Phase-shift keying

There are three major classes of digital modulation techniques used for transmission of digitally represented data: • Amplitude-shift keying (ASK) • Frequency-shift keying (FSK) • Phase-shift keying (PSK) All convey data by changing some aspect of a base signal, the carrier wave (usually a sinusoid), in response to a data signal. In the case of PSK, the phase is changed to represent the data signal. There are two fundamental ways of utilizing the phase of a signal in this way: • By viewing the phase itself as conveying the information, in which case the demodulator must have a reference signal to compare the received signal's phase against; or • By viewing the change in the phase as conveying information differential schemes, some of which do not need a reference carrier (to a certain extent). A convenient method to represent M-ary transmission PSK schemes is on a constellation diagram. This shows the points in the complex plane where, in this context, the real and imaginary axes are termed the in-phase and quadrature axes respectively due to their 90° separation. Such a representation on perpendicular axes lends itself to straightforward implementation. The amplitude of each point along the in-phase axis is used to modulate a cosine (or sine) wave and the amplitude along the quadrature axis to modulate a sine (or cosine) wave. By convention, in-phase modulates cosine and quadrature modulates sine. In PSK, the constellation points chosen are usually positioned with uniform angular spacing around a circle. This gives maximum phase-separation between adjacent points and thus the best immunity to corruption. They are positioned on a circle so that they can all be transmitted with the same energy. In this way, the moduli of the complex numbers they represent will be the same and thus so will the amplitudes needed for the cosine and sine waves. Two common examples are "binary phase-shift keying" (BPSK) which uses two phases, and "quadrature phase-shift keying" (QPSK) which uses four phases, although any number of phases may be used. Since the data to be conveyed are usually binary, the PSK scheme is usually designed with the number of constellation points being a power of two. Definitions For determining error-rates mathematically, some definitions will be needed: • E_b, energy per bit • E_s = nE_b, energy per symbol with n bits • T_b, bit duration • T_s, symbol duration • \frac{1}{2}N_0, noise power spectral density (W/Hz) • P_b, probability of bit-error • P_s, probability of symbol-error Q(x) 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 x. It is a scaled form of the complementary Gaussian error function: : 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. The error rates quoted here are those in additive white Gaussian noise (AWGN). These error rates are lower than those computed in fading channels, hence, are a good theoretical benchmark to compare with. ==Binary phase-shift keying (BPSK)==

example for BPSK BPSK (also sometimes called PRK, phase reversal keying, or 2PSK) is the simplest form of phase shift keying (PSK). It uses two phases which are separated by 180° and so can also be termed 2-PSK. It does not particularly matter exactly where the constellation points are positioned, and in this figure they are shown on the real axis, at 0° and 180°. Therefore, it handles the highest noise level or distortion before the demodulator reaches an incorrect decision. That makes it the most robust of all the PSKs. It is, however, only able to modulate at 1bit/symbol (as seen in the figure) and so is unsuitable for high data-rate applications. In the presence of an arbitrary phase-shift introduced by the communications channel, the demodulator (see, e.g. Costas loop) is unable to tell which constellation point is which. As a result, the data is often differentially encoded prior to modulation. BPSK is functionally equivalent to 2-QAM modulation. Implementation The general form for BPSK follows the equation: :s_n(t) = \sqrt{\frac{2E_b}{T_b}} \cos(2 \pi f t + \pi(1-n )),\quad n = 0,1. This yields two phases, 0 and π. In the specific form, binary data is often conveyed with the following signals: :P_b = Q\left(\sqrt{\frac{2 E_b}{N_0}}\right) or P_e = \frac{1}{2} \operatorname{erfc} \left( \sqrt{\frac{E_b}{N_0}}\right) Since there is only one bit per symbol, this is also the symbol error rate. ==Quadrature phase-shift keying (QPSK)==

