Editing Phase-shift keying (section)

===Demodulation===
[[File:DPSK BER curves.svg|thumb|right|280px|BER comparison between DBPSK, DQPSK and their non-differential forms using Gray coding and operating in white noise]]

For a signal that has been differentially encoded, there is an obvious alternative method of demodulation. Instead of demodulating as usual and ignoring carrier-phase ambiguity, the phase between two successive received symbols is compared and used to determine what the data must have been. When differential encoding is used in this manner, the scheme is known as differential phase-shift keying (DPSK). Note that this is subtly different from just differentially encoded PSK since, upon reception, the received symbols are ''not'' decoded one-by-one to constellation points but are instead compared directly to one another.

Call the received symbol in the <math>k</math><sup>th</sup> timeslot <math>r_k</math> and let it have phase <math>\phi_k</math>. Assume without loss of generality that the phase of the carrier wave is zero. Denote the [[additive white Gaussian noise]] (AWGN) term as <math>n_k</math>. Then

:<math>r_k = \sqrt{E_s}e^{j\phi_k} + n_k.</math>

The decision variable for the <math>k-1</math><sup>th</sup> symbol and the <math>k</math><sup>th</sup> symbol is the phase difference between <math>r_k</math> and <math>r_{k-1}</math>. That is, if <math>r_k</math> is projected onto <math>r_{k-1}</math>, the decision is taken on the phase of the resultant complex number:

:<math>r_kr_{k-1}^* = E_se^{j\left(\varphi_k - \varphi_{k-1}\right)} + \sqrt{E_s}e^{j\varphi_k}n_{k-1}^* + \sqrt{E_s}e^{-j\varphi_{k-1}}n_k + n_kn_{k-1}^*</math>

where superscript * denotes [[complex conjugation]]. In the absence of noise, the phase of this is <math>\phi_{k}-\phi_{k-1}</math>, the phase-shift between the two received signals which can be used to determine the data transmitted.

The probability of error for DPSK is difficult to calculate in general, but, in the case of DBPSK it is:

:<math>P_b = \frac{1}{2}e^{-\frac{E_b}{N_0}},</math><ref>{{cite journal |first=G.L. |last=Stüber |title=Soft Decision Direct-Sequence DPSK Receivers |journal=IEEE Transactions on Vehicular Technology |volume=37 |issue=3 |pages=151–157 |date=August 1988 |doi=10.1109/25.16541 }}</ref>

which, when numerically evaluated, is only slightly worse than ordinary BPSK, particularly at higher <math>E_b/N_0</math> values.

Using DPSK avoids the need for possibly complex carrier-recovery schemes to provide an accurate phase estimate and can be an attractive alternative to ordinary PSK.

In [[optical communications]], the data can be modulated onto the phase of a [[laser]] in a differential way. The modulation is a laser which emits a [[continuous wave]], and a [[Mach–Zehnder modulator]] which receives electrical binary data. For the case of BPSK, the laser transmits the field unchanged for binary '1', and with reverse polarity for '0'. The demodulator consists of a [[delay line interferometer]] which delays one bit, so two bits can be compared at one time. In further processing, a [[photodiode]] is used to transform the [[optical field]] into an electric current, so the information is changed back into its original state.

The bit-error rates of DBPSK and DQPSK are compared to their non-differential counterparts in the graph to the right. The loss for using DBPSK is small enough compared to the complexity reduction that it is often used in communications systems that would otherwise use BPSK. For DQPSK though, the loss in performance compared to ordinary QPSK is larger and the system designer must balance this against the reduction in complexity.