  Detecting fast radio bursts (FRBs) with frequency-dependent intensity remains
a challenge, as existing search algorithms do not account for the spectral
shape, potentially leading to non-detections. We propose a novel detection
statistic, which we call the Kalman detector, that improves the sensitivity of
FRB signal detection by incorporating spectral shape information. The detection
statistic is based on an optimal matched filter, marginalizing over all
possible intensity functions, weighted by a random walk probability
distribution, considering some decorrelation bandwidth. Our analysis of
previously detected FRBs demonstrates that the Kalman score provides a
comparable yet independent source of information for bursts with significant
spectral structure, and the sensitivity improvement is of the order 0%--200%
with a median improvement of 20%. We also applied the Kalman detector to
existing data from FRB 20201124A and detected two new repeat bursts that were
previously missed. Furthermore, we suggest a practical implementation for
real-time surveys by employing a low significance soft-trigger from initial
flux integration-based detection algorithms. The Kalman detector has the
potential to significantly enhance FRB detection capabilities and enable new
insights into the spectral properties of these enigmatic astrophysical
phenomena.
