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As a generalized Doppler filter bank processing, Radon-Fourier transform (RFT) has recently been proposed for long-time coherent integration detection of radar moving targets. The likelihood ratio test (LRT) detector is derived here for rectilinearly moving targets. It is found that the proposed LRT detector has the identical form as the existing RFT detector, which means that the RFT detector is an optimal detector for rectilinearly moving targets under the white Gaussian noise background. For the fast implementations of the RFT detector, instead of the joint 2-D trajectory searching and coherent integration in pulse-range domain, the 1-D fast Fourier transform (FFT)-based frequency bin RFT (FBRFT) method is proposed in the pulse-range frequency domain without loss of integration performance. Moreover, at the cost of a controllable performance loss, a suboptimal approach called subband RFT (SBRFT) is also proposed to reduce the storage memory. It is shown that not only the long-time coherent integration gain can be obtained via the proposed SBRFT, but also the computational complexity and memory cost can be reduced to the level of the conventional Doppler filter banks processing, e.g., moving target detection (MTD). Some numerical experiments are also provided to demonstrate the effectiveness of the proposed methods.