Experimental determination of impulse response function
You can determine the impulse response of a system using Gaussian random noise as a test signal, let's name it $x$. Then, you have to measure the system response $y$ to that noise. With a considerable amount of digital samples from both signals then you can compute the crosscorrelation between $x$ and $y$ as follows:

$$
R_{xy}(n\Delta t) = \frac{1}{N-n} \sum_{k=1}^{N-n-1} x[k] \cdot y(k-n\Delta t)
$$

If the white noise has a bandwidth that is much greater than the bandwidth of the system (for instance, 100 times) then the crosscorrelation is equal to the impulse response times the power spectrum density of the noise (which is constant) divided by 2. Of course, there is some uncertainty in the estimation, which decreases with large values for $N$. 

Source: Oliver-Cage: Electronic Measurements And Instrumentation