Bounds to the error probability of hybrid diversity two-dimensional signaling

In hybrid selection/maximal ratio combining the receiver selects the L branches with the largest signal-to-noise ratios from N available diversity branches and performs maximal ratio combining. This incurs a loss or penalty with respect to maximal ratio combining which can be defined as the increase in signal-to-noise ratio required for hybrid combining to achieve the same symbol error probability as maximal ratio combining. In this paper, we derive lower and upper bound expressions for the penalty of any two-dimensional signaling constellation with polygonal decision regions. Using these bounds with well known results for maximal ratio combining, the error probability performance of two-dimensional signaling with hybrid diversity can be approximated at small and large values of signal-to-noise ratio, to a high accuracy.