Bit error probability of differentially detected (G)MSK in unequalized mobile radio channels

We describe a new method for the analytical computation of the bit error probability BER of minimum shift keying (MSK) in indoor mobile radio channels. We include the following effects: (i) additive white Gaussian noise; (ii) time dispersion of the channel; (iii) Gaussian filtering of the data sequence (GMSK); (iv) receiver filtering. The channel is an N-path Rayleigh fading channel with arbitrary mean amplitudes of the paths and can also contain a line-of-sight component (i.e. Rician statistics). The sampling is done at a fixed but arbitrary instant. We first set up the general equations for occurrence of an error; direct evaluation of this equation requires the evaluation of a quadruple integral. We then develop closed-form approximate equations for the BER of unfiltered MSK that are valid for small BER. Finally, we develop a new mathematical formalism, which we call the two-path equivalent matrix (TPEM) method, which allows exact evaluation of the BER by solving a single well conditioned numerical integral; for small BERs we find closed-form expressions that are valid both for unfiltered and filtered (G)MSK. In order to prove the validity of our equations, we compare results to Monte-Carlo simulations and demonstrate excellent agreement