This paper presents the development of the digital acquisition chain for the upgrade of the Beam Position Monitor (BPM) system for the Super Proton Synchrotron (SPS) at CERN. The front-end electronics will be based on logarithmic amplifiers and, including the digitization, it will be located inside the accelerator tunnel, with optical latency deterministic transmission to surface back-end electronics, where the data stream is processed in a Field Programmable Gate Array. The paper focuses on the processing techniques for the position estimation out of beam data conditioned by logarithmic amplifiers. The elements of the acquisition chain have been tested in laboratory and with beam signals.