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Phase­shift keyingFrom Wikipedia, the free encyclopedia

Phase­shift 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. PSKuses a finite number of phases, each assigned a unique pattern of binary digits. Usually, each phaseencodes an equal number of bits. Each pattern of bits forms the symbol that is represented by theparticular phase. The demodulator, which is designed specifically for the symbol­set used by themodulator, determines the phase of the received signal and maps it back to the symbol it represents, thusrecovering the original data. This requires the receiver to be able to compare the phase of the receivedsignal to a reference signal — such a system is termed coherent (and referred to as CPSK).

Alternatively, instead of operating with respect to a constant reference wave, the broadcast can operatewith respect to itself. Changes in phase of a single broadcast waveform can be considered the significantitems. In this system, the demodulator determines the changes in the phase of the received signal ratherthan the phase (relative to a reference wave) itself. Since this scheme depends on the difference betweensuccessive phases, it is termed differential phase­shift keying (DPSK). DPSK can be significantlysimpler to implement than ordinary PSK since there is no need for the demodulator to have a copy of thereference signal to determine the exact phase of the received signal (it is a non­coherent scheme). Inexchange, it produces more erroneous demodulation.

Contents

1 Introduction1.1 Definitions

2 Applications3 Binary phase­shift keying (BPSK)

3.1 Implementation3.2 Bit error rate

4 Quadrature phase­shift keying (QPSK)4.1 Implementation4.2 Bit error rate4.3 Variants

4.3.1 Offset QPSK (OQPSK)4.3.2 π /4–QPSK4.3.3 SOQPSK4.3.4 DPQPSK

5 Higher­order PSK5.1 Bit error rate

6 Differential phase­shift keying (DPSK)6.1 Differential encoding

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6.2 Demodulation6.3 Example: Differentially encoded BPSK

7 Channel capacity8 See also9 Notes10 References

Introduction

There are three major classes of digital modulation techniques used for transmission of digitallyrepresented data:

Amplitude­shift keying (ASK)Frequency­shift keying (FSK)Phase­shift keying (PSK)

All convey data by changing some aspect of a base signal, the carrier wave (usually a sinusoid), inresponse to a data signal. In the case of PSK, the phase is changed to represent the data signal. There aretwo 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 musthave a reference signal to compare the received signal's phase against; orBy viewing the change in the phase as conveying information — differential schemes, some ofwhich do not need a reference carrier (to a certain extent).

A convenient method to represent PSK schemes is on a constellation diagram. This shows the points inthe complex plane where, in this context, the real and imaginary axes are termed the in­phase andquadrature axes respectively due to their 90° separation. Such a representation on perpendicular axeslends itself to straightforward implementation. The amplitude of each point along the in­phase axis isused to modulate a cosine (or sine) wave and the amplitude along the quadrature axis to modulate a sine(or cosine) wave. By convention, in­phase modulates cosine and quadrature modulates sine.

In PSK, the constellation points chosen are usually positioned with uniform angular spacing around acircle. This gives maximum phase­separation between adjacent points and thus the best immunity tocorruption. They are positioned on a circle so that they can all be transmitted with the same energy. Inthis way, the moduli of the complex numbers they represent will be the same and thus so will theamplitudes needed for the cosine and sine waves. Two common examples are "binary phase­shiftkeying" (BPSK) which uses two phases, and "quadrature phase­shift keying" (QPSK) which uses fourphases, 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 error­rates mathematically, some definitions will be needed:

= Energy­per­bit

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= Energy­per­symbol = with n bits per symbol = Bit duration = Symbol duration

= Noise power spectral density (W/Hz) = Probability of bit­error = Probability of symbol­error

will give the probability that a single sample taken from a random process with zero­mean andunit­variance Gaussian probability density function will be greater or equal to . It is a scaled form ofthe complementary Gaussian error function:

.

The error­rates quoted here are those in additive white Gaussian noise (AWGN). These error rates arelower than those computed in fading channels, hence, are a good theoretical benchmark to compare with.

