I had recently posted a question on applying powers on a sum of Gaussians (here) to enhance signal resolution artificially. The discussion has given rise to another query. Consider this problem. We have two Gaussian time functions $G_1$ and $G_2$. Suppose we wish sum these Gaussians and raise it to a power of 2.
Situation no. 1: $(G_1+G_2)^2$, we should get another function $G$= $G_1^2$ + $2G_1G_2$ + $G_2^2$. The product of two $G_1G_2$ is another Gaussian. We should see three peaks in this case.
Situation no. 2: We sum the two Gaussians element wise i.e., $S_i$ = $g(t)_{1i}$ + $g(t)_{2i}$ where $g(t)_i$ 's are the corresponding elements of $G_1$ + $G_2$. Now each summed element $S_i$ is raised to power 2. We get only two peaks as a result.
I discussed this issue with some of my colleagues including the person who had proposed this method; they seek agree that both operations are equivalent. It seems they are not, because in situation 2, we will never see $G_1G_2$. What would the discrete version of situation 1?
I feel there is some fallacy here and some mathematical insight should be able to settle this problem.
Thanks.
For Situation no. 2: Here is the MATLAB code.
t=[0:1/200:60]';
u1= 19; % mu the mean time peak i
u2= 22; % mu the mean time peak j
A= 250; %Area
s_ij=[0.84 0.87];; %s_ij represent the sigma of i,j pair; Choose elements by indexing
G_ij=A*normpdf(t, u1, s_ij(1,1))+ A*normpdf(t, u2, s_ij(1,2)); %Sum normpdf is a built in function for a unit area Gaussian from statistical package.
Power= (G_ij).^2;
subplot(2,1,1); plot(t,G_ij);title('Unresolved Gaussians');
subplot(2,1,2); plot (t, Power);title ('Gaussian Sum Raised to Power 2');
G_ij
in the code. $\endgroup$ – MBaz Sep 17 '19 at 13:24