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We propose and assess new algorithms for adaptive detection and tracking based on space-time data. At design stage we take into account possible spillover of target energy to adjacent range cells and assume a target kinematic model. Then, resorting to the generalized likelihood ratio test (GLRT) we derive track-before-detect (TBD) algorithms that can operate in scan-to-scan varying scenarios and, more important, that ensure the constant false track acceptance rate (CFTAR) property with respect to the covariance matrix of the disturbance. Moreover, we also propose CFTAR versions of the maximum likelihood-probabilistic data association (ML-PDA) algorithm capable of working with data from an array of sensors. The preliminary performance assessment, conducted resorting to Monte Carlo simulation, shows that the proposed TBD structures outperform the ML-PDA implementations especially in terms of probability of track detection (and for low signal-to-noise ratio (SNR) values).