Can all systems be described by differential/difference equations?

Question

All LCCODEs (linear, constant-coefficient ordinary differential equation) with initial rest conditions describe causal LTI systems (is this a correct statement?)

• Is the converse true, i.e. can all causal LTI systems be described by LCCODEs with initial rest? If not, what are the necessary/sufficient conditions?

It seems trivial that arbitrary differential/difference equations describe some sort of system (is that also true?)

• Again, is the converse true, i.e. can we describe any arbitrary systems with (possibly nonlinear/variable-coefficient/partial, without restriction on initial conditions) differential equations? If not, what are the necessary/sufficient conditions?

Answer 1

No the converse is not true in general.

Take for example the discrete-time ideal lowpass filter with impulse response

$$ h[n] = \frac{ \sin( \omega_c n) }{ \pi n } ~~~,~~~-\infty < n <\infty$$

which describes an LTI system but it does not correspond to a difference equation of any kind. Indeed $h[n]$ is derived based on the inverse discrete-time Fourier transform of the filter frequency response;

$$ H(\omega) = \begin{cases} {1 ~~~,~~~ |\omega| < \omega_c \\ 0 ~~~,~~~ \text{ otherwise } }\end{cases} $$

$$h[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} H(\omega) e^{j\omega n} d\omega $$