Applications

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

The wireless LAN standard, IEEE 802.11b­1999,[1][2] uses a variety of different PSKs depending on thedata rate required. At the basic rate of 1 Mbit/s, it uses DBPSK (differential BPSK). To provide theextended rate of 2 Mbit/s, DQPSK is used. In reaching 5.5 Mbit/s and the full rate of 11 Mbit/s, QPSK isemployed, but has to be coupled with complementary code keying. The higher­speed wireless LANstandard, IEEE 802.11g­2003,[1][3] has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and9 Mbit/s modes use OFDM modulation where each sub­carrier is BPSK modulated. The 12 and 18Mbit/s modes use OFDM with QPSK. The fastest four modes use OFDM with forms of quadratureamplitude modulation.

Because of its simplicity BPSK is appropriate for low­cost passive transmitters, and is used in RFIDstandards such as ISO/IEC 14443 which has been adopted for biometric passports, credit cards such asAmerican Express's ExpressPay, and many other applications.[4]

Bluetooth 2 will use ­DQPSK at its lower rate (2 Mbit/s) and 8­DPSK at its higher rate (3 Mbit/s)when the link between the two devices is sufficiently robust. Bluetooth 1 modulates with Gaussianminimum­shift keying, a binary scheme, so either modulation choice in version 2 will yield a higherdata­rate. A similar technology, IEEE 802.15.4 (the wireless standard used by ZigBee) also relies onPSK. IEEE 802.15.4 allows the use of two frequency bands: 868–915 MHz using BPSK and at 2.4 GHzusing OQPSK.

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

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Constellation diagram example forBPSK.

Included among the exceptions is HughesNet satellite ISP. For example, the model HN7000S modem(on KU­band satcom) uses 8­PSK modulation.

Binary phase­shift keying (BPSK)

BPSK (also sometimes called PRK, phase reversal keying, or 2PSK)is the simplest form of phase shift keying (PSK). It uses two phaseswhich are separated by 180° and so can also be termed 2­PSK. Itdoes not particularly matter exactly where the constellation pointsare positioned, and in this figure they are shown on the real axis, at0° and 180°. This modulation is the most robust of all the PSKssince it takes the highest level of noise or distortion to make thedemodulator reach an incorrect decision. It is, however, only able tomodulate at 1 bit/symbol (as seen in the figure) and so is unsuitablefor high data­rate applications.

In the presence of an arbitrary phase­shift introduced by thecommunications channel, the demodulator is unable to tell whichconstellation point is which. As a result, the data is oftendifferentially encoded prior to modulation.

BPSK is functionally equivalent to 2­QAM modulation.

Implementation

The general form for BPSK follows the equation:

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

for binary "0"

for binary "1"

where fc is the frequency of the carrier­wave.

Hence, the signal­space can be represented by the single basis function

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

course, arbitrary.

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Constellation diagram for QPSKwith Gray coding. Each adjacentsymbol only differs by one bit.

This use of this basis function is shown at the end of the next section in a signal timing diagram. Thetopmost signal is a BPSK­modulated cosine wave that the BPSK modulator would produce. The bit­stream that causes this output is shown above the signal (the other parts of this figure are relevant only toQPSK).

Bit error rate

The bit error rate (BER) of BPSK in AWGN can be calculated as:[5]

or

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

Quadrature phase­shift keying (QPSK)

Sometimes this is known as quadriphase PSK, 4­PSK, or 4­QAM.(Although the root concepts of QPSK and 4­QAM are different, theresulting modulated radio waves are exactly the same.) QPSK usesfour points on the constellation diagram, equispaced around a circle.With four phases, QPSK can encode two bits per symbol, shown inthe 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 todouble the data rate compared with a BPSK system whilemaintaining the same bandwidth of the signal, or to maintain thedata­rate of BPSK but halving the bandwidth needed. In this lattercase, the BER of QPSK is exactly the same as the BER of BPSK ­and deciding differently is a common confusion when consideringor describing QPSK. The transmitted carrier can undergo numbersof phase changes.

