Frequency response and impulse response of black box
If the system is continuous-time then you can't simulate an impulse (Dirac delta).

Generally people take a number of approaches:


Instead of using an impulse, find the step response of the system of interest.  This can be related to the integral of the impulse response as the unit step is the integral of the unit impulse.
Generate a set of spaced sinusoids and measure the output phase and amplitude of each.  The transfer function is found by interpolating these discrete frequency points.
Generate a swept sine wave (chirp) signal over a known frequency range and deconvolve the output to find the transfer function.
Generate a pseudorandom noise sequence, measure the response, and use it to estimate the transfer function.