plot(times, hilbert_transform, color="green", label="Hilbert transform")
    plt.plot(times, numpy.abs(analytic_signal), color="steelblue", label="Envelope")
    plt.legend()
    plt.tight_layout()
    plt.show()

def gibbs_oscillation_envelope():
    times = numpy.linspace(-1, 1, 512)
    values = numpy.zeros(len(times))
    for n in range(20):
        m = 2*n+1
        bm = 2/(m*π)
        values += bm*numpy.sin(2*π*m*times/10)

    analytic_signal = scipy.signal.hilbert(values)
    hilbert_transform = numpy.imag(analytic_signal)
    plt.plot(times, values, color="orange", label="Signal")
    plt.plot(times, hilbert_transform, color="green", label="Hilbert transform")
    plt.plot(times, numpy.abs(analytic_signal), color="steelblue", label="Envelope")
    plt.legend()
    plt.tight_layout()
    plt.show()




if __name__ == '__main__':
    sine_envelope()
    sinc_envelope()
    gibbs_oscillation_envelope()

Answer 1

The reason seems Hilbert transform works the best for signals like $x(t)=a\cos(\phi(t))$, where $a$ may change mildly. In your last case, this $a$ is not in this scenario.

You can try to apply your code to this

 values = numpy.zeros(len(times)) + 2

which simply shifts the whole function, but it should generate better result.