Phase-shift keying

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

• ${\displaystyle E_{b}}$, energy per bit
• ${\displaystyle E_{s}=nE_{b}}$, energy per symbol with n bits
• ${\displaystyle T_{b}}$, bit duration
• ${\displaystyle T_{s}}$, symbol duration
• ${\displaystyle {\frac {1}{2}}N_{0}}$, noise power spectral density (W/Hz)
• ${\displaystyle P_{b}}$, probability of bit-error
• ${\displaystyle P_{s}}$, probability of symbol-error

${\displaystyle Q(x)}$ 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 ${\displaystyle x}$. It is a scaled form of the complementary Gaussian error function:

${\displaystyle Q(x)={\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }e^{-{\frac {1}{2}}t^{2}}\,dt={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right),\ x\geq 0}$.

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.

The general form for BPSK follows the equation:

${\displaystyle s_{n}(t)={\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft+\pi (1-n)),\quad n=0,1.}$

This yields two phases, 0 and π. In the specific form, binary data is often conveyed with the following signals:

${\displaystyle s_{0}(t)={\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft+\pi )=-{\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft)}$ for binary "0"

${\displaystyle s_{1}(t)={\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft)}$ for binary "1"

where f is the frequency of the base band.

Hence, the signal space can be represented by the single basis function

${\displaystyle \phi (t)={\sqrt {\frac {2}{T_{b}}}}\cos(2\pi ft)}$

where 1 is represented by ${\displaystyle {\sqrt {E_{b}}}\phi (t)}$ and 0 is represented by ${\displaystyle -{\sqrt {E_{b}}}\phi (t)}$. This assignment is arbitrary.

This use of this basis function is shown at the end of the next section in a signal timing diagram. The topmost signal is a BPSK-modulated cosine wave that the BPSK modulator would produce. The bit-stream that causes this output is shown above the signal (the other parts of this figure are relevant only to QPSK). After modulation, the base band signal will be moved to the high frequency band by multiplying ${\displaystyle \cos(2\pi f_{c}t)}$.

The bit error rate (BER) of BPSK under additive white Gaussian noise (AWGN) can be calculated as:[8]

${\displaystyle P_{b}=Q\left({\sqrt {\frac {2E_{b}}{N_{0}}}}\right)}$ or ${\displaystyle P_{e}={\frac {1}{2}}\operatorname {erfc} \left({\sqrt {\frac {E_{b}}{N_{0}}}}\right)}$

Since there is only one bit per symbol, this is also the symbol error rate.

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:

${\displaystyle s_{n}(t)={\sqrt {\frac {2E_{s}}{T_{s}}}}\cos \left(2\pi f_{c}t+(2n-1){\frac {\pi }{4}}\right),\quad n=1,2,3,4.}$

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

${\displaystyle {\begin{aligned}\phi _{1}(t)&={\sqrt {\frac {2}{T_{s}}}}\cos \left(2\pi f_{c}t\right)\\\phi _{2}(t)&={\sqrt {\frac {2}{T_{s}}}}\sin \left(2\pi f_{c}t\right)\end{aligned}}}$

The first basis function is used as the in-phase component of the signal and the second as the quadrature component of the signal.

Hence, the signal constellation consists of the signal-space 4 points

${\displaystyle {\begin{pmatrix}\pm {\sqrt {\frac {E_{s}}{2}}}&\pm {\sqrt {\frac {E_{s}}{2}}}\end{pmatrix}}.}$

The factors of 1/2 indicate that the total power is split equally between the two carriers.

Comparing these basis functions with that for BPSK shows clearly how QPSK can be viewed as two independent BPSK signals. Note that the signal-space points for BPSK do not need to split the symbol (bit) energy over the two carriers in the scheme shown in the BPSK constellation diagram.

QPSK systems can be implemented in a number of ways. An illustration of the major components of the transmitter and receiver structure are shown below.

Conceptual transmitter structure for QPSK. The binary data stream is split into the in-phase and quadrature-phase components. These are then separately modulated onto two orthogonal basis functions. In this implementation, two sinusoids are used. Afterwards, the two signals are superimposed, and the resulting signal is the QPSK signal. Note the use of polar non-return-to-zero encoding. These encoders can be placed before for binary data source, but have been placed after to illustrate the conceptual difference between digital and analog signals involved with digital modulation.

Receiver structure for QPSK. The matched filters can be replaced with correlators. Each detection device uses a reference threshold value to determine whether a 1 or 0 is detected.

