Editing Phase-shift keying (section)

===Bit error rate===
For the general M-PSK there is no simple expression for the symbol-error probability if <math>M > 4</math>. Unfortunately, it can only be obtained from

:<math>P_s = 1 - \int_{-\pi/M}^{\pi/M} p_{\theta_r}\left(\theta_r\right)d\theta_r,</math>

where

:<math>\begin{align}
  p_{\theta_r} \left(\theta_r\right) &= \frac{1}{2\pi} e^{-2\gamma_s\sin^2\theta_r} \int_0^\infty V e^{-\frac{1}{2}\left(V - 2\sqrt{\gamma_s}\cos\theta_r\right)^2}\,dV,\\
                                   V &= \sqrt{r_1^2 + r_2^2},\\
                            \theta_r &= \tan^{-1}\left(\frac{r_2}{r_1}\right),\\
                            \gamma_s &= \frac{E_s}{N_0}
\end{align}</math>
and <math>r_1 \sim N\left(\sqrt{E_s}, \frac{1}{2}N_0\right)</math> and <math>r_2 \sim N\left(0, \frac{1}{2} N_0\right)</math> are each Gaussian [[random variable]]s.

[[File:PSK BER curves.svg|thumb|right|280px|Bit-error rate curves for BPSK, QPSK, 8-PSK and 16-PSK, additive white Gaussian noise channel]]
This may be approximated for high <math>M</math> and high <math>E_b/N_0</math> by:

:<math>P_s \approx 2Q\left(\sqrt{2\gamma_s}\sin\frac{\pi}{M}\right). </math>

The bit-error probability for <math>M</math>-PSK can only be determined exactly once the bit-mapping is known. However, when [[Gray coding]] is used, the most probable error from one symbol to the next produces only a single bit-error and

:<math>P_b \approx \frac{1}{k} P_s. </math>

(Using Gray coding allows us to approximate the [[Lee distance]] of the errors as the [[Hamming distance]] of the errors in the decoded bitstream, which is easier to implement in hardware.)

The graph on the right compares the bit-error rates of BPSK, QPSK (which are the same, as noted above), 8-PSK and 16-PSK. It is seen that [[higher-order modulation]]s exhibit higher error-rates; in exchange however they deliver a higher raw data-rate.

Bounds on the error rates of various digital modulation schemes can be computed with application of the [[union bound]] to the signal constellation.