Convolving two functions

Question

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 e^{-\frac{\left(a+p_1-x\right){}^2}{2 \left(q_1^2+\sigma ^2\right)}}+j_0 e^{-\frac{\left(b+p_1-x\right){}^2}{2 \left(q_1^2+\sigma ^2\right)}}\right)$$

The convolution is then this: $$ \int_{-\infty }^{\infty } e^{-\frac{{(x-p_2)}^2}{2 q_2^2}} \left(\left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 e^{-\frac{\left(a+p_1-x\right){}^2}{2 \left(q_1^2+\sigma ^2\right)}}+j_0 e^{-\frac{\left(b+p_1-x\right){}^2}{2 \left(q_1^2+\sigma ^2\right)}}\right)\text{/.}\, x\to z-x\right) \, dx $$

These are the assumptions: $\left(\sigma |a|b|i_0|i_1|j_0|j_1|p_0|p_1|q_0|q_1\right)\in \mathbb{R}\land \sigma >0\land q_1>0\land q_2>0$

Fiddling with Mathematica and intuition I've got to this: $$\sqrt{2 \pi } q_2 \sigma \sqrt{\frac{q_1^2+\sigma ^2}{2 q_2^2 \sigma ^2+q_1^2 \left(q_2^2+\sigma ^2\right)+\sigma ^4}} \times \left(i_0 e^{-\frac{\left(x-a-p_1-p_2\right){}^2}{2 \left(\sigma ^2+q_1^2+q_2^2\right)}}+j_0 e^{-\frac{\left(x-b-p_1-p_2\right){}^2}{2 \left(\sigma ^2+q_1^2+q_2^2\right)}}\right) \left(i_1 e^{-\frac{\left(x-a-p_2\right){}^2}{2 \left(\sigma ^2+q_2^2\right)}}+j_1 e^{-\frac{\left(x-b-p_2\right){}^2}{2 \left(\sigma ^2+q_2^2\right)}}\right)$$

It's seemingly very close, however it's not correct.

It doesn't quite match the correct curve:

Two questions:

• Any idea what's wrong?
• What's the correct way to do this, without relying on Mathematica and intuition?

Answer 1

What's wrong is that you did the convolution wrong. Your g(x) has two factors which I'll call g1(x) and g2(x). What you appear to have calculated is $(f(x)*g1(x))\times(f(x)*g2(x))$ where * is the convolution operation. That's just not correct. What you need to do is:

• Substitute $x\rightarrow(z-x)$ in either f(x) or g(x)
• Expand and multiply so that you get four terms, each a product of Gaussians
• Within each term, multiply the Gaussians together by adding the exponents and simplifying those exponents into tidy quadratics in x z.

Now you will have four Gaussians and you can perform the integral on each one separately. The necessary algebra is high-school level, but there will be a lot of it.