# How to prove property of transfer function for causal LTI system

For causal linear time-invariant systems the degree of denominator must be larger or equal than the degree of the numerator.

How can this be proven rigorously?

• Think of the simple, ideal differentiator: $s$. What's its impulse response? Can you obtain such a response in real life? Commented Sep 21, 2022 at 13:28
• This is the usual "explanation". I look for a more mathemtical srgument. Commented Sep 21, 2022 at 14:28
• I'm not sure about the meaning of you using quotes. Having a pure differentiator means pure derivative of the input. What if the input is discontinuous? How do you handle such case? The moment you mention causality you invite practicality, otherwise you'd be fine talking about acausal FIRs. Commented Sep 21, 2022 at 15:07
• Yes, a pure differentiator cannot be realized technically. But my question is about causality, not technical feasibility. Of course it is also clear, that there is no state space model for an improper H(s). I would regard an un-causal system to have an impulse response which doesn't vanish for t<0. However, what is the impulse response of H(s) = s²/(s+1) ? It includes derivation of Dirac impulse : h(t) = -δ(t) + δ'(t) + e^(-t) but where is this un-causal? It is zero for t<0... Commented Sep 22, 2022 at 10:23

OTOH, if you want to keep it theoretical, you can even have $$\s^n\$$ and nobody can stop you, not even the paper you're writing on. You can have as many Diracs, all null before $$\t=0\$$, and everything will work. But only on paper. Otherwise acceleration could easily be determined by a double differentiator but, in practice, not imposing a bandwidth will bring havok. So the real reason for the transfer function being proper is reality.