Obtaining impulse response by means of cross-correlation between input and output

Having a system with an input (sound emitter) that is defined as a unit-variance white noise sequence, and I get the output (sound receiver) so I can measure cross-correlation functions between them, how could I employ the correlation function to obtain the impulse response h[n], which defines the media?

• See this answer of mine for the method that can be used. Oct 9 '17 at 13:53

we can calculate it!

define $y=h*t$, $r_{xy} -$ is the cross correlation

$r_{xy}(\tau)=\int_{-\infty}^ \infty x(t)y(\tau+t)dt = (y*x^-)(\tau) = (x^-*h*x)(\tau) =(r_{xx}*h)(\tau)\underbrace{=}_{r_{xx}=\delta}h(\tau)$

so actually the cross correlation is the impulse response

• you are very welcome :) Oct 10 '17 at 7:50