Given that radio communication channels are allocated by agencies such as the Federal CommunicationCommission giving a prescribed (maximum) bandwidth, the advantage of QPSK over BPSK becomesevident: 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 thanthe 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 encodedQPSK is often used in practice.

Implementation

The implementation of QPSK is more general than that of BPSK and also indicates the implementationof higher­order PSK. Writing the symbols in the constellation diagram in terms of the sine and cosinewaves used to transmit them:

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This yields the four phases π/4, 3π/4, 5π/4 and 7π/4 as needed.

This results in a two­dimensional signal space with unit basis functions

The first basis function is used as the in­phase component of the signal and the second as the quadraturecomponent of the signal.

Hence, the signal constellation consists of the signal­space 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 twoindependent BPSK signals. Note that the signal­space 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 thetransmitter and receiver structure are shown below.

Conceptual transmitter structure for QPSK. The binary data stream is split into the in­phase andquadrature­phase 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 theresulting signal is the QPSK signal. Note the use of polar non­return­to­zero encoding. These encoderscan be placed before for binary data source, but have been placed after to illustrate the conceptualdifference between digital and analog signals involved with digital modulation.

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

Bit error rate

Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independentlymodulated quadrature carriers. With this interpretation, the even (or odd) bits are used to modulate thein­phase component of the carrier, while the odd (or even) bits are used to modulate the quadrature­phase component of the carrier. BPSK is used on both carriers and they can be independentlydemodulated.

As a result, the probability of bit­error for QPSK is the same as for BPSK:

However, in order to achieve the same bit­error probability as BPSK, QPSK uses twice the power (sincetwo bits are transmitted simultaneously).

The symbol error rate is given by:

.

If the signal­to­noise ratio is high (as is necessary for practical QPSK systems) the probability of symbolerror may be approximated:

The modulated signal is shown below for a short segment of a random binary data­stream. The twocarrier waves are a cosine wave and a sine wave, as indicated by the signal­space analysis above. Here,the odd­numbered bits have been assigned to the in­phase component and the even­numbered bits to thequadrature component (taking the first bit as number 1). The total signal — the sum of the twocomponents — is shown at the bottom. Jumps in phase can be seen as the PSK changes the phase oneach component at the start of each bit­period. The topmost waveform alone matches the descriptiongiven for BPSK above.

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Signal doesn't cross zero, becauseonly one bit of the symbol is changedat a time

Timing diagram for QPSK. The binary data stream is shown beneath the time axis. The two signalcomponents 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 bit­period boundaries.

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 in­phase component: 1 1 0 0 0 1 1 0The even bits, highlighted here, contribute to the quadrature­phase component: 1 1 0 0 0 1 1 0

Variants

Offset QPSK (OQPSK)

Offset quadrature phase­shift keying (OQPSK) is a variant ofphase­shift keying modulation using 4 different values of thephase to transmit. It is sometimes called Staggered quadraturephase­shift keying (SQPSK).

Taking four values of the phase (two bits) at a time to construct aQPSK symbol can allow the phase of the signal to jump by asmuch as 180° at a time. When the signal is low­pass filtered (asis typical in a transmitter), these phase­shifts result in largeamplitude fluctuations, an undesirable quality in communicationsystems. By offsetting the timing of the odd and even bits by onebit­period, or half a symbol­period, the in­phase and quadraturecomponents will never change at the same time. In theconstellation diagram shown on the right, it can be seen that thiswill limit the phase­shift to no more than 90° at a time. Thisyields much lower amplitude fluctuations than non­offset QPSKand is sometimes preferred in practice.

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

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Difference of the phase betweenQPSK and OQPSK

The modulated signal is shown below for a short segment of a random binary data­stream. Note the halfsymbol­period offset between the two component waves. The sudden phase­shifts occur about twice asoften 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 offset­QPSK. The binary data stream is shown beneath the time axis. The twosignal components with their bit assignments are shown the top and the total, combined signal at thebottom. Note the half­period offset between the two signal components.

