# Applications or physical interpretation of auto-convolution?

I wonder if anyone has any experience with auto-convolution. In particular i'm interested in understanding the physical interpretation of it. I understand what convolution, correlation and auto-correlation are, also i'm aware that the definition of auto-convolution will be something like $$f\ast f = \int_{-\infty}^{\infty} f(\tau)f(t-\tau)d\tau$$ but i still don't get what are the implications o the meaning of it. I've been looking for a while and so far i haven't found any good or detailed explanation (on constrast with auto-correlation). So, if anyone has any experience dealing with this topic or has any intuitive interpretation that could share, i'll appreciate it. Thanks.

Autoconvolution is used in signal detection, but the way you've written it is not correct. Suppose you're trying to detect a signal $f(t)$ by filtering with h(t).

$y(t) = f(t) \ast h(t)$

You want to maximize your response to the signal $f(t)$. We can do this by maximizing the correlation coefficient between $f$ and $h$. Here the correlation is time-varying , so we'll maximize the average autocorrelation coefficient. We'll assume the $f$ and $h$ signals have DC values of zero for simplicity.I'll use $\mu_y$ to denote average value of a the auto-correlation of response, $y$.

$h = argmax_{h} \ \ \mu_y\big(E_f E_h \big)^{-\frac{1}{2}}$

Let's take a differential of our performance $J \propto \mu_y\big(E_f E_h \big)^{-\frac{1}{2}}$ with respect to h

$\partial_{h(\tau)} J \propto \partial_{h(\tau)} \bigg((\mu_y\big(E_f E_h \big)^{-\frac{1}{2}}\bigg)$

$\ \ \ \ \ \ \ \ \ = \frac{1}{|dom(f)|}\bigg(\partial_{h(\tau)}\mu_y\bigg)\big(E_fE_h)^{-\frac{1}{2}} +\frac{1}{2}\mu_yE_f^{-\frac{1}{2}}E_h^{-\frac{3}{2}}\bigg(\partial_{h(\tau)}E_h\bigg)$

$\ \ \ \ \ \ \ \ \ \propto \bigg(\partial_{h(\tau)}\frac{1}{|dom(f)|}\int_{\infty}^{\infty}f(t-\tau)h(\tau)d\tau \bigg)\big(E_fE_h\big)^{-\frac{1}{2}} +\frac{1}{2}\mu_yE_f^{-\frac{1}{2}}E_h^{-\frac{3}{2}}\bigg(\partial_{h(\tau)}E_h\bigg)$

$\ \ \ \ \ \ \ \ \ = \bigg(\frac{1}{|dom(f)|}\int_{\infty}^{\infty} f(t-\tau)d\tau \bigg)\big(E_fE_h\big)^{-\frac{1}{2}} -\frac{1}{2}\mu_yE_f^{-\frac{1}{2}}E_h^{-\frac{3}{2}}\bigg(\partial_{h(\tau)}E_h\bigg)$

$\ \ \ \ \ \ \ \ \ = \big(E_fE_h\big)^{-\frac{1}{2}} \frac{1}{|dom(f)|}\int_{\infty}^{\infty} f(t-\tau)d\tau-\frac{1}{2}\mu_yE_f^{-\frac{1}{2}}E_h^{-\frac{3}{2}}\bigg(\partial_{h(\tau)}\int_{\infty}^{\infty} h^2(\tau)d\tau \bigg)$

$\ \ \ \ \ \ \ \ \ = \frac{1}{|dom(f)|}\big(E_fE_h\big)^{-\frac{1}{2}} \int_{\infty}^{\infty} f(t-\tau)d\tau-\mu_yE_f^{-\frac{1}{2}}E_h^{-\frac{3}{2}}\int_{\infty}^{\infty} h(\tau)d\tau$

$\ \ \ \ \ \ \ \ \ \propto \frac{1}{|dom(f)|}\int_{\infty}^{\infty} f(t-\tau)d\tau-\mu_yE_h^{-1}\int_{\infty}^{\infty} h(\tau)d\tau$

$\ \ \ \ \ \ \ \ \ = \int_{\infty}^{\infty} \bigg(\frac{1}{|dom(f)|}f(t-\tau)-h(\tau)\mu_yE_h^{-1}\bigg)d\tau$

At minimum $J$ we shouldn't assume $h(\tau) = 0$, we'll enforce $$\frac{1}{|dom(f)|}f(t-\tau)-h(\tau)\mu_yE_h^{-1} = 0$$

$$h(\tau) = \frac{E_h}{\mu_y|dom(f)|}f(t-\tau)$$

Or most importantly

$$h(\tau) \propto f(t-\tau) \text{where} \ \ \tau \ \ \text{is time and} \ \ t \ \ \text{is the delay}$$

or, using perhaps better "variable names"

$$h(t) \propto f(t_d-t) \text{where} \ \ t \ \ \text{is time and} \ \ t_d \ \ \text{is the delay}$$

That is, if we want to maximize the correlation between our signal detector with impulse h(t), we better pick h(t) to be a time-reversed and time-shifted version of our signal of interest. In practice $t_d$ would probably be set to zero, as it just represents whenever the $f$ part that you're looking for finally arrives.

Under this chosen $h(t) \propto f(t_d - t)$, your original question makes more sense. The autocorrelation signal becomes proportional to the accumulated energy of $f(t)$ that is seen by your filter.

$y(t) = f(t) \ast h(t)$

$y(t) \propto \int f(\tau) h(t-\tau)d\tau$

$y(t) = \int f(\tau) f(t_d-(t+\tau))d\tau$

$y(t) = \int f(\tau) f(\tau + t_d - t)d\tau$

$y(t) = \int P_f(\tau + t_d - t)d\tau$

• What's the difference between all the above and the concept of a matched filter ? – Dilip Sarwate Apr 2 '15 at 3:01
• Thanks, your answer is great. Also, the question mentioned by @DilipSarwate helped a lot too, so thanks also. One additional question, i wonder if you know any resource (articles, books) i could look at. – Alejandro Cruz Gtz Apr 2 '15 at 4:03
• @DilipSarwate ah yeah good call, but all the same, I enjoyed it as an exercise. – user27886 Apr 2 '15 at 4:08
• This answer really does not address the question that has been asked: where does autoconvolution show up in physical applications? A matched filter or a Wiener filter does not really correspond to autoconvolution except in special cases when $f$ is an even function (or a time-shifted even function). Thus, even the claim in the very first sentence "Autoconvolution is used in signal detection..." must be taken with a very large grain of salt. – Dilip Sarwate Apr 2 '15 at 14:17
• @DilipSarwate is correct on both counts. the OP said "I understand what convolution, correlation and auto-correlation are," differentiating from the term "auto-convolution" whereas the answer is about auto-correlation. – robert bristow-johnson Apr 2 '15 at 16:10