2810:) 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.
132:
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290:
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2799:
4237:
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792:
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194:
36:
5233:
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77:
3596:{\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}}}
2230:
648:, 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).
2690:
2696:
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.
2110:
5003:
6429:
5223:
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
2817:
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 OQPSK (since the signals no longer change together), but they are less severe. In other
651:
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
6607:
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.
4167:
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".
2912:
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
2849:
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
2695:
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
2220:
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
802:
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
2986:
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
4171:
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
2998:
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
1436:
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
2944:
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
1720:
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
4379:
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
749:
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
2242:
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
2519:
2209:
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.
1687:. (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
2534:
1909:
1926:
3031:
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
1147:
2706:
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
4324:(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
4750:
2825:
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
2201:
6795:
4151:
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
6281:
3192:
1654:
2813:
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°.
3015:- 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.
976:
656:. 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.
3915:
5850:
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.
1740:
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:
4380:
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
765:. 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
3036:(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.
3684:
2333:
1243:
2319:
1574:
5139:
1335:
2987:
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 (
2685:{\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)}
2225:
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.
3753:
777:) 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
4573:
4240:
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
1747:
4123:
3208:
2338:
2105:{\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}}}
1931:
2995:
3998:
991:
5420:
2822:
5622:
4998:{\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}^{*}}
2703:
1713:
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.
1427:
700:), 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:
5208:
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
2124:
1381:
6706:
5056:
2233:
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.
6133:
1483:
6424:{\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}
5267:
6026:
6691:
5938:
required to overcome differential modulation in coded systems, however, is larger – typically about 3 dB. The performance degradation is a result of
5936:
5894:
5557:
5184:
4318:
3830:
3076:
1579:
5445:
4459:
636:
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
5939:
5357:
5814:
5754:
5721:
5661:
4742:
4682:
6951:
Hill, Terrance J. (2000). "A non-proprietary, constant envelope, variant of shaped offset QPSK (SOQPSK) for improved spectral containment and detection efficiency".
6568:
5854:
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
5189:
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.
4361:
3068:
2974:
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2910:
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847:
7236:
6269:
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7043:
Saeed, N.; Elzanaty, A.; Almorad, H.; Dahrouj, H.; Al-Naffouri, T.Y.; Alouini, M.S. (2020). "Cubesat communications: Recent advances and future challenges".
2880:
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.
769:
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" (
6651:
of its definition. The curves of mutual information saturate to the number of bits carried by each symbol in the limit of infinite signal to noise ratio
2213:
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.
2514:{\displaystyle {\begin{aligned}P_{s}&=1-\left(1-P_{b}\right)^{2}\\&=2Q\left({\sqrt {\frac {E_{s}}{N_{0}}}}\right)-\left^{2}.\end{aligned}}}
95:
3609:
572:
2324:
However, in order to achieve the same bit-error probability as BPSK, QPSK uses twice the power (since two bits are transmitted simultaneously).
6801:
4196:
then computes the difference in phase between this received signal and the preceding one. The difference encodes the data as described above.
7262:
7184:
7172:
7160:
6894:
Nelson, T.; Perrins, E.; Rice, M. (2005). "Common detectors for Tier 1 modulations" (Document). International
Foundation for Telemetering.
1154:
4014:
The graph on the right compares the bit-error rates of BPSK, QPSK (which are the same, as noted above), 8-PSK and 16-PSK. It is seen that
2252:
1507:
750:
along the quadrature axis to modulate a sine (or cosine) wave. By convention, in-phase modulates cosine and quadrature modulates sine.
1698:
The mathematical analysis shows that QPSK can be used either to double the data rate compared with a BPSK system while maintaining the
536:
5067:
2846:), in the sense that an integrate-and-dump offset QPSK detector produces the same output no matter which kind of transmitter is used.
1265:
7404:
3689:
2115:
The first basis function is used as the in-phase component of the signal and the second as the quadrature component of the signal.
5224:
in performance compared to ordinary QPSK is larger and the system designer must balance this against the reduction in complexity.
5520:
is a binary "1". Otherwise it remains in its previous state. This is the description of differentially encoded BPSK given above.
4498:
2787:) is a variant of phase-shift keying modulation using four different values of the phase to transmit. It is sometimes called
1904:{\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.}
7027:
6986:"Performance of coded offset quadrature phase-shift keying (OQPSK) and MIL-STD shaped OQPSK (SOQPSK) with iterative decoding"
6248:
6953:
MILCOM 2000 Proceedings. 21st
Century Military Communications. Architectures and Technologies for Information Superiority
4050:
565:
481:
49:
5359:. Assume that the constellation diagram positions the symbols at ±1 (which is BPSK). The differential encoder produces:
1142:{\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)}
807:
reaches an incorrect decision. That makes it the most robust of all the PSKs. It is, however, only able to modulate at 1
803:
they are shown on the real axis, at 0° and 180°. Therefore, it handles the highest noise level or distortion before the
4203:
The modulated signal is shown below for both DBPSK and DQPSK as described above. In the figure, it is assumed that the
3950:
7240:
7385:
7366:
7347:
7313:
7089:
6968:
6927:
6878:
4320:. Furthermore, this analysis (and the graphical results below) are based on a system in which the only corruption is
1717:
460:
255:
237:
175:
153:
113:
63:
2243:
quadrature-phase component of the carrier. BPSK is used on both carriers and they can be independently demodulated.
219:
146:
6813:
4375:
BER comparison between DBPSK, DQPSK and their non-differential forms using Gray coding and operating in white noise
354:
7105:
6582:, a binary scheme, so either modulation choice in version 2 will yield a higher data rate. A similar technology,
6510:
6452:
5365:
4129:
4041:
3033:
541:
455:
427:
394:
369:
321:
5565:
2196:{\displaystyle {\begin{pmatrix}\pm {\sqrt {\frac {E_{s}}{2}}}&\pm {\sqrt {\frac {E_{s}}{2}}}\end{pmatrix}}.}
7331:
The notation and theoretical results in this article are based on material presented in the following sources:
558:
204:
6790:{\displaystyle {\textrm {MI}}\simeq \log _{2}\left({\sqrt {{\frac {4\pi }{e}}{\frac {E_{s}}{N_{0}}}}}\right).}
1386:
6644:
6436:
4462:
4321:
2977:
1702:
1498:
1343:
6612:
standard requires support for both QPSK and 8PSK. The chipsets used in new satellite set top boxes, such as
6247:
will give the probability that a single sample taken from a random process with zero-mean and unit-variance
4200:(SDQPSK) is like DQPSK, but encoding is symmetric, using phase shift values of −135°, −45°, +45° and +135°.
5015:
2528:
is high (as is necessary for practical QPSK systems) the probability of symbol error may be approximated:
6694:
6101:
5061:
The probability of error for DPSK is difficult to calculate in general, but, in the case of DBPSK it is:
359:
5942: – in this case it refers to the fact that tracking of the phase is completely ignored.
5058:, the phase-shift between the two received signals which can be used to determine the data transmitted.
4021:
Bounds on the error rates of various digital modulation schemes can be computed with application of the
7409:
7198:"Understanding the Requirements of ISO/IEC 14443 for Type B Proximity Contactless Identification Cards"
5205:
1440:
511:
331:
7161:
IEEE Std 802.11-1999: Wireless LAN Medium Access
Control (MAC) and Physical Layer (PHY) Specifications
6700:
At intermediate values of signal to noise ratios the mutual information (MI) is well approximated by:
6693:. On the contrary, in the limit of small signal to noise ratios the mutual information approaches the
5243:
2854:: SOQPSK has the same signal constellation as QPSK, however the phase of SOQPSK is always stationary.
6486:
5209:
5144:
which, when numerically evaluated, is only slightly worse than ordinary BPSK, particularly at higher
384:
4258:
Analysis shows that differential encoding approximately doubles the error rate compared to ordinary
3187:{\displaystyle P_{s}=1-\int _{-\pi /M}^{\pi /M}p_{\theta _{r}}\left(\theta _{r}\right)d\theta _{r},}
1649:{\displaystyle P_{e}={\frac {1}{2}}\operatorname {erfc} \left({\sqrt {\frac {E_{b}}{N_{0}}}}\right)}
521:
404:
140:
7273:
5988:
3944:
is used, the most probable error from one symbol to the next produces only a single bit-error and
340:
6654:
6578:
Mbit/s) when the link between the two devices is sufficiently robust. Bluetooth 1 modulates with
6516:
Because of its simplicity, BPSK is appropriate for low-cost passive transmitters, and is used in
5899:
5857:
5526:
5147:
4281:
4015:
3793:
418:
297:
215:
55:
7197:
2776:
Signal doesn't pass through the origin, because only one bit of the symbol is changed at a time.
