Phase-shift keying (PSK) is a digital modulation
process which conveys data
by changing (modulating) the phase
of a constant frequency
(the carrier wave
). The modulation is accomplished by varying the sine
inputs at a precise time. It is widely used for wireless LAN
Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases, each assigned a unique pattern of binary digit
s. 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 (and referred to as CPSK).
CPSK requires a complicated demodulator, because it must extract the reference wave from the received signal and keep track of it, to compare each sample to. Alternatively, the phase shift of each symbol sent can be measured with respect to the phase of the previous symbol sent. Because the symbols are encoded in the difference in phase between successive samples, this is called differential phase-shift keying (DPSK). DPSK can be significantly simpler to implement than ordinary PSK, as it is a 'non-coherent' scheme, i.e. there is no need for the demodulator to keep track of a reference wave. A trade-off is that it has more demodulation errors.
There are three major classes of digital modulation
techniques used for transmission of digital
ly represented data:
* Amplitude-shift keying
* Frequency-shift keying
* 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''
of which 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 in the complex plane
where, in this context, the real
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
For determining error-rates mathematically, some definitions will be needed:
, energy per bit
, energy per symbol with ''n'' bits
, bit duration
, symbol duration
, noise power spectral density
, probability of symbol-error
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
. It is a scaled form of the complementary Gaussian error function
The error rates quoted here are those in additive white Gaussian noise
(AWGN). These error rates are lower than those computed in fading channel
s, hence, are a good theoretical benchmark to compare with.
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
[IEEE Std 802.11-1999: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications](_blank)
the overarching IEEE 802.11 specification.
[IEEE Std 802.11b-1999 (R2003)](_blank)
the IEEE 802.11b specification.
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
[IEEE Std 802.11g-2003](_blank)
the IEEE 802.11g specification.
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 passport
s, credit cards such as American Express
, and many other applications.
-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: 868–915MHz
with BPSK and at 2.4GHz
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.
Binary phase-shift keying (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
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 following signals:
for binary "0"
for binary "1"
where ''f'' is the frequency of the base band.
Hence, the signal space can be represented by the single basis function
where 1 is represented by
and 0 is represented by
. 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
Bit error rate
The bit error rate
(BER) of BPSK under additive white Gaussian noise
(AWGN) can be calculated as:
Since there is only one bit per symbol, this is also the symbol error rate.
Quadrature phase-shift keying (QPSK)
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.
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 receiver structure are shown below.
thumb|600px|center|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.
Probability of error
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:
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:
If the signal-to-noise ratio
is high (as is necessary for practical QPSK systems) the probability of symbol error may be approximated:
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.
The binary data that is conveyed by this waveform is: 11000110
* The odd bits, highlighted here, contribute to the in-phase component: 1010
* The even bits, highlighted here, contribute to the quadrature-phase component: 1001
Offset QPSK (OQPSK)
''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'').
Taking four values of the phase (two bit
s) 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.
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.
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
This variant of QPSK uses two identical constellations which are rotated by 45° (
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
-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,
-QPSK lends itself to easy demodulation and has been adopted for use in, for example, TDMA cellular telephone
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.
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.
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
. Unfortunately, it can only be obtained from
are each Gaussian random variable
This may be approximated for high
The bit-error probability for
-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
(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 modulation
s 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