Envelope detection using matlab hilbert function

I am trying to detect the envelope of a high-frequency signal (similar to below) using Matlab Hilbert function This is my output It generally follows the envelope, but there is a high-frequency ripple in the envelope, see zoomed version below Here is my code:

t =out.tout';
x =out.Itxmod';
y = hilbert(x);
env = abs(y);
figure;
plot_param = {'Color', [0.6 0.1 0.2],'Linewidth',2};
plot(t,x)
hold on
plot(t,[-1;1]*env,plot_param{:})
hold off
xlim([0.001 0.002])
title('Hilbert Envelope')

Question: How can I get a smooth envelope the signal without this high-frequency ripple?

This must be an artifact of the implementation for the hilbert function (which to be clear is NOT the Hilbert Transform, but the analytic signal which consists of the signal plus it's Hilbert transform on the imaginary axis), in that any implementation of finite length will have a finite ripple. It doesn't appear the underlying filter using in the hilbert function can be modified, so there are three suggestions to obtain a smoothed demodulated result:

Method 1: Determine the analytic signal using the FFT: take the FFT of the original signal, set all the negative frequencies to zero (the upper half of the FFT) and double the positive frequencies leaving bin 0 (DC) as is (and as Overlord pointed out in the comments, leave the Nyquist bin at $$N/2$$ as is when the FFT length $$N$$ is even). The IFFT will be the analytic signal and taking the absolute value of this signal will be the desired envelope. (As @aconcernedcitizen pointed out in the comments, this IS the method specifically implemented by Octave using the hilbert function from the signal package.)

Method 2: Determine the analytic signal with quadrature phase tracking filters where the filter length versus ripple can be traded.

Method 3: Use a traditional AM demodulator by multiplying the modulated signal with the coherent carrier and then low pass filtering the result.

A demonstration of Method 1 is shown below, which is a suitable solution for the OP's case: Zooming in on the start of the signal shows the distortion limited to the start-up condition and the smooth envelope after that. The Matlab code for this is as follows:

N = length(sig);
sig_spectrum = fft(sig);
hilbert_spectrum = zeros(1, N);
hilbert_spectrum(1) = sig_spectrum(1);
hilbert_spectrum(2: ceil(N/2) - 1) = 2 * sig_spectrum(2:ceil(N/2) - 1);
if mod(N, 2) == 0
hilbert_spectrum(N/2) = sig_spectrum(N/2);
endif
hilbert_time = ifft(hilbert_spectrum);
envelope = abs(hilbert_time);
• Caveat, Nyquist bin shouldn't be doubled (even N). Apr 23 '21 at 20:44
• Good point! The bin at N/2 should not be doubled when N is even. Apr 24 '21 at 0:24
• I thought method 1 is the one that it's done by default with hilbert()? I know Octave uses it like that. Apr 24 '21 at 8:07
• @aconcernedcitizen you're right, I just looked at the code in Octave and it is exactly method 1 Apr 24 '21 at 11:01
• Hi Dan, I think MATLAB's hilbert() also uses method 1, according to the document Aug 27 '21 at 2:00