. Each adjacent symbol only differs by one bit. Sometimes this is known as quadriphase PSK, 4-PSK, or 4-QAM. (Although the root concepts of QPSK and 4-QAM are different, the resulting modulated radio waves are exactly the same.) QPSK uses four points on the constellation diagram, equispaced around a circle. With four phases, QPSK can encode two bits per symbol, shown in the diagram with Gray coding to minimize the bit error rate (BER) sometimes misperceived as twice the BER of BPSK. The mathematical analysis shows that QPSK can be used either to double the data rate compared with a BPSK system while maintaining the same bandwidth of the signal, or to maintain the data-rate of BPSK but halving the bandwidth needed. In this latter case, the BER of QPSK is exactly the same as the BER of BPSK and believing differently is a common confusion when considering or describing QPSK. The transmitted carrier can undergo numbers of phase changes. Given that radio communication channels are allocated by agencies such as the Federal Communications Commission giving a prescribed (maximum) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK - at the same BER. The engineering penalty that is paid is that QPSK transmitters and receivers are more complicated than the ones for BPSK. However, with modern electronics technology, the penalty in cost is very moderate. As with BPSK, there are phase ambiguity problems at the receiving end, and differentially encoded QPSK is often used in practice. Implementation The implementation of QPSK is more general than that of BPSK and also indicates the implementation of higher-order PSK. Writing the symbols in the constellation diagram in terms of the sine and cosine waves used to transmit them: :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. 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 This variant of QPSK uses two identical constellations which are rotated by 45° (\pi/4 radians, hence the name) with respect to one another. Usually, either the even or odd symbols are used to select points from one of the constellations and the other symbols select points from the other constellation. This also reduces the phase-shifts from a maximum of 180°, but only to a maximum of 135° and so the amplitude fluctuations of \pi/4-QPSK are between OQPSK and non-offset QPSK. One property this modulation scheme possesses is that if the modulated signal is represented in the complex domain, transitions between symbols never pass through 0. In other words, the signal does not pass through the origin. This lowers the dynamical range of fluctuations in the signal which is desirable when engineering communications signals. On the other hand, \pi/4-QPSK lends itself to easy demodulation and has been adopted for use in, for example, TDMA cellular telephone systems. The modulated signal is shown below for a short segment of a random binary data-stream. The construction is the same as above for ordinary QPSK. Successive symbols are taken from the two constellations shown in the diagram. Thus, the first symbol (1 1) is taken from the "blue" constellation and the second symbol (0 0) is taken from the "green" constellation. Note that magnitudes of the two component waves change as they switch between constellations, but the total signal's magnitude remains constant (constant envelope). The phase-shifts are between those of the two previous timing-diagrams. DPQPSK Dual-polarization quadrature phase shift keying (DPQPSK) or dual-polarization QPSK - involves the polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by a factor of 2. This is a cost-effective alternative to utilizing 16-PSK, instead of QPSK to double the spectral efficiency. ==Higher-order PSK==

Any number of phases may be used to construct a PSK constellation but 8-PSK is usually the highest order PSK constellation deployed. With more than 8 phases, the error-rate becomes too high and there are better, though more complex, modulations available such as quadrature amplitude modulation (QAM). Although any number of phases may be used, the fact that the constellation must usually deal with binary data means that the number of symbols is usually a power of 2 to allow an integer number of bits per symbol. Bit error rate For the general M-PSK there is no simple expression for the symbol-error probability if M > 4. Unfortunately, it can only be obtained from :P_s = 1 - \int_{-\pi/M}^{\pi/M} p_{\theta_r}\left(\theta_r\right)d\theta_r, where :\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} and r_1 \sim N\left(\sqrt{E_s}, \frac{1}{2}N_0\right) and r_2 \sim N\left(0, \frac{1}{2} N_0\right) are each Gaussian random variables. This may be approximated for high M and high E_b/N_0 by: :P_s \approx 2Q\left(\sqrt{2\gamma_s}\sin\frac{\pi}{M}\right). The bit-error probability for M-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 :P_b \approx \frac{1}{k} P_s. (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 modulations 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. Spectral efficiency Bandwidth (or spectral) efficiency of M-PSK modulation schemes increases with increasing of modulation order M (unlike, for example, M-FSK): :\rho = \frac{\log_2M}{2} \quad [\text{bits}/\text{s} \cdot \text{Hz}] The same relationship holds true for M-QAM. ==Differential phase-shift keying (DPSK)==

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 M 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 t = 0. Analysis shows that differential encoding approximately doubles the error rate compared to ordinary M-PSK but this may be overcome by only a small increase in E_b/N_0. 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 \pi/4-DQPSK and 8-DPSK were implemented. Demodulation 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 kth timeslot r_k and let it have phase \phi_k. Assume without loss of generality that the phase of the carrier wave is zero. Denote the additive white Gaussian noise (AWGN) term as n_k. Then :r_k = \sqrt{E_s}e^{j\phi_k} + n_k. The decision variable for the k-1th symbol and the kth symbol is the phase difference between r_k and r_{k-1}. That is, if r_k is projected onto r_{k-1}, the decision is taken on the phase of the resultant complex number: :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}^* where superscript * denotes complex conjugation. In the absence of noise, the phase of this is \phi_{k}-\phi_{k-1}, 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: :P_b = \frac{1}{2}e^{-\frac{E_b}{N_0}}, which, when numerically evaluated, is only slightly worse than ordinary BPSK, particularly at higher E_b/N_0 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 At the k^{\textrm{th}} time-slot call the bit to be modulated b_k, the differentially encoded bit e_k and the resulting modulated signal m_k(t). Assume that the constellation diagram positions the symbols at ±1 (which is BPSK). The differential encoder produces: :\,e_k = e_{k-1} \oplus b_k where \oplus{} indicates binary or modulo-2 addition. So e_k only changes state (from binary "0" to binary "1" or from binary "1" to binary "0") if b_k 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 e_k = \pm 1 and then the differential decoder reverses the encoding procedure and produces :b_k = e_k \oplus e_{k-1}, since binary subtraction is the same as binary addition. Therefore, b_k=1 if e_k and e_{k-1} differ and b_k=0 if they are the same. Hence, if both e_k and e_{k-1} are inverted, b_k 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 E_b/N_0 to overcome this is small. The increase in E_b/N_0 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 \pi/4-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==

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 E_s/N_0. 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: :\textrm{MI} \simeq \log_2\left(\sqrt{\frac{4\pi}{e}\frac{E_s}{N_0}}\right). The mutual information of PSK over the AWGN channel is generally lower than that of QAM modulation formats. ==See also==