Deconvolution Using Response to an Heavy Side I'm measuring a "charge" signal in function of time from an amplifier.
Here is a measured signal (x-axis is the time in some arbitrary units, y-axis is the charge in ADU):


I would like to get the "true", original signal, by deconvolving the instrument's response from the measured signal.

Here the response of the instrument to an Heaviside step function:


I guess this should help me characterize the instrument's response, and do the deconvolution. 
I read about SVD and Bayesian unfolding, but these don't seem to do the trick (I need to built a transfer matrix first).
Could you suggest me any tool/algorithm to do this?

Thanks in advance for the help.
If we can assume no noise (Or the SNR is very high) you can get the response by applying the inverse filter in frequency domain.

Lets say $ y [n] $ are the signal samples.
Given $ x [n] $ the samples of the ideal signal you can apply on both of them the DFT to get $ Y [k] $ and $ X [k] $.

The response is given by the Inverse DFT of the division $ \frac{Y[k]}{X[k]} $.

If there's some noise you need to regulate the result.
Easy choice would be the Wiener Filter or more specifically Wiener Deconvolution.