# Phase-shift keying

Phase-shift keying (PSK) is the overarching name for a number of digital modulation schemes that convey data by changing (modulating) the phase of a reference signal (the carrier wave).

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## Introduction

As with all modulation schemes, PSK conveys 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 (modulated or keyed) 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).

As for many digital modulation schemes, the constellation diagram is a useful representation and is relied upon in this article.

In PSK, the constellation points chosen are usually positioned with uniform angular spacing around a circle which gives maxmimum 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 cos and sin 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 2 (2,4,8,...).

### Definitions

For determining error-rates we will need some definitions:

• [itex]E_b[itex] = Energy-per-bit
• [itex]E_s[itex] = Energy-per-symbol = [itex]kE_b[itex] with k bits per symbol
• [itex]N_0[itex] = Noise power spectral density (W/Hz)
• [itex]P_b[itex] = Probability of bit-error
• [itex]P_s[itex] = Probability of symbol-error
• [itex]Q(x) = \frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt,\ x\geq{}0

[itex].

[itex]Q(x)[itex] is related to the complementary Gaussian error function by: [itex]Q(x) = \frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)[itex], which is the probability that x will be under the tail of the Gaussian PDF towards positive infinity.

The error-rates quoted here are those in additive white Gaussian noise (AWGN).

## Binary Phase-shift Keying (BPSK)

Missing image
BPSK_Gray_Coded.png
Constellation diagram for BPSK.

BPSK is the simplest form of PSK. It uses just 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°. This modulation is the most robust of all the PSKs since it takes serious distortion to make the demodulator reach an incorrect decision. In exchange though, it is only able to modulate at 1bit/symbol (as seen in the figure) and so is unsuitable for high data-rate applications.

The bit error rate (BER) of BPSK in AWGN can be calculated as:

[itex]P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)[itex]

which, since there is only one bit per symbol, is also the symbol error rate.

In the presence of an arbitrary phase-shift introduced by the communications channel, the demodulator is unable to tell which constellation point is which. As a result, the data is often differentially encoded prior to modulation.

Missing image
QPSK_Gray_Coded.png
Constellation diagram for QPSK with Gray coding.

Sometimes known as quaternary PSK or 4-PSK, 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 BER — twice the rate of BPSK. Analysis shows that this may be used either to double the data rate compared to a BPSK system while maintaining the bandwidth of the signal or to maintain the data-rate of BPSK but halve the bandwidth needed.

Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers with, say, the even bits (0,2,4,...) modulating the I-carrier and the odd bits (1,3,5,...) modulating the Q-carrier. BPSK may then be used on each carrier and, since they can be independently demodulated, the probability of bit-error is the same as for BPSK:

[itex]P_b = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)[itex].

However, with two bits per symbol, the symbol error rate is increased:

[itex]P_s = 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\left[1 - \frac{1}{2}Q\left(\sqrt{\frac{2E_b}{N_0}}\right)\right][itex].

As with BPSK, there are phase ambiguity problems at the receiver and differentially encoded QPSK is more normally used in practise.

### Offset QPSK (OQPSK)

Taking two bits at a time to construct a QPSK symbol can, as seen easily in the diagram, allow the phase of the signal to jump by as much as 180° at a time. This produces undesirably large amplitude fluctuations in the signal. By offsetting the timing of the odd and even bits by one bit-period (half a symbol-period), the two channels will never change at the same time. The constellation diagram shows that this will limit the phase-shift to no more than 90° at a time. This yields much lower amplitude fluctuation than non-offset QPSK and is often preferred as a result.

### [itex]\pi/4[itex]–QPSK

This final variant of QPSK uses two identical constellations which are rotated by 45° ([itex]\pi/4[itex] radians, hence the name) with respect to one another. Usually, either the even or odd data bits are used to select points from one of the constellations and the other bits 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 [itex]\pi/4[itex]–QPSK are between OQPSK and non-offset QPSK.

On the other hand, [itex]\pi/4[itex]–QPSK lends itself to easy demodulation and has been adopted for use in, for example, TDMA cellular telephone systems.

## Higher-order PSK

Missing image
8PSK_Gray_Coded.png
Constellation diagram for 8-PSK with Gray coding.

