Coherent differential phase-shift keying

Phase-shift keying (PSK) is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal (the carrier wave).

Any digital modulation scheme uses a finite number of distinct signals to represent digital data. In the case of PSK, a finite number of phases are used. Each of these phases is assigned a unique pattern of binary bits. 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.

Alternatively, instead of using the bit patterns to set the phase of the wave, it can instead be used to change it by a specified amount. The demodulator then determines the changes in the phase of the received signal rather than the phase itself. Since this scheme depends on the difference between successive phases, it is termed differential phase-shift keying (DPSK). DPSK can be significantly simpler to implement than ordinary PSK since there is no need for the demodulator to have a copy of the reference signal to determine the exact phase of the received signal (it is a non-coherent scheme). In exchange, it produces more erroneous demodulations. The exact requirements of the particular scenario under consideration determine which scheme is used.

Introduction

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 way to represent PSK schemes is on a constellation diagram. This shows the points in the Argand 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.

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 2.

Definitions

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

• E_b = Energy-per-bit
  
• E_s = Energy-per-symbol = kE_b with k bits per symbol
  
• T_b = Bit duration
  
• T_s = Symbol duration
  
• N₀ = 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 x is under the tail of the Gaussian probability density function towards positive infinity. It is a scaled form of the complementary Gaussian error function:

.

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

Applications

Owing to PSK's simplicity, particularly when compared with its competitor quadrature amplitude modulation, it is widely used in existing technologies.

The most popular wireless LAN standard, IEEE 802.11b^[1][2][3], uses a variety of different PSKs depending on the data-rate required. At the basic-rate of 1 Mbit/s, it uses DBPSK. To provide the extended-rate of 2 Mbit/s, DQPSK is used. In reaching 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is employed, but has to be coupled with complementary code keying. The higher-speed wireless LAN standard, IEEE 802.11g^[1][4] has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and 9 Mbit/s modes use BPSK. The 12 and 18 Mbit/s modes use QPSK. The fastest four modes use forms of quadrature amplitude modulation.

The recently-standardised Bluetooth 2^[5] will use π / 4-DQPSK at its lower rate (2 Mbit/s) and 8-DPSK at its higher rate (3 Mbit/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, ZigBee^[6] (also known as IEEE 802.15.4) also relies on PSK. ZigBee operates in two frequency bands: 868–915MHz where it employs BPSK and at 2.4GHz where it uses OQPSK.

Notably absent from these various schemes is 8-PSK. This is because its error-rate performance is close to that of 16-QAM — it is only about 0.5dB better — but its data rate is only three-quarters that of 16-QAM. Thus 8-PSK is often omitted from standards and, as seen above, schemes tend to 'jump' from QPSK to 16-QAM (8-QAM is possible but difficult to implement).

Binary Phase-shift Keying (BPSK)

BPSK is the simplest form of 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°. This modulation is the most robust of all the PSKs since it takes serious distortion to make the demodulator reach an incorrect decision. It is, however, 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:

Since there is only one bit per symbol, this 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.

Implementation

Binary data is often conveyed with the following signals:

for binary "0"

for binary "1"

where f_c is the frequency of the carrier-wave.

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

where 0 is represented by and 1 is represented by . This assignment is, of course, arbitrary.

The use of this basis function is shown at the end of the next section in a signal timing diagram. The topmost wave is PSK modulating a cosine wave, and is the signal that the BPSK modulator would produce. The bit-stream that causes this output is shown above the wave (the other parts of this figure are relevant only to QPSK).

Quadrature Phase-shift Keying (QPSK)

Sometimes known as quaternary or quadriphase 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 Coherent differential phase-shift keying 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:

.

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

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

As with BPSK, there are phase ambiguity problems at the receiver and differentially encoded QPSK is more normally 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:

.

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

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

.

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 structure are shown below.

QPSK signal in the time domain

The binary data that is conveyed by this waveform is: 1 1 0 0 0 1 1 0.

• The odd bits, highlighted here, contribute to the in-phase component: 1 1 0 0 0 1 1 0
  
• The even bits, highlighted here, contribute to the quadrature-phase component: 1 1 0 0 0 1 1 0

Offset QPSK (OQPSK)

Taking 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. This produces large amplitude fluctuations in the signal; 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, 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 the non-offset QPSK and is often preferred in practice.

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. 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.

Note that the signals depicted here are only for a particular implementation of OQPSK. If another set of basis functions are used, or if a different pattern of constellation points are used, the waveforms would look different. In this case, any abrupt transitions pass through 0. It is possible to design OQPSK systems so that these transitions do not pass through 0. Sometimes this is preferable to improve the signal reception at the receiver.

π / 4–QPSK

This final variant of QPSK uses two identical constellations which are rotated by 45° (π / 4 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 π / 4–QPSK are between OQPSK and non-offset QPSK.

On the other hand, π / 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. The phase-shifts are between those of the two previous timing-diagrams.

Higher-order PSK

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 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.

The modulated signal is shown below for both DBPSK and DQPSK as described above. 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 / N₀. Furthermore, this analysis (and the graphical results below) are based on a system in which the only corruption is additive white Gaussian noise. 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.

Example: Differentially encoded BPSK

At the k^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:

where 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 = ±1 and then the differential decoder reverses the encoding procedure and produces:

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 / N₀ to overcome this is small.

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 this 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.

Call the received symbol in the k^th timeslot r_k and let it have phase φ_k. Assume without loss of generality that the phase of the carrier wave is zero. Denote the AWGN term as n_k. Then

.

The decision variable for the k − 1^th symbol and the k^th 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:

where superfix * denotes complex conjugation. In the absence of noise, the phase of this is θ_k − θ_{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:

,

which, when numerically evaluated, is only slightly worse than ordinary BPSK, particularly at higher E_b / N₀ 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.

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.

See also

• Modulation — for an overview of all modulation schemes
  
• Phase modulation (PM) — the analogue equivalent of PSK

Notes

1. ^ IEEE Std 802.11-1999: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications — the overarching IEEE 802.11 specification.

2. ^ IEEE Std 802.11b-1999 (R2003) — the IEEE 802.11b specification.

3. ^ A description of the technical features of IEEE 802.11b.

4. ^ IEEE Std 802.11g-2003 — the IEEE 802.11g specification.

5. ^ A description of the technical features of Bluetooth 2 — the official standards documents are only available by subscription to the Bluetooth Special Interest Group.

6. ^ A description of the technical features of ZigBee — the official standards documents are only available by registering with the ZigBee alliance or the IEEE.

References

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

• Proakis, John G. (1995). Digital Communications, Singapore: McGraw Hill. ISBN 0071138145
  
• Couch, Leon W. III (1997). Digital and Analog Communications, Upper Saddle River, NJ: Prentice-Hall Inc. ISBN 0130812234
  
• Haykin, Simon (1988). Digital Communications, Toronto, Canada: John Wiley & Sons, Inc. ISBN 0471629472

Electronics Topics

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