Recover filter coefficients from filtered noise I have a digital signal which may be represented as noise filtered with an FIR (finite impulse response) filter.  Let us suppose that the noise consists of pulses (nonzero samples on a zero background), and there is an equal probability of a pulse at any sample (so for example the number of pulses in an interval follows the poisson distribution).  Further suppose the strength of each pulse is equal.
The question: Can the coefficients of the FIR filter be recovered from the filtered signal, and how?  
Now assume the noise is impulsive, but may be somewhat correlated with itself (it doesn't have perfect Poisson statistics).  Also assume that the strength of each pulse may vary.  Can the coefficients be recovered approximately if the exact distribution of noise is not known?
Formal notation.
Noise N[i] where i=0..n, N[i] = 1 with some probability p, 0 otherwise; each N[i] is independent of other N[i]
Filter F[k] where k=0..m, m << n, unknown
Signal S[i] where i=0..n
S = N * F, where * is convolution
Given S, estimate F.  N and p are also unknown.
To give this some practical background, the average period between noise pulses can be 20-40 samples, the FIR filter can have a few hundred nonzero coefficients (so the filtered signals resulting from each pulse overlap significantly), and the total signal is a few thousand samples.

As I already noted it is possible (though unlikely) that there is no noise at all and thus you do not learn anything about the signal. 
A more practical problem is that unless $N[0]=1$ you will not be able to determine if $F[0] =0$. For if you take a $N$ with $N[0]=0$ and translate that vector to the left, setting $N[n]=0$, and translate $F$ to the right, setting $F[0]=0$, you will leave the output unchanged. 