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 — this allows an equal number of bits-per-symbol.

For the general [itex]M[itex]-PSK there is no simple expression for the symbol-error probability if [itex]M>4[itex]. Unfortunately, it can only be obtained from:

[itex]

P_s = 1 - \int_{\frac{\pi}{M}}^{\frac{\pi}{M}}p_{\Theta_{r}}\left(\Theta_{r}\right)d\Theta_{r} [itex]

where

[itex]p_{\Theta_{r}}\left(\Theta_r\right) = \frac{1}{2\pi}e^{-2\gamma_{s}\sin^{2}\Theta_{r}}\int_{0}^{\infty}Ve^{-\left(V-\sqrt{4\gamma_{s}}\cos\Theta{r}\right)^{2}/2}[itex],
[itex]V = \sqrt{r_1^2 + r_2^2}[itex],
[itex]\Theta_r = \tan^{-1}\left(r_2/r_1\right)[itex],
[itex]\gamma_{s} = \frac{E_{s}}{N_{0}}[itex] and
[itex]r_1 \sim{} N\left(\sqrt{E_s},N_{0}/2\right)[itex] and [itex]r_2 \sim{} N\left(0,N_{0}/2\right)[itex] are jointly-Gaussian random variables.

This may be approximated for high [itex]M[itex] and high [itex]E_b/N_0[itex] by:

[itex]P_s \approx 2Q\left(\sqrt{2\gamma_s}\sin\frac{\pi}{M}\right)[itex].

The bit-error probability for [itex]M[itex]-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

[itex]P_b \approx \frac{1}{k}P_s[itex].

## Differential Encoding

As mentioned for BPSK and QPSK there is an ambiguity of phase if the constellation is rotated by some effect in the communications channel the signal passes through. 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. In differentially-encoded QPSK, 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 [itex]M[itex] 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.

Analysis shows that differential encoding approximately doubles the error rate compared to ordinary [itex]M[itex]-PSK but that this actually translates into needing only a small increase in [itex]E_b/N_0[itex].

### Example: Differentially encoded BPSK

Missing image
Differential_Codec.png
Differential encoding/decoding system diagram.

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

[itex]\,e_k = e_{k-1}\oplus{}b_k[itex]

So [itex]e_k[itex] only changes state (from binary '0' to binary '1' or from binary '1' to binary '0') if [itex]b_k[itex] 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 [itex]e_k=[itex]±1 and then the differential decoder reverses the encoding procedure and produces:

[itex]\,b_k = e_{k}\oplus{}e_{k-1}[itex] since binary subtraction is the same as binary addition.

Therefore, [itex]b_k=1[itex] if [itex]e_k[itex] and [itex]e_{k-1}[itex] differ and [itex]b_k=0[itex] if they are the same. Hence, if both [itex]e_k[itex] and [itex]e_{k-1}[itex] are inverted, [itex]b_k[itex] 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.

## Differential Phase-shift Keying (DPSK)

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 it is subtly different to 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.

We will not go into the mathematical detail here since it can become quite complicated, but a summary is given nevertheless:

Let the received symbol in the [itex]k^{\textrm{th}}[itex] symbol period be [itex]r_k[itex] and let its phase, [itex]\angle{}r_k = \phi_k[itex]. Then

[itex]r_{k}r_{k-1}^{*} = f\left(\phi_{k}-\phi_{k-1}\right)[itex] (* denotes complex conjugation),

a function of 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, but, in the case of DBPSK it is:

[itex]P_b = \frac{1}{2}e^{-E_b/N_0}[itex],

which, when numerically evaluated, is only slightly worse than ordinary BPSK, particulatly at higher [itex]E_b/N_0[itex] 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 fact, 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.

## References

These results can be found in any good communications textbook, but the notation used here has mainly (but not exclusively) been taken from:

• John G. Proakis, "Digital Communications, 3rd Edition", McGraw-Hill Book Co., 1995. ISBN 0071138145
• Leon W. Couch III, "Digital and Anlaog Communication Systems, 6th Edition", Prentice-Hall, Inc., 2001. ISBN 0130812234

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