π /4–QPSK

This variant of QPSK uses two identical constellations which are rotated by 45° ( radians, hence thename) with respect to one another. Usually, either the even or odd symbols are used to select points fromone of the constellations and the other symbols select points from the other constellation. This alsoreduces the phase­shifts from a maximum of 180°, but only to a maximum of 135° and so the amplitudefluctuations of –QPSK are between OQPSK and non­offset QPSK.

One property this modulation scheme possesses is that if the modulated signal is represented in thecomplex domain, it does not have any paths through the origin. In other words, the signal does not passthrough the origin. This lowers the dynamical range of fluctuations in the signal which is desirable whenengineering communications signals.

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Dual constellation diagram for π/4­QPSK. This shows the two separateconstellations with identical Graycoding but rotated by 45° withrespect to each other.

On the other hand, –QPSK lends itself to easy demodulation and has been adopted for use in, forexample, TDMA cellular telephone systems.

The modulated signal is shown below for a short segment of a random binary data­stream. Theconstruction is the same as above for ordinary QPSK. Successive symbols are taken from the twoconstellations shown in the diagram. Thus, the first symbol (1 1) is taken from the 'blue' constellationand the second symbol (0 0) is taken from the 'green' constellation. Note that magnitudes of the twocomponent waves change as they switch between constellations, but the total signal's magnitude remainsconstant (constant envelope). The phase­shifts are between those of the two previous timing­diagrams.

Timing diagram for π/4­QPSK. The binary data stream is shown beneath the time axis. The two signalcomponents 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.

SOQPSK

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Constellation diagram for 8­PSKwith Gray coding.

The license­free shaped­offset QPSK (SOQPSK) is interoperable with Feher­patented QPSK(FQPSK), in the sense that an integrate­and­dump offset QPSK detector produces the same output nomatter which kind of transmitter is used.[6]

These modulations carefully shape the I and Q waveforms such that they change very smoothly, and thesignal stays constant­amplitude even during signal transitions. (Rather than traveling instantly from onesymbol to another, or even linearly, it travels smoothly around the constant­amplitude circle from onesymbol to the next.)

The standard description of SOQPSK­TG involves ternary symbols.

DPQPSK

Dual­polarization quadrature phase shift keying (DPQPSK) or dual­polarization QPSK ­ involvesthe polarization multiplexing of two different QPSK signals, thus improving the spectral efficiency by afactor of 2. This is a cost­effective alternative, to utilizing 16­PSK instead of QPSK to double thespectral efficiency.

Higher­order PSK

Any number of phases may be used to construct a PSK constellationbut 8­PSK is usually the highest order PSK constellation deployed.With more than 8 phases, the error­rate becomes too high and thereare better, though more complex, modulations available such asquadrature amplitude modulation (QAM). Although any number ofphases may be used, the fact that the constellation must usually dealwith binary data means that the number of symbols is usually apower of 2 to allow an integer number of bits per symbol.

Bit error rate

For the general ­PSK there is no simple expression for thesymbol­error probability if . Unfortunately, it can only beobtained from:

where

,

,

,

and

and are jointly Gaussian random variables.

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Bit­error rate curves for BPSK, QPSK, 8­PSKand 16­PSK, AWGN channel.

This may be approximated for high and high by:

.

The bit­error probability for ­PSK can only bedetermined exactly once the bit­mapping is known.However, when Gray coding is used, the most probableerror from one symbol to the next produces only asingle bit­error and

.

(Using Gray coding allows us to approximate the Leedistance of the errors as the Hamming distance of the errors in the decoded bitstream, which is easier toimplement in hardware.)

The graph on the left compares the bit­error rates of BPSK, QPSK (which are the same, as noted above),8­PSK and 16­PSK. It is seen that higher­order modulations exhibit higher error­rates; in exchangehowever they deliver a higher raw data­rate.

Bounds on the error rates of various digital modulation schemes can be computed with application of theunion bound to the signal constellation.

