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In this paper, we propose an experimental scheme for a compact X-ray source which can produce X-ray pulses with 10 femtosecond duration. This novel X-ray source is based on the nonlinear Thomson backscattering of terawatt ultrashort infra-red laser from MeV plasma accelerated electrons, which are now routinely generated from the laser-wakefield accelerators. The synchronization between the TW laser and e-beam can simply be achieved via a jitter-less optical delay. At KERI, a 2 TW 700 fs laser system has been installed and the experiment on e-beam generation is ongoing. Here we show some simulation results for the typical ultrashort e-beam generation from the LWFA and the basic calculations for the X-ray generation.