# Linear Time-Invariant system without State-space form

I would like to know whether there is any LTI system that does not have state-space representation, but do have a convolution representation. and vice-versa. Can you name some examples for me to research further on this?

This seems like a homework problem, but I'll bite.

The thing with a state-space representation is that it needs to have finite dimension:

$$x_{k+1} = \mathbf{A} x_k + \mathbf{B} u_k\\ y_k = \mathbf{C} x_k + \mathbf{D} u_k$$

where $$\mathbf{A} \in \mathbb{R}^{n\times n}$$ and the other matrices and vectors have appropriate dimensions.

Here, the dimension, $$n$$, must be finite.

So an LTI system that can be represented as a transcendental impulse response cannot be written that way.

For example: $$y_k = {\rm sinc}(0.1 k) \star x_k$$ will not be representable in a state-space formulation.