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:

${\displaystyle P_{b}=Q\left({\sqrt {\frac {2E_{b}}{N_{0}}}}\right)}$

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:

${\displaystyle {\begin{aligned}P_{s}&=1-\left(1-P_{b}\right)^{2}\\&=2Q\left({\sqrt {\frac {E_{s}}{N_{0}}}}\right)-\left[Q\left({\sqrt {\frac {E_{s}}{N_{0}}}}\right)\right]^{2}.\end{aligned}}}$

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

${\displaystyle P_{s}\approx 2Q\left({\sqrt {\frac {E_{s}}{N_{0}}}}\right)=\operatorname {erfc} \left({\sqrt {\frac {E_{s}}{2N_{0}}}}\right)=\operatorname {erfc} \left({\sqrt {\frac {E_{b}}{N_{0}}}}\right)}$

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 – the sum of the two components – 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.

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: 11000110.

• The odd bits, highlighted here, contribute to the in-phase component: 11000110
• The even bits, highlighted here, contribute to the quadrature-phase component: 11000110

Signal doesn't pass through the origin, because only one bit of the symbol is changed at a time.

Offset quadrature phase-shift keying (OQPSK) is a variant of phase-shift keying modulation using four different values of the phase to transmit. It is sometimes called staggered quadrature phase-shift keying (SQPSK).

Difference of the phase between QPSK and OQPSK

Taking four values of the phase (two bits) at a time to construct a QPSK symbol can allow the phase of the signal to jump by as much as 180° at a time. When the signal is low-pass filtered (as is typical in a transmitter), these phase-shifts result in large amplitude fluctuations, an undesirable quality in communication systems. By offsetting the timing of the odd and even bits by one bit-period, or half a symbol-period, the in-phase and quadrature components will never change at the same time. In the constellation diagram shown on the right, it can be seen that this will limit the phase-shift to no more than 90° at a time. This yields much lower amplitude fluctuations than non-offset QPSK and is sometimes preferred in practice.

The picture on the right shows the difference in the behavior of the phase between ordinary QPSK and OQPSK. It can be seen that in the first plot the phase can change by 180° at once, while in OQPSK the changes are never greater than 90°.

The modulated signal is shown below for a short segment of a random binary data-stream. Note the half symbol-period offset between the two component waves. The sudden phase-shifts occur about twice as often as for QPSK (since the signals no longer change together), but they are less severe. In other words, the magnitude of jumps is smaller in OQPSK when compared to QPSK.

Timing diagram for offset-QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown the top and the total, combined signal at the bottom. Note the half-period offset between the two signal components.

The license-free shaped-offset QPSK (SOQPSK) is interoperable with Feher-patented QPSK (FQPSK), in the sense that an integrate-and-dump offset QPSK detector produces the same output no matter which kind of transmitter is used.[9]

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.[10][11]

The standard description of SOQPSK-TG involves ternary symbols.[12] SOQPSK is one of the most spread modulation schemes in application to LEO satellite communications.[13]

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° (${\displaystyle \pi /4}$ 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 ${\displaystyle \pi /4}$-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, ${\displaystyle \pi /4}$-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.

Timing diagram for π/4-QPSK. The binary data stream is shown beneath the time axis. The two signal components with their bit assignments are shown the top and the total, combined signal at the bottom. Note that successive symbols are taken alternately from the two constellations, starting with the "blue" one.

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.

For the general M-PSK there is no simple expression for the symbol-error probability if ${\displaystyle M>4}$. Unfortunately, it can only be obtained from

${\displaystyle P_{s}=1-\int _{-\pi /M}^{\pi /M}p_{\theta _{r}}\left(\theta _{r}\right)d\theta _{r},}$

where

${\displaystyle {\begin{aligned}p_{\theta _{r}}\left(\theta _{r}\right)&={\frac {1}{2\pi }}e^{-2\gamma _{s}\sin ^{2}\theta _{r}}\int _{0}^{\infty }Ve^{-{\frac {1}{2}}\left(V-2{\sqrt {\gamma _{s}}}\cos \theta _{r}\right)^{2}}\,dV,\\V&={\sqrt {r_{1}^{2}+r_{2}^{2}}},\\\theta _{r}&=\tan ^{-1}\left({\frac {r_{2}}{r_{1}}}\right),\\\gamma _{s}&={\frac {E_{s}}{N_{0}}}\end{aligned}}}$

and ${\displaystyle r_{1}\sim N\left({\sqrt {E_{s}}},{\frac {1}{2}}N_{0}\right)}$ and ${\displaystyle r_{2}\sim N\left(0,{\frac {1}{2}}N_{0}\right)}$ are each Gaussian random variables.

