Automate the length of samples in the envelope function to get the optimal peak envelope from various signals

The optimal extraction of peak envelope with the 'envelope' function (https://nl.mathworks.com/help/signal/ref/envelope.html) necessitates the optimal selection of the sample length ("The envelopes are determined using spline interpolation over local maxima separated by at least np samples."). While manual optimization of the length is the most common route, it does not allow a rapid and accurate envelope extraction from different types of signals (with different time - frequency relationships). Is there a way to automate the selection of sample length for obtaining the optimal envelope? How to describe the optimal peak envelope (one definition would be touching all the peaks with minimal difference)?

[Mathieu NOE]

hello

do you have some code and data to share ?

[Shiva Vikram Bhagavatula]

Hi, I have attached the sample code which illustrates the issue. Changing the variable "factor" leads to either a better or worse envelope with an optimal value.

Thank you for the reply.

[Star Strider]

The code works, and appears to be correct —

type('envelope_fmcw_testing.m')

clear all
close all

t=0:1e-9:1e-6;% time
fc=1e6;       % carrier 
del_f=300e3;  % frequency deviation
mod_f=100e3;  % modulating frequency

std_dev=10;   %introduce randomness in amplitude of carrier and frequency deviation
amp=(std_dev*randn(length(t),1))';            
mod_signal=1e-2*amp.*(del_f*cos(2*pi*mod_f*t)); % modulating signal with randomized frequency deviation 
fmcw_mod=5*amp.*cos(2*pi*fc*t+mod_signal);      % fmcw modulated signal

figure
plot(t,fmcw_mod);

factor=2^9
env_len=round(length(fmcw_mod)/factor);%length over which the function executes
[up,lo]=envelope(fmcw_mod,env_len,'peak');

hold all
plot(t,up);% upper envelope
plot(t,lo);% lower envelope

clear all
close all

t=0:1e-9:1e-6;% time
fc=1e6;       % carrier 
del_f=300e3;  % frequency deviation
mod_f=100e3;  % modulating frequency

std_dev=10;   %introduce randomness in amplitude of carrier and frequency deviation
amp=(std_dev*randn(length(t),1))';            
mod_signal=1e-2*amp.*(del_f*cos(2*pi*mod_f*t)); % modulating signal with randomized frequency deviation 
fmcw_mod=5*amp.*cos(2*pi*fc*t+mod_signal);      % fmcw modulated signal

figure
plot(t,fmcw_mod);

factor=2^9

factor = 512

env_len=round(length(fmcw_mod)/factor);%length over which the function executes
[up,lo]=envelope(fmcw_mod,env_len,'peak');

hold all
plot(t,up);% upper envelope
plot(t,lo);% lower envelope

figure
plot(t,fmcw_mod);
hold on
plot(t,up);% upper envelope
plot(t,lo);% lower envelope
hold off
xlim([0.4 0.6]*1E-6)
title('Detail')

.

[Shiva Vikram Bhagavatula]

Thanks for looking through the code. The code works fine, but it still misses a lot of peaks and valleys . It is preferred if the upper (or lower) envelope touches all the peaks (or valleys). In this case, the specific choice of sample length was manually optimized. The choice of length needs to be automated to adjust to variations in carrier frequency as well. The automated optimization with find peaks and an error metric (as suggested by @Star Strider) would be a plausible solution to this.

[Star Strider]

This is the sort of approach I had in mind when I wrote my Answer —

clear all
close all

t=0:1e-9:1e-6;% time
fc=1e6;       % carrier 
del_f=300e3;  % frequency deviation
mod_f=100e3;  % modulating frequency

std_dev=10;   %introduce randomness in amplitude of carrier and frequency deviation
amp=(std_dev*randn(length(t),1))';            
mod_signal=1e-2*amp.*(del_f*cos(2*pi*mod_f*t)); % modulating signal with randomized frequency deviation 
fmcw_mod=5*amp.*cos(2*pi*fc*t+mod_signal);      % fmcw modulated signal

figure
plot(t,fmcw_mod);

factor=2^9

factor = 512

% env_len=round(length(fmcw_mod)/factor);%length over which the function executes
[pks,locs] = findpeaks(fmcw_mod, 'MinPeakProminence',10)        % Use 'MinPeakProminence' To Select The Peaks To Detect 

pks = 1×287

57.2126 50.1938 119.7309 48.5453 13.4690 22.4986 -1.5904 38.7909 74.1662 29.8850 42.2083 79.4538 17.2369 67.5221 113.0527 2.8986 57.9184 35.6831 29.1597 0.3300 18.3431 22.4078 26.7490 36.7620 8.1993 8.4524 45.4026 55.4935 38.1078 15.8680

locs = 1×287

3 5 9 17 19 21 24 28 32 35 38 45 48 52 57 60 63 65 69 75 77 82 85 90 94 96 101 108 110 113

env_len = floor(median(diff(locs)))                             % Create An Appropriate Metric From The Returned Peak Indices

env_len = 3

[up,lo]=envelope(fmcw_mod,env_len,'peak');

hold all
plot(t,up);% upper envelope
plot(t,lo);% lower envelope

figure
plot(t,fmcw_mod);
hold on
plot(t,up);% upper envelope
plot(t,lo);% lower envelope
hold off
xlim([0.4 0.6]*1E-6)
title('Detail')

.

Answer 1

[Star Strider]

Consider first using either findpeaks or islocalmax with find, and then calculate the appropriate distances (using min, max, mean, median or some other metric on the differences of the peak locations) to get the appropriate parameter for the second envelope argument. This is likely easiest, although another option would be to calculate the one-sided fft, find the peaks of the absolute values, and then calculate the inverse of the frequency at that peak location.