Differential phase­shift keying (DPSK)

Differential encoding

Differential phase shift keying (DPSK) is a common form of phase modulation that conveys data bychanging the phase of the carrier wave. As mentioned for BPSK and QPSK there is an ambiguity ofphase if the constellation is rotated by some effect in the communications channel through which thesignal 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 thecurrent phase and a binary '0' by adding 0° to the current phase. Another variant of DPSK is SymmetricDifferential Phase Shift keying, SDPSK, where encoding would be +90° for a '1' and −90° for a '0'.

In differentially encoded QPSK (DQPSK), the phase­shifts 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 non­differentialPSK but the phase ambiguities can be ignored. Thus, each received symbol is demodulated to one of the

points in the constellation and a comparator then computes the difference in phase between thisreceived signal and the preceding one. The difference encodes the data as described above. SymmetricDifferential 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, itis assumed that the signal starts with zero phase, and so there is a phase shift in both signals at .

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BER comparison between DBPSK, DQPSK andtheir non­differential forms using gray­codingand operating in white noise.

Timing diagram for DBPSK and DQPSK. The binary data stream is above the DBPSK signal. Theindividual bits of the DBPSK signal are grouped into pairs for the DQPSK signal, which only changesevery Ts = 2Tb.

Analysis shows that differential encoding approximately doubles the error rate compared to ordinary ­PSK but this may be overcome by only a small increase in . Furthermore, this analysis (and thegraphical results below) are based on a system in which the only corruption is additive white Gaussiannoise(AWGN). However, there will also be a physical channel between the transmitter and receiver inthe communication system. This channel will, in general, introduce an unknown phase­shift to the PSKsignal; in these cases the differential schemes can yield a better error­rate than the ordinary schemeswhich rely on precise phase information.

Demodulation

For a signal that has been differentially encoded, thereis an obvious alternative method of demodulation.Instead of demodulating as usual and ignoring carrier­phase ambiguity, the phase between two successivereceived symbols is compared and used to determinewhat the data must have been. When differentialencoding is used in this manner, the scheme is knownas differential phase­shift keying (DPSK). Note that thisis subtly different from just differentially encoded PSKsince, upon reception, the received symbols are notdecoded one­by­one to constellation points but areinstead compared directly to one another.

Call the received symbol in the th timeslot and let ithave phase . Assume without loss of generality thatthe phase of the carrier wave is zero. Denote theAWGN term as . Then

.

The decision variable for the th symbol and the th symbol is the phase difference between and. That is, if is projected onto , the decision is taken on the phase of the resultant complex

number:

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where superscript * denotes complex conjugation. In the absence of noise, the phase of this is , the phase­shift 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 values.

Using DPSK avoids the need for possibly complex carrier­recovery schemes to provide an accuratephase 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. Themodulation is a laser which emits a continuous wave, and a Mach­Zehnder modulator which receiveselectrical binary data. For the case of BPSK for example, the laser transmits the field unchanged forbinary '1', and with reverse polarity for '0'. The demodulator consists of a delay line interferometer whichdelays one bit, so two bits can be compared at one time. In further processing, a photodiode is used totransform the optical field into an electric current, so the information is changed back into its originalstate.

The bit­error rates of DBPSK and DQPSK are compared to their non­differential counterparts in thegraph to the right. The loss for using DBPSK is small enough compared to the complexity reduction thatit is often used in communications systems that would otherwise use BPSK. For DQPSK though, theloss in performance compared to ordinary QPSK is larger and the system designer must balance thisagainst the reduction in complexity.

Example: Differentially encoded BPSK

Differential encoding/decoding system diagram.

At the time­slot call the bit to be modulated , the differentially encoded bit and the resultingmodulated signal . Assume that the constellation diagram positions the symbols at ±1 (which isBPSK). The differential encoder produces:

where indicates binary or modulo­2 addition.

So only changes state (from binary '0' to binary '1' or from binary '1' to binary '0') if is a binary '1'.Otherwise it remains in its previous state. This is the description of differentially encoded BPSK givenabove.