Bit-error rate curves for BPSK, QPSK, 8-PSK and 16-PSK, additive white Gaussian noise channel

This may be approximated for high ${\displaystyle M}$ and high ${\displaystyle E_{b}/N_{0}}$ by:

${\displaystyle P_{s}\approx 2Q\left({\sqrt {2\gamma _{s}}}\sin {\frac {\pi }{M}}\right).}$

The bit-error probability for ${\displaystyle M}$-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

${\displaystyle P_{b}\approx {\frac {1}{k}}P_{s}.}$

(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 left 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.

Bandwidth (or spectral) efficiency of M-PSK modulation schemes increases with increasing of modulation order M (unlike, for example, M-FSK):[14]

${\displaystyle \rho ={\frac {\log _{2}M}{2}}\quad [{\text{bits}}/{\text{s}}\cdot {\text{Hz}}]}$

The same relationship holds true for M-QAM.[15]

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. 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 ${\displaystyle M}$ 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 ${\displaystyle t=0}$.

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_s = 2T_b.

Analysis shows that differential encoding approximately doubles the error rate compared to ordinary ${\displaystyle M}$-PSK but this may be overcome by only a small increase in ${\displaystyle E_{b}/N_{0}}$. 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 ${\displaystyle \pi /4}$-DQPSK and 8-DPSK were implemented.

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 ${\displaystyle k}$^th timeslot ${\displaystyle r_{k}}$ and let it have phase ${\displaystyle \phi _{k}}$. Assume without loss of generality that the phase of the carrier wave is zero. Denote the additive white Gaussian noise (AWGN) term as ${\displaystyle n_{k}}$. Then

${\displaystyle r_{k}={\sqrt {E_{s}}}e^{j\phi _{k}}+n_{k}.}$

The decision variable for the ${\displaystyle k-1}$^th symbol and the ${\displaystyle k}$^th symbol is the phase difference between ${\displaystyle r_{k}}$ and ${\displaystyle r_{k-1}}$. That is, if ${\displaystyle r_{k}}$ is projected onto ${\displaystyle r_{k-1}}$, the decision is taken on the phase of the resultant complex number:

${\displaystyle r_{k}r_{k-1}^{*}=E_{s}e^{j\left(\varphi _{k}-\varphi _{k-1}\right)}+{\sqrt {E_{s}}}e^{j\varphi _{k}}n_{k-1}^{*}+{\sqrt {E_{s}}}e^{-j\varphi _{k-1}}n_{k}+n_{k}n_{k-1}^{*}}$

where superscript * denotes complex conjugation. In the absence of noise, the phase of this is ${\displaystyle \phi _{k}-\phi _{k-1}}$, 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:

${\displaystyle P_{b}={\frac {1}{2}}e^{-{\frac {E_{b}}{N_{0}}}},}$[16]

which, when numerically evaluated, is only slightly worse than ordinary BPSK, particularly at higher ${\displaystyle E_{b}/N_{0}}$ 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.

Differential encoding/decoding system diagram

At the ${\displaystyle k^{\textrm {th}}}$ time-slot call the bit to be modulated ${\displaystyle b_{k}}$, the differentially encoded bit ${\displaystyle e_{k}}$ and the resulting modulated signal ${\displaystyle m_{k}(t)}$. Assume that the constellation diagram positions the symbols at ±1 (which is BPSK). The differential encoder produces:

${\displaystyle \,e_{k}=e_{k-1}\oplus b_{k}}$

where ${\displaystyle \oplus {}}$ indicates binary or modulo-2 addition.

BER comparison between BPSK and differentially encoded BPSK operating in white noise

So ${\displaystyle e_{k}}$ only changes state (from binary "0" to binary "1" or from binary "1" to binary "0") if ${\displaystyle b_{k}}$ 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 ${\displaystyle e_{k}=\pm 1}$ and then the differential decoder reverses the encoding procedure and produces

${\displaystyle b_{k}=e_{k}\oplus e_{k-1},}$

since binary subtraction is the same as binary addition.

Therefore, ${\displaystyle b_{k}=1}$ if ${\displaystyle e_{k}}$ and ${\displaystyle e_{k-1}}$ differ and ${\displaystyle b_{k}=0}$ if they are the same. Hence, if both ${\displaystyle e_{k}}$ and ${\displaystyle e_{k-1}}$ are inverted, ${\displaystyle b_{k}}$ 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 ${\displaystyle E_{b}/N_{0}}$ to overcome this is small. The increase in ${\displaystyle E_{b}/N_{0}}$ 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.