5428:
5193:
4437:
4153:
815:
682:
676:
531:
526:
364:
349:
157:
6466:, uses a variety of different PSKs depending on the data rate required. At the basic rate of 1
5326:
6985:
6648:
5786:
5726:
5693:
5633:
4714:
4654:
4384:
decoded one-by-one to constellation points but are instead compared directly to one another.
3766:
Bit-error rate curves for BPSK, QPSK, 8-PSK and 16-PSK, additive white
Gaussian noise channel
2850:
from one symbol to the next.) SOQPSK modulation can be represented as the hybrid of QPSK and
2525:
795:
754:
734:
516:
450:
6623:
such as the Bell 201, 208, and 209 and the CCITT V.26, V.27, V.29, V.32, and V.34 used PSK.
6545:
4338:
3047:
2951:
2916:
2887:
2206:
The factors of 1/2 indicate that the total power is split equally between the two carriers.
971:{\displaystyle s_{n}(t)={\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft+\pi (1-n)),\quad n=0,1.}
211:
91:
6579:
6191:
6155:
6069:
6037:
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5823:
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4468:
4461:. Assume without loss of generality that the phase of the carrier wave is zero. Denote the
4410:
2851:
374:
311:
306:
17:
6221:
8:
6818:
5009:
4581:
4210:
4146:
4033:
3910:{\displaystyle P_{s}\approx 2Q\left({\sqrt {2\gamma _{s}}}\sin {\frac {\pi }{M}}\right).}
811:
bit/symbol (as seen in the figure) and so is unsuitable for high data-rate applications.
778:
399:
268:
This article is about a type of digital modulation. For the PSK encryption protocol, see
6697:, which is the supremum among all possible choices of symbol statistical distributions.
5212:
which delays one bit, so two bits can be compared at one time. In further processing, a
3044:
For the general M-PSK there is no simple expression for the symbol-error probability if
822:) is unable to tell which constellation point is which. As a result, the data is often
7052:
6933:
6640:
6525:
6254:
5493:
only changes state (from binary "0" to binary "1" or from binary "1" to binary "0") if
5452:
4607:
4390:
4261:
4175:
3923:
3773:
2980:
2222:
665:
595:
432:
409:
5220:
into an electric current, so the information is changed back into its original state.
2976:-QPSK lends itself to easy demodulation and has been adopted for use in, for example,
640:. Usually, each phase encodes an equal number of bits. Each pattern of bits forms the
7381:
7362:
7343:
7337:
7309:
7085:
7023:
6964:
6923:
6874:
6490:
6463:
4018:
exhibit higher error-rates; in exchange however they deliver a higher raw data-rate.
2988:
379:
6937:
6800:
The mutual information of PSK over the AWGN channel is generally lower than that of
5462:
BER comparison between BPSK and differentially encoded BPSK operating in white noise
4332:
7140:
7062:
7015:
6956:
6915:
6895:
6839:
6829:
6529:
6139:
4156:
through which the signal passes. This problem can be overcome by using the data to
4011:
of the errors in the decoded bitstream, which is easier to implement in hardware.)
4008:
746:
599:
506:
501:
326:
316:
7218:
6908:"Common detectors for shaped offset QPSK (SOQPSK) and Feher-patented QPSK (FQPSK)"
652:
are encoded in the difference in phase between successive samples, this is called
6631:
6616:'s 7000 series support 8PSK and are backward compatible with the older standard.
5847:
will still be decoded correctly. Thus, the 180° phase ambiguity does not matter.
5448:
5201:
3940:-PSK can only be determined exactly once the bit-mapping is known. However, when
3756:
2884:
This variant of QPSK uses two identical constellations which are rotated by 45° (
2862:
1918:
441:
7019:
6919:
7066:
6960:
6509:
Mbit/s modes use OFDM with QPSK. The fastest four modes use OFDM with forms of
6440:
6272:
5559:
and then the differential decoder reverses the encoding procedure and produces
2858:
2740:
The even bits, highlighted here, contribute to the quadrature-phase component:
1692:
1494:
1257:
766:
712:
must have a reference signal to compare the received signal's phase against; or
269:
4328:
error-rate than the ordinary schemes which rely on precise phase information.