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BER comparison between BPSK anddifferentially encoded BPSK with gray­codingoperating in white noise.

Given a fixed bandwidth, channel capacity vs. SNRfor some common modulation schemes

The received signal is demodulated to yield ±1 and then the differential decoder reverses theencoding procedure and produces:

since binary subtraction is the same as binary addition.

Therefore, if and differ and ifthey are the same. Hence, if both and areinverted, will still be decoded correctly. Thus, the180° phase ambiguity does not matter.

Differential schemes for other PSK modulations may bedevised along similar lines. The waveforms for DPSKare the same as for differentially encoded PSK givenabove since the only change between the two schemesis at the receiver.

The BER curve for this example is compared toordinary BPSK on the right. As mentioned above,whilst the error­rate is approximately doubled, theincrease needed in to overcome this is small.The increase in required to overcomedifferential modulation in coded systems, however, islarger ­ typically about 3 dB. The performance degradation is a result of noncoherent transmission ­ inthis case it refers to the fact that tracking of the phase is completely ignored.

Channel capacity

Like all M­ary modulation schemes with M = 2bsymbols, when given exclusive access to a fixedbandwidth, the channel capacity of any phase shiftkeying modulation scheme rises to a maximum of bbits per symbol as the signal­to­noise ratio increases.

See also

Differential codingModulation — for an overview of allmodulation schemesPhase modulation (PM) — the analogueequivalent of PSKPolar modulationPSK31PSK63Binary offset carrier modulation

Notes

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1. ^ a b IEEE Std 802.11­1999: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)Specifications (http://standards.ieee.org/getieee802/download/802.11­1999.pdf) — the overarching IEEE802.11 specification.

2. ^ IEEE Std 802.11b­1999 (R2003) (http://standards.ieee.org/getieee802/download/802.11b­1999.pdf) — theIEEE 802.11b specification.

3. ^ IEEE Std 802.11g­2003 (http://standards.ieee.org/getieee802/download/802.11g­2003.pdf) — the IEEE802.11g specification.

4. ^ Understanding the Requirements of ISO/IEC 14443 for Type B Proximity Contactless Identification Cards(http://www.atmel.com/dyn/resources/prod_documents/doc2056.pdf), Application Note, Rev. 2056B–RFID–11/05, 2005, ATMEL

5. ^ Communications Systems, H. Stern & S. Mahmoud, Pearson Prentice Hall, 2004, p2836. ^ Tom Nelson, Erik Perrins, and Michael Rice. "Common detectors for Tier 1 modulations"

(http://people.eecs.ku.edu/~esp/publications/c2005ItcCommon.pdf). T. Nelson, E. Perrins, M. Rice."Common detectors for shaped offset QPSK (SOQPSK) and Feher­patented QPSK (FQPSK)"(http://www.researchgate.net/publication/4213516_Common_detectors_for_shaped_offset_QPSK_(SOQPSK)_and_Feher­patented_QPSK_(FQPSK)) Nelson, T.; Perrins, E.; Rice, M. (2005). "Common detectors forshaped offset QPSK (SOQPSK) and Feher­patented QPSK (FQPSK)". GLOBECOM '05. IEEE GlobalTelecommunications Conference, 2005. pp. 5 pp. doi:10.1109/GLOCOM.2005.1578470(http://dx.doi.org/10.1109%2FGLOCOM.2005.1578470). ISBN 0­7803­9414­3. ISBN 0­7803­9414­3

References

The notation and theoretical results in this article are based on material presented in the followingsources:

Proakis, John G. (1995). Digital Communications. Singapore: McGraw Hill. ISBN 0­07­113814­5.Couch, Leon W. II (1997). Digital and Analog Communications. Upper Saddle River, NJ:Prentice­Hall. ISBN 0­13­081223­4.Haykin, Simon (1988). Digital Communications. Toronto, Canada: John Wiley & Sons. ISBN 0­471­62947­2.

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Categories: Quantized radio modulation modes Data transmission

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