Weighted time-frequency and time-scale transforms for non-stationary signal detection

Maximum likelihood detectors of narrowband, non-stationary random echos in Gaussian noise can be efficiently implemented in the time-frequency domain. When the transmitted signals have large time-bandwidth products, the natural implementation of estimators and detectors is the time-scale or wavelet transform domain implementation. This paper extends the wavelet transform implementations to include weighted time-frequency or time-scale (TF/TS) transforms. 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. We show how weighted transforms are used in the estimator-correlator detection statistic for complex scattering environments in conjunction with cascaded scattering functions so that the resulting detection statistic is much more robust. The weighted TF/TS transform turns out to be a natural transform for solving nonstationary detection, estimation, and filtering problems and has important applications to transient signal estimation in multipath channels with colored non-stationary Gaussian noise.