Define LTI system from a single input output pair

Question

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).

Answer 1

To find the response of a system, you need to know the input, the output

The equation is this (where y(s) is the output):

So if you know the input and output functions, you can take the laplace tranform and find the transfer function.

Shifting simply adds a delay, and won't change the transfer function, as the delay will happen in the output and will be canceled out.

\$\mathcal{L}U(t) f(t-a)=e^{-as}F(s) \$

Scaling will scale, and will also be canceled out, as multiplying the input by c will also multiply the output.

To get the response of the system, you need something that operates on all frequencies, like a dirac delta function, a sine sweep or a step input function.

The method for estimating the transfer function with data is called system identification

Answer 2

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).

No.

For example, if the test input signal is a pure sine wave at some particular frequency, that won't give you enough information to reconstruct the impulse response.

The reason is that no matter how you shift and sum a pure sine wave signal, you always end up with a new sine wave of the same frequency, possibly shifted in phase and changed in amplitude. So you can't use it to reconstruct all possible input functions.

What you're looking for is called an orthonormal basis set of functions that can be formed just by shifting and scaling a single base function. Other than the Dirac delta, I don't know of any such set. Wavelet transform techniques use something similar, but they require the ability to both time-shift and time-dilate the "mother wavelet" to form the basis set.