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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?

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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

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  • $\begingroup$ you are very welcome :) $\endgroup$ – Cherny Oct 10 '17 at 7:50

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