Suppose that there is a FIR filter F and a signal S. The filtered signal is the convolution of F and S, F * S.
The problem: how to calculate a signal S' such that F * S' = S' (the filtered version is the same as itself, up to a fixed time shift), and S' minimizes norm(S, S') (where norm is some kind of distance metric, for example sum of squared differences).
Loosely: how to calculate the most similar signal to a given signal which is not affected by a filter?
EDIT: The original problem may be over constrained because there may be very few (or no) signals for which F*S = S exactly. An alternate formulation is to find S' which minimizes the combined error alpha*norm( F*S' - S') + beta*norm( S' - S ), for some weights alpha and beta.
EDIT: The specific filter I have in this case is F=[ 0.00097656, -0.00976562, 0.06347656, -0.15625 , 0.18554688, 0.83203125, 0.18554688, -0.15625 , 0.06347656, -0.00976562, 0.00097656] which is a fairly modest lowpass filter. I was interested in the general case as well though.