# Exact formula for 8-PSK BER

Is there an exact formula for the probability of bit error (or bit error rate, BER) for 8-PSK (in the literature, course slides, etc.)? I am not referring to SEP (Symbol Error Probability) but BER.

There is already a thread for 8-PSK but it does not present the BER. Bit Error Probability for 8PSK

• You forgot to state which noise model you're assuming; but seeing you're referring to that other post, is additive uncorrelated circularly normal noise the right assumption? I ask because you want something that is pretty much not that useful; the approximation given in the other answer is "good enough" for any sensible SNR, and if your system doesn't assume good SNR, or that estimate isn't good enough for that, then it's probably a PSK transmission system that doesn't actually experience that type of noise. so, is the noise model I'm assuming correct?Please edit your question to state! Commented Nov 19, 2021 at 22:01
• I've got an answer for that noise model nearly ready (it still has two minor factual errors), but let's discuss expectations here: the formula you'll get involves integrals over functions that cannot analytically be integrated. There cannot be an easier formula, sadly. Does this really help you? Commented Nov 19, 2021 at 22:02
• Hello Marcus, I am looking for a formula that uses the Q function or the erfc function, just like in the case of BPSK and QPSK. I am not looking for a mathematical expression in the form of an integral, an infinite series, a bound, or an approximation. Is there a such a formula published in an article, textbook, or course slides involving the Q function or the erfc function? Commented Nov 19, 2021 at 23:08
• For BPSK and QPSK, $$p_{BER}=Q\left(\sqrt{\frac{2E_b}{N_o}}\right)$$ I want to know if a formula such as this exist in the literature for 8-PSK. Again, I am not looking for bounds, pairwise probability of error, SER, approximation, infinite sums, integrals, etc., but a formula involving the $Q$ function or the $erfc$ function. Commented Nov 19, 2021 at 23:48
• Gray mapping or not. Commented Nov 19, 2021 at 23:57

The exact error probability for the M-PSK constellation is derived in "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" by JW Craig, a well-known technique, especially for fading channels. Eq. (6) in the reference denotes the probability of error as given below: $$\mathrm{P}_{\mathrm{M}}=\frac{1}{\pi} \int_{0}^{\pi-\Psi} \exp \left[-\frac{\bar{\gamma} \sin ^{2}(\Psi)}{\sin ^{2}(\theta)}\right] \mathrm{d} \theta$$
where $$\bar{\gamma}$$ is the SNR and $$\Psi= \pi/M$$.
• I think you are looking for an expression including some well-defined functions like $sin$, $tan$ $log$, etc. Actually, these functions can not be expressed exactly without using infinite sum as well. For example for binary case (BPSK), the BEP exists in terms of the $Q$ function, which is nothing but a special representation of an integral! In M-PSK it is assumed that the symbols are gray mapped, therefore, a symbol error results in a bit error. Commented Nov 19, 2021 at 22:38
• For BPSK and QPSK, $$p_{BER}=Q\left(\sqrt{\frac{2E_b}{N_o}}\right)$$ I want to know if a formula such as this exist in the literature for 8-PSK. Again, I am not looking for bounds, pairwise probability of error, SER, approximation, infinite sums, integrals, etc., but a formula involving the $Q$ function or the $erfc$ function. Commented Nov 19, 2021 at 23:51
• Hello, $\theta$ is the dummy variable that the integral is taken over. There was a typo, $\bar{\gamma}$ is the SNR which is defined as $E_s/N_o$ where $E_s$ is the average symbol energy, and $N_o$ is the power spectral density of the additive noise. Commented Nov 25, 2021 at 21:12