2994:
1716:
Given that radio communication channels are allocated by agencies such as the
7398:
6833:
6583:
6521:
5950:
For determining error-rates mathematically, some definitions will be needed:
5217:
2837:
738:
705:
603:
7131:
Stüber, G.L. (August 1988). "Soft
Decision Direct-Sequence DPSK Receivers".
2246:
As a result, the probability of bit-error for QPSK is the same as for BPSK:
1659:
Since there is only one bit per symbol, this is also the symbol error rate.
289:
7301:
6459:
6136:
5458:
4236:
4004:
3679:{\displaystyle r_{1}\sim N\left({\sqrt {E_{s}}},{\frac {1}{2}}N_{0}\right)}
2991:). The phase-shifts are between those of the two previous timing-diagrams.
1728:
As with BPSK, there are phase ambiguity problems at the receiving end, and
693:
669:
637:
622:
610:
546:
486:
465:
6451:
Owing to PSK's simplicity, particularly when compared with its competitor
4036:
of M-PSK modulation schemes increases with increasing of modulation order
2876:
2772:
1709:
but halving the bandwidth needed. In this latter case, the BER of QPSK is
6180:
6094:
4022:
3941:
2818:
words, the magnitude of jumps is smaller in OQPSK when compared to QPSK.
1722:
1695:(BER) – sometimes misperceived as twice the BER of BPSK.
1688:
1672:
982:
819:
804:
742:
709:
645:
641:
618:
7109:
6899:
6626:
6473:/s, it uses DBPSK (differential BPSK). To provide the extended rate of 2
2861:. SOQPSK is one of the most spread modulation schemes in application to
6823:
6533:
5213:
4193:
2821:
281:
7106:"Link Budget Analysis: Digital Modulation, Part 3 (www.AtlantaRF.com)"
4744:, the decision is taken on the phase of the resultant complex number:
2719:
The odd bits, highlighted here, contribute to the in-phase component:
2702:
2118:
Hence, the signal constellation consists of the signal-space 4 points
7144:
6539:
5232:
1238:{\displaystyle s_{1}(t)={\sqrt {\frac {2E_{b}}{T_{b}}}}\cos(2\pi ft)}
630:
614:
607:
491:
6590:) also relies on PSK using two frequency bands: 868 MHz and 915
2216:
222:. Statements consisting only of original research should be removed.
7057:
6613:
6062:
2798:
2314:{\displaystyle P_{b}=Q\left({\sqrt {\frac {2E_{b}}{N_{0}}}}\right)}
1569:{\displaystyle P_{b}=Q\left({\sqrt {\frac {2E_{b}}{N_{0}}}}\right)}
697:
278:
6505:
modulation where each sub-carrier is BPSK modulated. The 12 and 18
4371:
3023:
1667:
791:
3762:
5134:{\displaystyle P_{b}={\frac {1}{2}}e^{-{\frac {E_{b}}{N_{0}}}},}
2229:
1914:
This yields the four phases π/4, 3π/4, 5π/4 and 7π/4 as needed.
1330:{\displaystyle \phi (t)={\sqrt {\frac {2}{T_{b}}}}\cos(2\pi ft)}
6620:
6609:
6587:
5200:
in a differential way. The modulation is a laser which emits a
762:
729:
of which do not need a reference carrier (to a certain extent).
726:
7163: – the overarching IEEE 802.11 specification.
814:
In the presence of an arbitrary phase-shift introduced by the
773:) which uses two phases, and "quadrature phase-shift keying" (
692:
All convey data by changing some aspect of a base signal, the
6912:
6849:
6844:
5197:
3748:{\displaystyle r_{2}\sim N\left(0,{\frac {1}{2}}N_{0}\right)}
830:
758:
496:
7010:
Sahin, C.; Perrins, E. (2011). "The capacity of SOQPSK-TG".
6826: – for an overview of all modulation schemes
7042:
6517:
6502:
6470:
6443:, hence, are a good theoretical benchmark to compare with.
