Cross Correlation of output with input is equivalent to impulse response of a large bandwidth input (LTI System)?
There are many different ways. Here is one


Excite the system with a periodic broad band noise signal that has a period of larger than the length of the impulse response. Call it $N$ samples. 
Capture the output of the system after it's stable and periodic (discard the first $N$ samples). Make sure that excitation and acquisition are using the same clock. 
Grab any $N$ samples of your excitation and acquisition 
FFT them and divide the acquisition spectrum by the excitation spectrum
Inverse FFT the result: you have an Impulse Response of the systems


Bonus points for:


Using coherent averaging to improve signal to noise ratio
Shape the spectrum of your excitation to match the noise floor in the frequency range of interest so get flat SNR.