# How to calculate the accurate BER expression of circular 8-QAM constellation modulation?

It is easy to understand the BER derivation of square QAM, but the circular 8-QAM? I'm quite confused. The decision region is not in a rectangular shape anymore, meaning regular way does not work here. The constellation is like follows (from wikipedia)

Any suggestion or reference?

• An exact formula is very challenging to obtain. The good news is that you don't need it: an approximation based on the union bound is easy to calculate and usually pretty accurate (at least for moderate to large SNR). – MBaz Sep 16 '18 at 2:10
• As @MBaz says, an exact expression for the BER is quite messy to calculate but to even begin to approach the problem, you need to specify what the bit labels for the eight points are. Yes, the BER will vary depending on what labeling you choose. For example, if the point $(1,1)$ is assigned the label 000 and the point $(1,-1)$ is assigned the label 001 while the point $(1+\sqrt{3}/2,0)$ is assigned the label 111, the BER will be larger than if you assigned the label 100 to the point $(1+\sqrt{3}/2,0)$. Also, the three bits will have different BERs. – Dilip Sarwate Sep 16 '18 at 15:10