This is a (more general) question related to "Estimated solution to system of equations with phase-shifted functions".
Given this system of two equations and two unknown functions:
$$ y_1(t) = \mu_{1a} f_a(t-\phi_{1a}) + \mu_{1b} f_b(t-\phi_{1b}) $$ $$ y_2(t) = \mu_{2a} f_a(t-\phi_{2a}) + \mu_{2b} f_b(t-\phi_{2b}) $$
A solution (I think) is:
$$ f_a(t) - \frac { \mu_{1b} \mu_{2a} } { \mu_{1a} \mu_{2b} } f_a \left(t + \phi_{1a} - \phi_{1b} - \phi_{2a} + \phi_{2b} \right) = \frac 1 {\mu_{1a}} \left[ y_1 \left( t + \phi_{1a} \right) - \frac {\mu_{1b}} {\mu_{2b}} y_2 \left( t + \phi_{1a} - \phi_{1b} + \phi_{2b} \right) \right] $$
The solution to the recurrence relation:
$$ f_a(t) = \frac 1 {\mu_{1a}} \sum_{i=0}^\infty \left( \frac { \mu_{1b} \mu_{2a} } { \mu_{1a} \mu_{2b} } \right)^i \left[ y_1 \left( t + \phi_{1a} + i \left( \phi_{1a} - \phi_{1b} - \phi_{2a} + \phi_{2b} \right) \right) - \frac {\mu_{1b}} {\mu_{2b}} y_2 \left( t + \phi_{1a} - \phi_{1b} + \phi_{2b} + i \left( \phi_{1a} - \phi_{1b} - \phi_{2a} + \phi_{2b} \right) \right) \right] $$
This is based on the fact (?) that, given some original $f(t)$ and $g(t) = f(t) + \mu f(t-\phi)$, the original can be restored via
$$f(t) = \sum_{i=0}^\infty (-\mu)^i g(t-\phi i)$$
- Is the above correct?
- How do I generalize this process for a large number of similar equations?