Editing Phase-shift keying (section)

===Probability of error===
Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even (or odd) bits are used to modulate the in-phase component of the carrier, while the odd (or even) bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated.

As a result, the probability of bit-error for QPSK is the same as for BPSK:
:<math>P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)</math>

However, in order to achieve the same bit-error probability as BPSK, QPSK uses twice the power (since two bits are transmitted simultaneously).

The symbol error rate is given by:

:<math>
\begin{align}
 P_s &= 1 - \left( 1 - P_b \right)^2 \\
     &= 2Q\left( \sqrt{\frac{E_s}{N_0}} \right) - \left[ Q \left( \sqrt{\frac{E_s}{N_0}} \right) \right]^2.
\end{align}
</math>

If the [[signal-to-noise ratio]] is high (as is necessary for practical QPSK systems) the probability of symbol error may be approximated:

:<math>P_s \approx 2 Q \left( \sqrt{\frac{E_s}{N_0}} \right ) = \operatorname{erfc} \left( \sqrt{\frac{E_s}{2N_0}} \right) = \operatorname{erfc} \left( \sqrt{\frac{E_b}{N_0}} \right)</math>

The modulated signal is shown below for a short segment of a random binary data-stream. The two carrier waves are a cosine wave and a sine wave, as indicated by the signal-space analysis above. Here, the odd-numbered bits have been assigned to the in-phase component and the even-numbered bits to the quadrature component (taking the first bit as number 1). The total signal{{snd}} the sum of the two components{{snd}} is shown at the bottom. Jumps in phase can be seen as the PSK changes the phase on each component at the start of each bit-period. The topmost waveform alone matches the description given for BPSK above.
<br />[[File:QPSK timing diagram.png|frame|center|Timing diagram for QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown at the top, and the total combined signal at the bottom. Note the abrupt changes in phase at some of the bit-period boundaries.]]

The binary data that is conveyed by this waveform is: <span style="letter-spacing:0.5em;">11000110</span>.
* The odd bits, highlighted here, contribute to the in-phase component: <span style="letter-spacing:0.5em;">{{bg|lightblue|1}}1{{bg|lightblue|0}}0{{bg|lightblue|0}}1{{bg|lightblue|1}}0</span>
* The even bits, highlighted here, contribute to the quadrature-phase component: <span style="letter-spacing:0.5em;">1{{bg|lightblue|1}}0{{bg|lightblue|0}}0{{bg|lightblue|1}}1{{bg|lightblue|0}}</span>