A filter for separating time-limited modal signals

A two-stage deterministic filter is derived which can separate two sets of modal signals when presented with their arbitrary mixed sum on a fixed finite time interval. The assumptions are 1) that the mode locations in the complex plane are known for one set (for example, a finite Fourier series has known modes relative to a given time interval) and 2) the two sets of modes are disjoint. Using integral operations on the mixed sum over the fixed time interval, the filter first estimates the modes in the second set, then estimates the amplitudes of the modes in the first set, thus providing a complete reconstruction of the signal comprised of the known modes. The filter estimate is theoretically exact under the above assumptions and provides a particular kind of least squares estimate when the actual signals deviate from the assumed modal representations. Results of a computer simulation are discussed which illustrate strengths and weaknesses of the filter.