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?

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;

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

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

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    $\begingroup$ Why is there no difference equation? There is one with infinitely many terms. Is it implementable? No, but it does exist. $\endgroup$ – Rodrigo de Azevedo Mar 2 '19 at 8:05
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    $\begingroup$ Indeed so! I just probably referred to te general acceptance. As one usually either solves an LCCDE to get the impulse response or uses it to compute the output from input but in this case you neither solve for $h[n]$ which is first obtained from the Fourier transform inversion nor use such LCCDE to make any computations, but merely construct an uncomputable (infinite order!) LCDDE. Probably that's why it's accepted that there are no corresponding LCCDE for such infinetely-long yet FIR (finite impulse response) filters... $\endgroup$ – Fat32 Mar 4 '19 at 16:04
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    $\begingroup$ I understand your point. However, I would say that existence (in a Platonist's Heaven) is one thing, and realizability (using pencil & paper, hardware or software) is another. But this is obviously a matter of taste. $\endgroup$ – Rodrigo de Azevedo Mar 9 '19 at 23:26
  • $\begingroup$ @RodrigodeAzevedo probably so... $\endgroup$ – Fat32 Mar 10 '19 at 2:25

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