# Cross Correlation of output with input is equivalent to impulse response of a large bandwidth input (LTI System)?

What is the practical way to found the impulse response of a system ? As impulse function has a very large amplitude at zero time (analog signal).

So the cross corelation of output of a large bandwidth input signal (with relatively small amplitude) gives the impulse response of a LTI system .How is these two equivalent?

It is not exactly true (perhaps only partly true) that the cross-correlation of the response of an LTI system to a broadband input with the broadband input itself gives the impulse response of the system. It helps a little if the broadband signal has a flat spectrum over almost all of its bandwidth but generating such a signal to use as an input to the LTI system can be problematic. For a discussion of some of the issues involved and a specific way of generating a broadband signal that is very convenient and is probably built-in already to whatever gear you have on your workbench or into your MATLAB toolbox, see this answer of mine.

There are many different ways. Here is one

1. 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.
2. 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.
3. Grab any $$N$$ samples of your excitation and acquisition
4. FFT them and divide the acquisition spectrum by the excitation spectrum
5. Inverse FFT the result: you have an Impulse Response of the systems

Bonus points for:

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