how to robustly estimate low and up envelope of signal with trend, few level constant steps and noise

Michal Kvasnicka (view profile)

on 19 Sep 2018
Latest activity Commented on by Michal Kvasnicka

Michal Kvasnicka (view profile)

on 19 Sep 2018
I am looking for robust estimation method of low and up envelope of the signal consisting from smooth trend component, constant steps between few fixed levels and additive noise (+ outliers of course). This question raised during my current research work on real life signal processing.
Typical example of signal is produced by following script:
%%signal definition
% number of samples
Ns = 10000;
% sampling period [secs]
Ts = 1;
time = (1:Ns)*Ts;
% trend component
a = 2;
b = 0;
T = 1e4;
slope = time/Ts * 0.0005;
trend = a * sin(2*pi*time/T) + b + slope;
% steps component (4 constant levels)
step = [zeros(1,1000),linspace(0,1,6),1*ones(1,500),linspace(1,0,6),zeros(1,500),linspace(0,-2,10), -2*ones(1,1200),linspace(-2,1,15),1*ones(1,3000),linspace(1,0,6), zeros(1,3000),linspace(0,-1,6), -1*ones(1,751)];
% noise component (normal noise)
noise = 0.2*randn(1,Ns);
% noise = 0.5*(rand(1,Ns)-0.5);
%
%%show signals component
close all
figure
plot(time,trend,'r-')
hold on
plot(time,trend+step,'g-')
plot(time,trend - 2, 'b--')
plot(time,trend + 1, 'k--')
plot(time,trend+step+noise,'mo')
legend('trend','trend+steps','lowenvelope', 'upenvelope','trend+steps+noise')
title('smooth trend signal with constant steps between 4 levels and noise')
xlabel('time [sec]')
ylabel('value [-]')
hold off
See the following image
The separate signal components are unknown! Steps are always constant and between small number of fixed levels (typically < 4 or 5), so the estimated envelopes should be parallel to the trend signal. Noise is approximated by normal distribution with sigma ~0.1
Any idea how to solve this surprisingly difficult problem? Any relevant references or matlab code links?

Michal Kvasnicka

Michal Kvasnicka (view profile)

on 19 Sep 2018
In principle I am looking for any robust estimation of trend signal component which is nearly parallel to real trend signal.