# BER performance of a space-time code over Rayleigh channel model

For a given space-time code, and in general terms, how would one proceed in the evaluation of the BER performance of this code ?

In order to be more clear, say i have 16 code matrices(who's elements belong to QPSK modulation) of size 2x2 that could be sent over a Rayleigh flat fading channel by a MIMO system with 2 transmit antennas and 2 receive antennas. How would one proceed in the evaluation of the BER of such system ?

Thank you!

If $$X_0$$ and $$X$$ are two different code matrices, then their pairwise error probability is the probability that the receiver mistakes $$X$$ for $$X_0$$, when $$X_0$$ is transmitted. It is written $$P(X_0 \rightarrow X)$$.
Assuming all code matrices are transmitted with equal probability, and that the code is $$\mathcal{X}$$, then the average error rate is upper bounded by $$P_{\text{err}} \leq \frac{1}{|\mathcal{X}|} \sum_{X_n \in \mathcal{X}} \sum_{X_k \in \mathcal{X},k \neq x} P(X_n \rightarrow X_k)$$ which is just an application of the union bound.
The calculation of BER from $$P_{\text{err}}$$ for specific channels can get complicated and in fact in many cases no closed form formula can be found. I would suggest the book "Space-Time Block Coding for Wireless Communications" by Larsson and Stoica for more details, in particular section 4, "Error Probability Analysis".