In this work we present an application of the Goertzel Filter for the channelization of multi-tonal signals, typically used for the read-out of cryogenic sensors which are multiplexed in the frequency domain (FDM), by means of Microwave Superconducting Quantum Interference Device (SQUID) Multiplexer ($μ$MUX). We demonstrate how implementing a bank of many of these filters, can be used to perform a channelization of the multi-tonal input signal to retrieve the data added by the sensors. We show how this approach can be implemented in a resource-efficient manner in a Field Programmable Gate Array (FPGA) within the state-of-the-art, which allows great scalability for reading thousands of sensors; as is required by Radio Telescopes in Cosmic Microwave Background Radiation (CMB) surveys using cryogenic bolometers, particles detection like Neutrino mass estimation using cryogenic calorimeters or Quantum Computing.