  The Compact Muon Solenoid (CMS) experiment prepares its Phase-2 upgrade for
the high-luminosity era of the LHC operation (HL-LHC). Due to the increase of
occupancy, trigger latency and rates, the full electronics of the CMS Drift
Tube (DT) chambers will need to be replaced. In the new design, the time bin
for the digitisation of the chamber signals will be of around 1~ns, and the
totality of the signals will be forwarded asynchronously to the service cavern
at full resolution. The new backend system will be in charge of building the
trigger primitives of each chamber. These trigger primitives contain the
information at chamber level about the muon candidates position, direction, and
collision time, and are used as input in the L1 CMS trigger. The added
functionalities will improve the robustness of the system against ageing. An
algorithm based on analytical solutions for reconstructing the DT trigger
primitives, called Analytical Method, has been implemented both as a software
C++ emulator and in firmware. Its performance has been estimated using the
software emulator with simulated and real data samples, and through hardware
implementation tests. Measured efficiencies are 96 to 98\% for all qualities
and time and spatial resolutions are close to the ultimate performance of the
DT chambers. A prototype chain of the HL-LHC electronics using the Analytical
Method for trigger primitive generation has been installed during Long Shutdown
2 of the LHC and operated in CMS cosmic data taking campaigns in 2020 and 2021.
Results from this validation step, the so-called Slice Test, are presented.
