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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

Marcus Müller
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Loran
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  • 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! – Marcus Müller Nov 19 '21 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? – Marcus Müller Nov 19 '21 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? – Loran Nov 19 '21 at 23:08
  • For BPSK and QPSK, \begin{equation} p_{BER}=Q\left(\sqrt{\frac{2E_b}{N_o}}\right) \end{equation} 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. – Loran Nov 19 '21 at 23:48
  • Gray mapping or not. – Loran Nov 19 '21 at 23:57
  • but 8-PSK is not BPSK and not QPSK, the decision regions aren't rectangulat, so the decision and hence, the areas you integrate about aren't as easy, so, sorry, the formula is more complicated. – Marcus Müller Nov 20 '21 at 11:59
  • Try looking at a paper "Computation of the Bit Error Rate of Coherent M-ary PSK with Gray Code Bit Mapping" by P.Lee, IEEE Transactions on Communications, May 1986. I have not read the paper myself (it is behind IEEE's paywall) but the abstract says "..... A closed-form expression for the exact BER of 8-ary PSK is presented...." – Dilip Sarwate Nov 24 '21 at 03:42

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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 $\Psi= \pi/M$.

Okan Erturk
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  • I am not asking for pairwise probability of error approximations, BER approximations from SER, the use of bounds, infinite sums, ets. Is there an exact formula for the BER of 8-PSK in the literature that you are aware of? I have never encountered it. – Loran Nov 19 '21 at 22:32
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    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. – Okan Erturk Nov 19 '21 at 22:38
  • Thanks Okan. I want to know if the exact formula using the Q function or complementary error function exists in the literature --functions that have existed for some time in mathematical packages, just like exp, sin, tan, ln, etc. – Loran Nov 19 '21 at 22:50
  • For BPSK and QPSK, \begin{equation} p_{BER}=Q\left(\sqrt{\frac{2E_b}{N_o}}\right) \end{equation} 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. – Loran Nov 19 '21 at 23:51
  • Gray mapping or not. – Loran Nov 19 '21 at 23:59
  • @Loran sorry, see my comment on your question: the decision regions of 8-PSK are not rectangular like in BPSK and QPSK, so the formula is more complicated then that. Did you want an easy formula? Your question said exact! I haven't gone through the paper Okan cites, but I trust it – so, this is the right answer, and I would have expected the formula to be much much longer. – Marcus Müller Nov 20 '21 at 12:00
  • I would not consider $Q$ to be exact, whether it's in a math package or not. To find a numerical value of it a package evaluates a similar (but simpler) version of the integral that @okanerturk gives you. While there's probably some highly optimized arithmetic behind calculating it in a math package, it's still a numerical solution to a messy nonlinear integration, and will never, ever, be "exact". – TimWescott Nov 20 '21 at 16:57
  • What is $\theta$ in the formula? What is $\bar{\gamma}$? What is $\gamma_3$? None of these symbols seem to be defined in this answer. Also, $P_M$ sounds more like a symbol error probability, not a bit error probability, which is what the OP is looking for. – Dilip Sarwate Nov 24 '21 at 16:38
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    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. – Okan Erturk Nov 25 '21 at 21:12
  • -1 This answer gives a formula for the symbol error probability, and not the bit error probability (neither the specific values of BER for the three bits (not all the same), nor the average BER). – Dilip Sarwate Dec 02 '21 at 16:54
  • Hello, it is assumed that the error occurs only when the consecutive symbol is decoded. Since the constellation is Gray coded, the number of bit error is proportional to $1/(\log_2(M)$ symbol error. This is a quite tight approximation and practically acceptable. – Okan Erturk Dec 04 '21 at 10:11