Phase-shift keying (PSK) is a digital modulation process which conveys data by changing (modulating) the phase of a constant frequency carrier wave. The modulation is accomplished by varying the sine and cosine inputs at a precise time. It is widely used for wireless LANs, RFID and Bluetooth communication.
Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular phase. The demodulator, which is designed specifically for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data. This requires the receiver to be able to compare the phase of the received signal to a reference signal such a system is termed coherent (and referred to as CPSK).
CPSK requires a complicated demodulator, because it must extract the reference wave from the received signal and keep track of it, to compare each sample to. Alternatively, the phase shift of each symbol sent can be measured with respect to the phase of the previous symbol sent. Because the symbols are encoded in the difference in phase between successive samples, this is called differential phase-shift keying (DPSK). DPSK can be significantly simpler to implement than ordinary PSK, as it is a 'non-coherent' scheme, i.e. there is no need for the demodulator to keep track of a reference wave. A trade-off is that it has more demodulation errors.
Introduction
There are three major classes of digital modulation techniques used for transmission of digitally represented data:
:<math>s_n(t) = \sqrt{\frac{2E_s}{T_s \cos \left(2 \pi f_c t + (2n - 1)\frac{\pi}{4}\right),\quad n = 1, 2, 3, 4.</math>
This yields the four phases π/4, 3π/4, 5π/4 and 7π/4 as needed.
This results in a two-dimensional signal space with unit basis functions
These modulations carefully shape the I and Q waveforms such that they change very smoothly, and the signal stays constant-amplitude even during signal transitions. (Rather than traveling instantly from one symbol to another, or even linearly, it travels smoothly around the constant-amplitude circle from one symbol to the next.) SOQPSK modulation can be represented as the hybrid of QPSK and MSK: SOQPSK has the same signal constellation as QPSK, however the phase of SOQPSK is always stationary.
The standard description of SOQPSK-TG involves ternary symbols. SOQPSK is one of the most spread modulation schemes in application to LEO satellite communications.
π/4-QPSK
thumb|right|Dual constellation diagram for π/4-QPSK. This shows the two separate constellations with identical Gray coding but rotated by 45° with respect to each other.
This variant of QPSK uses two identical constellations which are rotated by 45° (<math>\pi/4</math> radians, hence the name) with respect to one another.
:<math>\rho = \frac{\log_2M}{2} \quad [\text{bits}/\text{s} \cdot \text{Hz}]</math>
The same relationship holds true for M-QAM.
Differential phase-shift keying (DPSK)
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Differential encoding
Differential phase shift keying (DPSK) is a common form of phase modulation that conveys data by changing the phase of the carrier wave. 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.
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>.
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> = 2T<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 standard where <math>\pi/4</math>-DQPSK and 8-DPSK were implemented.
Demodulation
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-1n_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>
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
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 or modulo-2 addition.
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 typically about 3 dB. The performance degradation is a result of noncoherent transmission in this case it refers to the fact that tracking of the phase is completely ignored.
Applications
Owing to PSK's simplicity, particularly when compared with its competitor quadrature amplitude modulation, it is widely used in existing technologies.
The wireless LAN standard, IEEE 802.11b-1999, uses a variety of different PSKs depending on the data rate required. At the basic rate of 1Mbit/s, it uses DBPSK (differential BPSK). To provide the extended rate of 2Mbit/s, DQPSK is used. In reaching 5.5Mbit/s and the full rate of 11Mbit/s, QPSK is employed, but has to be coupled with complementary code keying. The higher-speed wireless LAN standard, IEEE 802.11g-2003, has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54Mbit/s. The 6 and 9Mbit/s modes use OFDM modulation where each sub-carrier is BPSK modulated. The 12 and 18Mbit/s modes use OFDM with QPSK. The fastest four modes use OFDM with forms of quadrature amplitude modulation.
Because of its simplicity, BPSK is appropriate for low-cost passive transmitters, and is used in RFID standards such as ISO/IEC 14443 which has been adopted for biometric passports, credit cards such as American Express's ExpressPay, and many other applications.
Bluetooth 2 uses <math>\pi/4</math>-DQPSK at its lower rate (2Mbit/s) and 8-DPSK at its higher rate (3Mbit/s) when the link between the two devices is sufficiently robust. Bluetooth 1 modulates with Gaussian minimum-shift keying, a binary scheme, so either modulation choice in version 2 will yield a higher data rate. A similar technology, IEEE 802.15.4 (the wireless standard used by Zigbee) also relies on PSK using two frequency bands: 868 MHz and 915MHz with BPSK and at 2.4GHz with OQPSK.
Both QPSK and 8PSK are widely used in satellite broadcasting. QPSK is still widely used in the streaming of SD satellite channels and some HD channels. High definition programming is delivered almost exclusively in 8PSK due to the higher bitrates of HD video and the high cost of satellite bandwidth. The DVB-S2 standard requires support for both QPSK and 8PSK. The chipsets used in new satellite set top boxes, such as Broadcom's 7000 series support 8PSK and are backward compatible with the older standard.
Historically, voice-band synchronous modems such as the Bell 201, 208, and 209 and the CCITT V.26, V.27, V.29, V.32, and V.34 used PSK.
Mutual information with additive white Gaussian noise
300px|right|thumb|Mutual information of PSK over the AWGN channel
The mutual information of PSK can be evaluated in additive Gaussian noise by numerical integration of its definition. The curves of mutual information saturate to the number of bits carried by each symbol in the limit of infinite signal to noise ratio <math>E_s/N_0</math>. On the contrary, in the limit of small signal to noise ratios the mutual information approaches the AWGN channel capacity, which is the supremum among all possible choices of symbol statistical distributions.
At intermediate values of signal to noise ratios the mutual information (MI) is well approximated by: