The adaptive identification of the impulse response of an innovation filter is considered. The impulse response is a finite sum of known basis functions with unknown coefficients. These unknown coefficients are estimated using a pseudolinear regression. This estimate is implemented using a square root algorithm based on a displacement rank structure. When the initial conditions have low displacement rank, the filter update is $O(n)$. If the filter architecture is chosen to be triangular input balanced, the estimation problem is well-conditioned and a simple, low rank initialization is available.