Editing Phase-shift keying (section)

==Differential phase-shift keying (DPSK)==

===Differential encoding===
{{Main article|differential coding}}

Differential phase shift keying (DPSK) is a common form of phase modulation that conveys data by changing the phase of the carrier wave. As mentioned for BPSK and QPSK there is an ambiguity of phase if the constellation is rotated by some effect in the [[communications channel]] through which the signal passes. This problem can be overcome by using the data to ''change'' rather than ''set'' the phase.

For example, in differentially-encoded BPSK a binary "1" may be transmitted by adding 180° to the current phase and a binary "0" by adding 0° to the current phase. {{anchor|SDPSK}}
Another variant of DPSK is symmetric differential phase shift keying, SDPSK, where encoding would be +90° for a "1" and −90° for a "0".

In differentially-encoded QPSK (DQPSK), the phase-shifts are 0°, 90°, 180°, −90° corresponding to data "00", "01", "11", "10". This kind of encoding may be demodulated in the same way as for non-differential PSK but the phase ambiguities can be ignored. Thus, each received symbol is demodulated to one of the <math>M</math> points in the constellation and a [[comparator]] then computes the difference in phase between this received signal and the preceding one. The difference encodes the data as described above. ''Symmetric differential quadrature phase shift keying'' (SDQPSK) is like DQPSK, but encoding is symmetric, using phase shift values of −135°, −45°, +45° and +135°.

The modulated signal is shown below for both DBPSK and DQPSK as described above. In the figure, it is assumed that the ''signal starts with zero phase'', and so there is a phase shift in both signals at <math>t = 0</math>.

[[File:DBQPSK timing diag fixed.png|center|thumb|600px|Timing diagram for DBPSK and DQPSK. The binary data stream is above the DBPSK signal. The individual bits of the DBPSK signal are grouped into pairs for the DQPSK signal, which only changes every ''T<sub>s</sub>'' = 2''T<sub>b</sub>''.]]

Analysis shows that differential encoding approximately doubles the error rate compared to ordinary <math>M</math>-PSK but this may be overcome by only a small increase in <math>E_b/N_0</math>. Furthermore, this analysis (and the graphical results below) are based on a system in which the only corruption is [[additive white Gaussian noise]] (AWGN). However, there will also be a physical channel between the transmitter and receiver in the communication system. This channel will, in general, introduce an unknown phase-shift to the PSK signal; in these cases the differential schemes can yield a ''better'' error-rate than the ordinary schemes which rely on precise phase information.

One of the most popular applications of DPSK is the [[Bluetooth#Implementation|Bluetooth standard]] where <math>\pi/4</math>-DQPSK and 8-DPSK were implemented.

===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.

===Example: Differentially-encoded BPSK===
[[File:Differential Codec.png|center|500px|thumb|Differential encoding/decoding system diagram]]

At the <math>k^{\textrm{th}}</math> time-slot call the bit to be modulated <math>b_k</math>, the differentially encoded bit <math>e_k</math> and the resulting modulated signal <math>m_k(t)</math>. Assume that the constellation diagram positions the symbols at ±1 (which is BPSK). The differential encoder produces:

:<math>\,e_k = e_{k-1} \oplus b_k</math>

where <math>\oplus{}</math> indicates [[binary addition|binary]] or [[modular arithmetic|modulo-2]] addition.

[[File:Diff enc BPSK BER curves.svg|thumb|right|280px|BER comparison between BPSK and differentially encoded BPSK operating in white noise]]

So <math>e_k</math> only changes state (from binary "0" to binary "1" or from binary "1" to binary "0") if <math>b_k</math> is a binary "1". Otherwise it remains in its previous state. This is the description of differentially encoded BPSK given above.

The received signal is demodulated to yield <math>e_k = \pm 1</math> and then the differential decoder reverses the encoding procedure and produces

:<math>b_k = e_k \oplus e_{k-1},</math>

since binary subtraction is the same as binary addition.

Therefore, <math>b_k=1</math> if <math>e_k</math> and <math>e_{k-1}</math> differ and <math>b_k=0</math> if they are the same. Hence, if both <math>e_k</math> and <math>e_{k-1}</math> are ''inverted'', <math>b_k</math> will still be decoded correctly. Thus, the 180° phase ambiguity does not matter.

Differential schemes for other PSK modulations may be devised along similar lines. The waveforms for DPSK are the same as for differentially encoded PSK given above since the only change between the two schemes is at the receiver.

The BER curve for this example is compared to ordinary BPSK on the right. As mentioned above, whilst the error rate is approximately doubled, the increase needed in <math>E_b/N_0</math> to overcome this is small.  The increase in <math>E_b/N_0</math> required to overcome differential modulation in coded systems, however, is larger{{snd}} typically about 3&nbsp;dB. The performance degradation is a result of [[noncoherent transmission]]{{snd}} in this case it refers to the fact that tracking of the phase is completely ignored.

===Definitions===
For determining error-rates mathematically, some definitions will be needed:

* <math>E_b</math>,            energy per [[bit]]
* <math>E_s = nE_b</math>,     energy per symbol with ''n'' bits
* <math>T_b</math>,            [[bit rate|bit duration]]
* <math>T_s</math>,            [[symbol rate|symbol duration]]
* <math>\frac{1}{2}N_0</math>, [[signal noise|noise]] [[spectral density|power spectral density]] ([[watt|W]]/[[Hz]])
* <math>P_b</math>,            [[probability]] of ''bit-error''
* <math>P_s</math>,            probability of symbol-error

<math>Q(x)</math> will give the probability that a single sample taken from a random process with zero-mean and unit-variance [[Gaussian probability density function]] will be greater or equal to <math>x</math>. It is a scaled form of the [[Error function|complementary Gaussian error function]]:

: <math>Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^\infty e^{-\frac{1}{2}t^2} \,dt = \frac{1}{2} \operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right),\ x \geq 0</math>.

The error rates quoted here are those in [[additive white Gaussian noise]] (AWGN). These error rates are lower than those computed in [[fading channel]]s, hence, are a good theoretical benchmark to compare with.