It really isn't, and I don't understand the level of confusion that leads to this assertion.
Is it because the elementary school algorithm for multiplication decomposes a number in it's decimal places before multiplying?
> so why not start with that?
Because it would be wrong, misguided, and it would fail to demonstrate the logic behind convolution.
Let's take a step back and think about it. Convolution is mainly and primarily used as a convenient way to covert signals between time and frequency domain, within the scope of Fourier transforms. Who in their right mind associates elementary school multiplications with conversions to/from the frequency domain?
> Convolution is mainly and primarily used as a convenient way to covert signals between time and frequency domain
Btw that isn't true. That operation is called a "transform" as in "Fourier transform" and "z-transform". The property of those transforms is that element-wise multiplication in one domain becomes convolution in the transformed domain. Element-wise multiplication in "time" domain is convolution in "frequency" domain and element-wise multiplication in "frequency" domain is convolution in "time" domain.
Further to that, if you replace x with complex variable z^-1, you get what is usually called the z-transform. Set z to a specific complex root of unity exp(i2pi/N) and you have your discrete FT.
One of the algorithms for multiplying large arbitrary precision numbers uses multiplication of the discrete transform of the digit sequences (in a different base) iirc.
It really isn't, and I don't understand the level of confusion that leads to this assertion.
Is it because the elementary school algorithm for multiplication decomposes a number in it's decimal places before multiplying?
> so why not start with that?
Because it would be wrong, misguided, and it would fail to demonstrate the logic behind convolution.
Let's take a step back and think about it. Convolution is mainly and primarily used as a convenient way to covert signals between time and frequency domain, within the scope of Fourier transforms. Who in their right mind associates elementary school multiplications with conversions to/from the frequency domain?