I am currently studying signals and systems, and am learning about LTI systems right now. I know any LTI system whose impulse response is known can be completely defined through the use of the convolution sum.
I am trying to find out if given an arbitrary input signal, I can obtain the impulse signal by adding shifted and scaled versions of the original input signal( just as we can obtain any signal by adding shifted scaled versions of the impulse). That would allow me to find the impulse response of the system in question, because the system is given to be LTI. Then I could define the system completely based on a single input output pair. Viewing a signal as a function, this would also mean I could represent any function in terms of sums of shifted scaled versions of any other function, because I can represent any function in terms of the delta function via convolution, and I could represent the delta function in terms of any function(given what I am asking about was true)
Please tell me if this is possible. Is there a way to mathematically prove that this can (or cannot be done)? If there exists such a proof, does it extend to all functions, discrete, as well as continuous?
I apologize for any lack of rigor, if I violated the etiquette of this forum somehow, please let me know(I am a new user, sorry).