If you know the actual transfer function of the filter then passing the signal through an amplifier with a filter with transfer function equivalent to the original filter in the negative feedback path 'should' return the original signal, subject to any non-idealities encountered along the way.
If you know what any non idealities are in the original system, then if you can model them you can also inversely apply them s above. Some such will be impossible - eg random noise is random (natch) and cannot be systematically undone.
To get more than general advice you will need to describe your overall requirement and not just reveal parts of it that you think are important while squirreling away stuff that may be of value.
eg the statement
" ... Since the information of important parameters can only be extracted from the unfiltered signal, I would like to undo the filtering ..."
MAY be true but is suspect. It may be that adding a transformation to your extraction process is a more correct approach than "restoring" the signal BUT as we do not know what you are doing we cannot advise appropriately.
We need to know what the processes were that were used to transform the input signal to the current signal. As 'The Phton' rightly point out in a comment to this answer, processing may have been applied to the signal which caused irrecoverable loss. If you want a quality answer then you must improve the equality of your question - especially in terms of signal and system description and what information you consider has been lost.
Digitally or analoguely is not a primary issue. There may be 2nd order effects introduced by the technology used but they are incidental to the task proper.
Based on the additional step by step revealed information:
What is the pulse duration?
What is the minimum and maximum allowed RC time constant?
Is the filter input or output loaded and if so with what?
How is the filter implemented (analogue opamps, steam turbines , ...?)
Why did they use a Bessel filter?
What was the filter meant to do?
Why use an I->V rater than measuring V across a matched load?
Why not resist te temptation to be mysterious and give us a diagram with ALL* the pertinent stuff on it? (*-ALL!)
Rather than the death by 1000 mini questionlets and progressively revealing factlets approach so beloved by so many, why not tell us ALL there is to know?
IF (as you say) the system samples at 20 kHz you'd hope the designer assumed an absolute absolute absolute max f_component of 10 kHz and hopefully less.
At 20 kHz fclk/2 = 10 kHz max so Bessel filter cutoff is less than 2 octaves below fmax (10 kHz / 3.2 KHz < (2 x 2)) so roll off at 3 dB/pole/octave is < 2 octaves x 3 dB x 2 poles = 12 dB down or about 25% of 10 kHz components will still be there . Bessel is very gentle in stop band characteristics. By placing an equivalent Bessel transfer function in the negtive feedback path of an opamp you have a reasonable chance of getting a half decent facsimile of the input signal. ... – Russell McMahon 18 secs ago edit
| Your fitting algorithms [tm] need to be defitted and as we have no clue waht that means we have no clue as to what defitting entails. If you know then telling us is probably a REALLY good idea. If you are processing the result in software then doing the above in software probably makes good sense. There may be reasons this is not so but if so you haven't told us.