6439:(AWGN). These error rates are lower than those computed in
6143:
2221:
resulting signal is the QPSK signal. Note the use of polar
626:
4568:{\displaystyle r_{k}={\sqrt {E_{s}}}e^{j\phi _{k}}+n_{k}.}
4278:-PSK but this may be overcome by only a small increase in
6601:
6594:
5981:
5627:
since binary subtraction is the same as binary addition.
4135:
2807:
1917:
This results in a two-dimensional signal space with unit
1684:
1256:
Hence, the signal space can be represented by the single
774:
719:
in the phase as conveying information –
6493:, has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54
6147:
5227:
1662:
981:
This yields two phases, 0 and π. In the specific form,
708:
itself as conveying the information, in which case the
2711:
The binary data that is conveyed by this waveform is:
2133:
6709:
6657:
6627:
Mutual information with additive white
Gaussian noise
6548:
6485:
Mbit/s, QPSK is employed, but has to be coupled with
6284:
6257:
6224:
6194:
6158:
6104:
6072:
6040:
5991:
5959:
5902:
5860:
5826:
5789:
5762:
5729:
5696:
5669:
5636:
5568:
5529:
5499:
5472:
5431:
5368:
5329:
5302:
5275:
5246:
5150:
5070:
5018:
4753:
4717:
4690:
4657:
4630:
4610:
4584:
4501:
4471:
4440:
4413:
4393:
4341:
4284:
4264:
4213:
4178:
4053:
3953:
3926:
3841:
3796:
3776:
3692:
3612:
3206:
3079:
3050:
2954:
2919:
2890:
2537:
2336:
2255:
2127:
1929:
1750:
1582:
1510:
1443:
1389:
1346:
1268:
1157:
994:
850:
770:
733:
A convenient method to represent PSK schemes is on a
4331:
One of the most popular applications of DPSK is the
4198:
Symmetric differential quadrature phase shift keying
2842:(SOQPSK) is interoperable with Feher-patented QPSK (
7012:
4207:, and so there is a phase shift in both signals at
4118:{\displaystyle \rho ={\frac {\log _{2}M}{2}}\quad }
2945:desirable when engineering communications signals.
784:
86:
may be too technical for most readers to understand
7187: – the IEEE 802.11g specification.
7175: – the IEEE 802.11b specification.
6789:
6685:
6562:
6423:
6263:
6239:
6207:
6171:
6127:
6085:
6053:
6020:
5972:
5930:
5888:
5839:
5808:
5775:
5748:
5715:
5682:
5655:
5616:
5551:
5512:
5485:
5439:
5414:
5351:
5315:
5288:
5261:
5178:
5133:
5050:
4997:
4736:
4703:
4676:
4643:
4616:
4596:
4567:
4484:
4453:
4426:
4399:
4355:
4312:
4270:
4225:
4184:
4117:
3992:
3932:
3909:
3824:
3782:
3747:
3678:
3595:
3186:
3062:
2968:
2933:
2904:
2684:
2513:
2313:
2195:
2104:
1903:
1725:technology, the penalty in cost is very moderate.
1648:
1568:
1477:
1421:
1375:
1329:
1237:
1141:
970:
6905:
6893:
3993:{\displaystyle P_{b}\approx {\frac {1}{k}}P_{s}.}
644:that is represented by the particular phase. The
7396:
5196:, the data can be modulated onto the phase of a
5012:. In the absence of noise, the phase of this is
4003:(Using Gray coding allows us to approximate the
3027:Constellation diagram for 8-PSK with Gray coding
1432:This use of this basis function is shown at the
841:The general form for BPSK follows the equation:
621:inputs at a precise time. It is widely used for
613:. The modulation is accomplished by varying the
6635:Mutual information of PSK over the AWGN channel
3009:Dual-polarization quadrature phase shift keying
2857:The standard description of SOQPSK-TG involves
1675:. Each adjacent symbol only differs by one bit.
7237:"Low-Cost Satellite Set-top Box SoC — BCM7325"
6455:, it is widely used in existing technologies.
3070:. Unfortunately, it can only be obtained from
2802:Difference of the phase between QPSK and OQPSK
985:is often conveyed with the following signals:
7306:Principles and Practice of Information Theory
7296:
7294:
5236:Differential encoding/decoding system diagram
2941:-QPSK are between OQPSK and non-offset QPSK.
