# Element-wise powering vs. discrete version of binomial theorem on a sum of Gaussians

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');

• i might make a stab at answering this question, but i still don't see this question described adequately. there is more to a random signal in the time domain than it's p.d.f. (which is gaussian). there are also the spectral characteristics which are often described with a power spectral density or with the autocorrelation of a stochastic signal. anyway, to be consistent with electrical engineering convention, you should denote the time functions as "$g_1(t)$" and "$g_2(t)$". and define how they are related to each other (if they are at all), but we need the p.d.f. and autocorrelation at least – robert bristow-johnson Sep 17 '19 at 3:10
• @robertbristow-johnson, I am afraid this is not as complicated. I am a chemist, and the sum of two simple Gaussians are just a simulation of two chromatogram peaks. There is no correlation concept involved here, as far I can understand and the position of two peaks is not all related -just two arbitrary peaks. – M. Farooq Sep 17 '19 at 3:27
• I don't see a definition for G_ij in the code. – MBaz Sep 17 '19 at 13:24
• Corrected. G_ij – M. Farooq Sep 17 '19 at 13:32
• In the first plot you have the sum of two Gaussians. In the second, you have the sum squared. I don't understand why you think the two should be the same? – MBaz Sep 17 '19 at 14:59

([a b c] .+ [d e f]).^2 = [a+d b+e c+f].^2
= [(a+d)^2 (b+e)^2 (c+f)^2]


I suspect that:

• The problem may be in your assumptions: "We should see three peaks in this case" sounds like a hypothesis that may be false.
• You may have a bug in your code, which might be why "We get only two peaks as a result".

Here is some code to illustrate that (A.+B).^2 = A.^2 .+ 2A.*B .+ B.^2:

t=0:1/200:60;
gauss1 = 250*normpdf(t,19,0.84);
gauss2 = 250*normpdf(t,22,0.87);
m1 = gauss1.^2+2*gauss1.*gauss2+gauss2.^2;
m2 = (gauss1+gauss2).^2;
fprintf('Difference: %1.2f\n', sum(abs(m1-m2)));
subplot(2,1,1);plot(m1);title('Method 1');
subplot(2,1,2);plot(m2);title('Method 2');


I get:

Difference: 0.00


and So, the math is correct. You may still be wondering, how come the third Gaussian $$G_1G_2$$ is not apparent in the plot? This is the point where it's useful to challenge your assumption that the third Gaussian should be visible. If we plot $$G_1G_2$$ by itself we obtain:

plot(2*gauss1.*gauss2); Look at the amplitude and the position in the x-axis: this third Gaussian is not apparent in the plot because it is completely swamped by the other two, much larger, surrounding peaks.

• I am afraid there is no fancy code, I am just doing it in Excel or Matlab. I think there is some other fundamental fallacy. – M. Farooq Sep 17 '19 at 2:15
• I doubt it. The problem is not complex enough to hide a fallacy. If you share your code I'll take a look at it. – MBaz Sep 17 '19 at 2:40
• added the code for the sake of completeness. Thanks. – M. Farooq Sep 17 '19 at 2:55
• I was just about to post an answer when I saw that @MBaz nailed it. I got the same thing, as expected. I already upvoted this answer. So I will e-mail what I have separately to M. Farooq. – Ed V Sep 17 '19 at 15:16
• So we do generate a new peak when we square element-wise. Thanks for double checking and confirming this problem. – M. Farooq Sep 17 '19 at 17:18