If we are able to obtain the impulse response of an LTI system, then given a recording of the system output, we can recover the system "input" that corresponds to this output. This is accomplished by deconvolving the system impulse response from the recorded output signal. This process is commonly referred to as system inversion.
Here is a rough schematic of my experimental recording setup. I recorded the input and output signals simultaneously so I wouldn't have to worry about resynchronizing the signals later.
