Can We Uniquely Recover Coefficients from a Multi-Frequency Signal?

No, it is not possible to recover a1, a2, a3, and a4 uniquely from r(t).

The reason is that r(t) contains multiple frequency components, and each of the four terms (a1u(t)cos(2Πfct), a2u(t)sin(2Πfct), a3u(t)cos(4Πfct), and a4u(t)cos(64Πfct)) contribute to different frequency components.

When we pass r(t) through a filter to extract a specific frequency component, we will obtain a combination of a1, a2, a3, and a4. Therefore, to recover a1, a2, a3, and a4 uniquely, we need to know the frequency components of each term and separate them accordingly.

Without any additional information, it is impossible to determine the exact frequency components of each term, and thus, it is impossible to recover a1, a2, a3, and a4 uniquely from r(t).