On majority-logic diversity combining in Rayleigh fading

An interesting relationship between the bit error rate (BER) performance of majority-logic diversity combining and that of maximal-ratio combining in Rayleigh fading is derived. It is shown that, for the commonly-used binary modulation formats BPSK, DPSK, and BFSK, the BER of an L-diversity system using maximal-ratio combining (or equal-gain combining for the noncoherent modulation) identical to that of a (2L-1)-diversity system using majority-logic combining. The comparison is made based on equal branch signal-to-noise ratios (E_c /N₀), which implies that to achieve the same performance, a system using majority-logic combining requires approximately 3 dB greater total energy-per-bit-to-noise-density-ratio, E_b/N₀, than a system using maximal-ratio combining. As a related side note, it is also shown that for majority-logic combining, diversity of order 2L achieves the same performance as diversity of order 2L-1, assuming an unbiased tie resolution (i.e., when a tie occurs, the final output is randomly chosen)