# What optical system could perform a multiplication/convolution of several rectangular pulses with different widths?

Statement

There is a following multiplication/convolution of $$n$$ rectangular pulses with different widths $$f(x)=\mathrm{rect}(c_1x)\ast\mathrm{rect}(c_2x)\ast\ldots\ast\mathrm{rect}(c_mx)$$ or $$f(x)=\prod\limits_{n=1}^m f_n(x).$$

Question What optical system could perform a convolution of several rectangular pulses with different widths in general sense? Means there is no pre-requirements in the statement.

P.S. Problem has mathematical roots and optical system is needed to be an illustrative example.

UPD.1 19.01.2019

Clarification. Let's focus on the implementation of multiplication operation. Let us want to implement mathematical multiplication operation of two where a) $$m=2$$ and b) $$m=3$$ functions.

We come to the following question:

What optical system implements the multiplication of intensities of coherent beams of light come from different directions? How does the schematic diagram look like?

UPD.2 20.01.2019

In fact the question is about optical multiplier of intensities principal scheme.

• Optical systems are good at Fourier transforms. Can you perform your convolution in the Fourier plane, then convert it back? – D Duck Jan 19 at 14:12
• @DDuck, thank your for the feedback. I've updated the question. – Oleg Kravchenko Jan 19 at 19:50