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A novel field-programmable gate array (FPGA)-based online tracking algorithm for helix track reconstruction in a solenoidal field, developed for the Antiproton Annihilations at Darmstadt spectrometer, is described. Employing the straw tube tracker detector with 4636 straw tubes, the algorithm includes a complex track finder and a track fitter. Implemented in VHDL, the algorithm is tested on a Xilinx Virtex-4 FX60 FPGA chip with different types of events, at different event rates. A processing time of 7~\mu \texts per event for an average of six charged tracks is obtained. The momentum resolution is about 3% (4%) for pt ( pz ) at 1 GeV/c. Comparing with the algorithm running on a CPU chip (single core Intel Xeon E5520 at 2.26 GHz), an improvement of three orders of magnitude in processing time is obtained. The algorithm can handle severe overlapping of events, which are typical for interaction rates above 10 MHz. [ABSTRACT FROM AUTHOR]