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We propose a novel experimental scheme for a compact and tunable X-ray source capable of producing X-ray pulses with a few tens of femtosecond duration. This new source is simple; it does not require an RF-based accelerator or injector. It is based on the Thomson back-scattering of a terawatt femtosecond laser from the megaelectronvolt-level plasma-accelerated electrons. Here, we present particle-in-cell simulation results for the ultrashort e-beam generation from the laser-wakefield accelerators with a plasma-density transition and the basic characteristics of the X-ray generation. Using a 20-TW 40-fs laser system, we are going to perform an experiment for generating ultrashort X-rays via the Thomson scattering.