When I was learning about LTI systems, I noticed that LTI system is said to be BIBO stable if and only if its square sum of impulse response is finite. This expression is found on many textbook of signal processing.
I continued study and found this expression LTI system is stable if and only if its impulse response is exponentially bounded. I could not understand this at first, but by further research, I elaborately acquire this interpretation. However, I could not find explicit description about this. So my question is "is these things true ?",
- There are two types of stability (of the system), internal stability and external stability.
- BIBO stability is equivalent to external stability.
- If the system is controllable and observable, internal and external stability is equivalent.
- If the system is linear, "exponentially bounded" is equivalent to internal stability.
and if 4. is true, why the function must be "exponential decaying" ? How is this restricted to be "exponential" ?