  This paper presents the design, calibration, and survey strategy of the Fast
Radio Burst (FRB) digital backend and its real-time data processing pipeline
employed in the Tianlai Cylinder Pathfinder array. The array, consisting of
three parallel cylindrical reflectors and equipped with 96 dual-polarization
feeds, is a radio interferometer array designed for conducting drift scans of
the northern celestial semi-sphere. The FRB digital backend enables the
formation of 96 digital beams, effectively covering an area of approximately 40
square degrees with 3 dB beam. Our pipeline demonstrates the capability to make
automatic search of FRBs, detecting at quasi-real-time and classify FRB
candidates automatically. The current FRB searching pipeline has an overall
recall rate of 88\%. During the commissioning phase, we successfully detected
signals emitted by four well-known pulsars: PSR B0329+54, B2021+51, B0823+26,
and B2020+28. We report the first discovery of an FRB by our array, designated
as FRB 20220414A. We also investigate the optimal arrangement for the digitally
formed beams to achieve maximum detection rate by numerical simulation.
