Multipath estimation problem

We have some measurements, they are slightly noisy but lets assume noise free for now. Lets also assume the measured signal is the output from a LTI system:

y(t) = a_0 p(t) + a_1 p(t-tau_1)

Here are two versions the problem statement:

P1) Estimate (a_0, a_1, tau_1, p(t)) that best explains y(t) in a least-squares sense

P2) Simpler problem: Lets say I have a model for p(t) that it is of some form (say it is the step-response of a B Hz low pass filter). Then estimate (a_0, a_1, tau_1) assuming p(t) is sort of known.

Not so interested in recovering p(t) -- its more like a unknown nuisance parameter.

Hint (maybe not): constraints on p(t) may be easier to impose in the frequency domain