1
$\begingroup$

I have an observed signal which is the norm of the sum of two 3-vectors: $$ S(t) = || \mathbf{A}(t) + \mathbf{B}(t) || $$ Now, I have full knowledge of the vector $\mathbf{A}(t)$, $$ \mathbf{A}(t) = \pmatrix{A_1(t) \\ A_2(t) \\ A_3(t)} $$ i.e. I know each of the components $A_i(t)$. The signal $\mathbf{B}(t)$ is unknown, and $S(t)$ is what I observe. Now the signal $\mathbf{B}(t)$ has peaks, particularly in the $B_1(t)$ component, and my goal is to determine the position of these peaks. I don't care about the amplitude of the peaks, I only want the position (time).

So what I need to do is somehow filter out the signal $\mathbf{A}(t)$ from the observed signal $S(t)$ and then apply a peak detection algorithm on the residual signal. My question is whether it is possible to do this?

As a simple example, I made a matlab script to construct synthetic data as follows: $$ \mathbf{A}(t) = \pmatrix{ X(t) \\ Y(t) \\ 0} $$ $$ \mathbf{B}(t) = \pmatrix{ G(t) \\ 0 \\ 0} $$ where $X(t) = \sin{(2 \pi (f/4) t)}$, $Y(t) = \sin{(2 \pi f t)}$, and $G(t)$ is a single Gaussian peak. The signals and matlab script are given below. As you can see, it is not easy to tell that there is a peak in my observed signal at $t = 500$ as there is in the unknown signal $G(t)$.

So is it possible to filter $S(t)$ to remove $\mathbf{A}(t)$ and clearly find the peaks in $\mathbf{B}(t)$? enter image description here

N = 1000;
t = [1:N];

f = 20;
X = sin(2*pi*(f/4)*t/N);
Y = sin(2*pi*f*t/N);

G = gausswin(N, 100.0);
Z = zeros(1,N);

A = [ X ; Y ; Z];
B = [ G' ; Z ; Z ];
C = A + B;

signal = vecnorm(C);

figure

subplot(4,1,1);
plot(t,X);
title('Known X');

subplot(4,1,2);
plot(t,Y);
title('Known Y');

subplot(4,1,3);
plot(t,G);
title('Unknown signal G');

subplot(4,1,4);
plot(t,signal)
title('Observed signal');
$\endgroup$
1
$\begingroup$

Generally, this should not be possible because:

  • you project a 3D information onto a 1D signal, and the system is underdetermined in general,
  • you add a non-linearity (the norm), that further hinder restoration.

However, using the triangle inequality, you can bound $B$ from below:

$$ \|A+B\| \le \|A\|+\|B\|$$

hence

$$ B_{\inf} =\max(\|A+B\| -\|A\|,0)\le \|B\|$$

In the example you gave, if the signals $A$ and $B$ are weakly related, outcomes can become interesting:

signal = (sum(C.^2)).^(0.5);
signalInf = max((signal-(sum(A.^2)).^(0.5)),0);
signalG = (sum(B.^2)).^(0.5);

plot(t,[signal',signalInf',signalG'])
title('Observed signal');
legend('A','Binf','B')

norm inequality

From here, you can recover putative peaks from $ B_{\inf}$. However, it is unlikely to work every time.

Could additional filtering help on $S(t)$? Without knowledge about the relative spectra of $A$ or $B$, I don't know (yet).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.