# Exact average error probability of QPSK in Rayleigh fading channel

In Rayleigh fading channel(SISO), I know the instantaneous SER of QPSK for a given channel realization.

However, I do not know the exact average error probability.

It is very hard to calculate the average error probability because of the square of q function.

Are there any results on the exact average error probability of QPSK in Rayleigh fading channel?

From Tse and Viswanath, "Fundamentals of Wireless Communication" (which I highly recommend): $$p_e = \frac{1}{2}\left(1-\sqrt{\frac{\text{SNR}}{2+\text{SNR}}}\right)$$ Note that this is the probability of symbol error.