566:
7272:. Black Box Network Services. Archived from
7009:
6868:
5896:to overcome this is small. The increase in
5523:The received signal is demodulated to yield
7156:
7154:
7045:IEEE Communications Surveys & Tutorials
5415:{\displaystyle \,e_{k}=e_{k-1}\oplus b_{k}}
757:chosen are usually positioned with uniform
64:Learn how and when to remove these messages
7291:
6906:Nelson, T.; Perrins, E.; Rice, M. (2005).
6489:. The higher-speed wireless LAN standard,
5617:{\displaystyle b_{k}=e_{k}\oplus e_{k-1},}
2913:135° and so the amplitude fluctuations of
1729:
823:
720:
573:
559:
7133:IEEE Transactions on Vehicular Technology
7056:
6435:The error rates quoted here are those in
6360:
5369:
3408:
256:Learn how and when to remove this message
238:Learn how and when to remove this message
176:Learn how and when to remove this message
114:Learn how and when to remove this message
98:, without removing the technical details.
7151:
6630:
6574:Mbit/s) and 8-DPSK at its higher rate (3
6215:, probability of symbol-error
5457:
5231:
4370:
4235:
4140:
3761:
3022:
2993:
2875:
2820:
2797:
2771:
2701:
2228:
2215:
1666:
1422:{\displaystyle -{\sqrt {E_{b}}}\phi (t)}
790:
139:This article includes a list of general
7335:
6983:
6955:. Vol. 1. IEEE. pp. 347–352.
5269:time-slot call the bit to be modulated
4624:symbol is the phase difference between
2789:staggered quadrature phase-shift keying
2237:
1437:the high frequency band by multiplying
1376:{\displaystyle {\sqrt {E_{b}}}\phi (t)}
14:
7397:
7375:
7300:
7178:
7166:
7130:
7079:
6873:. Pearson Prentice Hall. p. 283.
6477:Mbit/s, DQPSK is used. In reaching 5.5
4136:Differential phase-shift keying (DPSK)
4028:
2767:
654:differential phase-shift keying (DPSK)
7356:
7207:. ATMEL. 2005. Rev. 2056B–RFID–11/05.
6619:Historically, voice-band synchronous
6273:complementary Gaussian error function
6249:Gaussian probability density function
5756:if they are the same. Hence, if both
5051:{\displaystyle \phi _{k}-\phi _{k-1}}
4128:The same relationship holds true for
2806:Taking four values of the phase (two
96:make it understandable to non-experts
7219:"How Communications Satellites Work"
6950:
5228:Example: Differentially-encoded BPSK
4363:-DQPSK and 8-DPSK were implemented.
2781:Offset quadrature phase-shift keying
1671:Constellation diagram for QPSK with
1663:Quadrature phase-shift keying (QPSK)
668:techniques used for transmission of
187:
125:
70:
29:
6128:{\displaystyle {\frac {1}{2}}N_{0}}
5323:and the resulting modulated signal
3039:
3018:
2327:The symbol error rate is given by:
1735:
1253:is the frequency of the base band.
829:BPSK is functionally equivalent to
24:
6325:
4192:points in the constellation and a
4034:Bandwidth (or spectral) efficiency
3770:This may be approximated for high
3324:
145:it lacks sufficient corresponding
25:
7421:
7359:Digital and Analog Communications
5296:, the differentially encoded bit
1718:Federal Communications Commission
1488:
1478:{\displaystyle \cos(2\pi f_{c}t)}
836:
664:There are three major classes of
45:This article has multiple issues.
7405:Quantized radio modulation modes
6993:Interplanetary Network Prog. Rep
6984:Li, Lifang; Simon, M.K. (2004).
6814:Binary offset carrier modulation
5262:{\displaystyle k^{\textrm {th}}}
4387:Call the received symbol in the
1732:QPSK is often used in practice.
1429:. This assignment is arbitrary.
785:Binary phase-shift keying (BPSK)
288:
192:
130:
75:
34:
7255:
7229:
7211:
7190:
6869:Stern, H.; Mahmoud, S. (2004).
6586:(the wireless standard used by
6536:, and many other applications.
