Derivation of Log-Likelihood Ratio for M-Ary Non-Orthogonal FSK Wireless System

In a soft decision Viterbi decoder, the log-likelihood ratio (LLR) metric is used to improve the performance of the error correction. On linear systems under additive white gaussian noise channel, this is straightforward as the error distribution at the input of the Viterbi decoder is also gaussian. In non-linear systems such as the M-ary non-coherent non-orthogonal frequency shift keying (FSK), this is generally not true. However, non-orthogonal non-coherent FSK LLR has not yet been proposed in literature. In this paper we derived the LLR for the M-ary non-orthogonal FSK mathematically and verified it using simulations.