Weighted time-frequency and time-scale transforms in reproducing kernel Hilbert spaces

Motivated by the time-frequency, time-scale (TF/TS) implementation of a maximum likelihood detector for transient signals in nonstationary Gaussian noise, we define weighted TF/TS transforms using reproducing kernel Hilbert space (RKHS) inner products. Inverses of these weighted TF/TS transforms are also given. The particular case of the weight being the inverse noise covariance is presented. The weighted TF/TS transforms turn out to be natural transforms for solving nonstationary detection, estimation, and filtering problems, and have important applications to transient signal estimation in multipath channels with colored nonstationary Gaussian noise.