6511:quadrature amplitude modulation
6453:quadrature amplitude modulation
6446:
4366:
4085:
3034:quadrature amplitude modulation
1873:
952:
737:. This shows the points in the
659:
53:or discuss these issues on the
7124:
7098:
7073:
7036:
7003:
6977:
6944:
6887:
6862:
6481:Mbit/s and the full rate of 11
6294:
6288:
6234:
6228:
5945:
5346:
5340:
4578:The decision variable for the
4112:
4086:
3920:The bit-error probability for
2034:
2028:
1950:
1944:
1852:
1837:
1767:
1761:
1707:maintain the data-rate of BPSK
1472:
1450:
1416:
1410:
1370:
1364:
1324:
1309:
1278:
1272:
1232:
1217:
1174:
1168:
1136:
1121:
1075:
1054:
1011:
1005:
946:
943:
931:
910:
867:
861:
13:
1:
7326:
7173:IEEE Std 802.11b-1999 (R2003)
6580:Gaussian minimum-shift keying
6437:additive white Gaussian noise
6271:. It is a scaled form of the
6028:, energy per symbol with
4463:additive white Gaussian noise
4322:additive white Gaussian noise
4205:signal starts with zero phase
4025:to the signal constellation.
1501:(AWGN) can be calculated as:
1499:additive white Gaussian noise
818:, the demodulator (see, e.g.
602:by changing (modulating) the
6251:will be greater or equal to
6021:{\displaystyle E_{s}=nE_{b}}
5008:where superscript * denotes
741:where, in this context, the
7:
7020:10.1109/MILCOM.2011.6127730
6920:10.1109/GLOCOM.2005.1578470
6832:(PM) – the
6807:
6686:{\displaystyle E_{s}/N_{0}}
6643:of PSK can be evaluated in
6570:-DQPSK at its lower rate (2
6524:which has been adopted for
5931:{\displaystyle E_{b}/N_{0}}
5889:{\displaystyle E_{b}/N_{0}}
5552:{\displaystyle e_{k}=\pm 1}
5179:{\displaystyle E_{b}/N_{0}}
4313:{\displaystyle E_{b}/N_{0}}
3825:{\displaystyle E_{b}/N_{0}}
2868:
2762:
1721:BPSK. However, with modern
1679:Sometimes this is known as
1433:
218:the claims made and adding
10:
7426:
7357:Couch, Leon W. II (1997).
7239:. Broadcom. Archived from
7067:10.1109/COMST.2020.2990499
7014:. IEEE. pp. 555–560.
6961:10.1109/MILCOM.2000.904973
4144:
2865:satellite communications.
1340:where 1 is represented by
482:Capacity-approaching codes
267:
7336:Proakis, John G. (1995).
7263:"Local and Remote Modems"
6487:complementary code keying
5440:{\displaystyle \oplus {}}
5216:is used to transform the
5210:delay line interferometer
4454:{\displaystyle \phi _{k}}
3003:
2830:
6855:
5980:, energy per
5940:noncoherent transmission
5352:{\displaystyle m_{k}(t)}
4016:higher-order modulations
1383:and 0 is represented by
688:Phase-shift keying (PSK)
6645:additive Gaussian noise
6528:, credit cards such as
5809:{\displaystyle e_{k-1}}
5749:{\displaystyle b_{k}=0}
5716:{\displaystyle e_{k-1}}
5656:{\displaystyle b_{k}=1}
4737:{\displaystyle r_{k-1}}
4677:{\displaystyle r_{k-1}}
1434:end of the next section
419:Hierarchical modulation
160:more precise citations.
7378:Digital Communications
7376:Haykin, Simon (1988).
7339:Digital Communications
7084:. Wiley. p. 368.
6871:Communications Systems
6791:
6687:
6636:
6564:
6563:{\displaystyle \pi /4}
6425:
6265:
6241:
6209:
6173:
6140:power spectral density
6129:
6087:
6055:
6022:
5974:
5932:
5890:
5841:
5810:
5777:
5750:
5717:
5684:
5657:
5618:
5553:
5514:
5487:
5463:
5441:
5416:
5353:
5317:
5290:
5263:
5237:
5206:Mach–Zehnder modulator
5194:optical communications
5180:
5135:
5052:
4999:
4738:
4705:
4678:
4645:
4618:
4598:
4569:
4486:
4455:
4434:and let it have phase
4428:
4401:
4376:
4357:
4356:{\displaystyle \pi /4}
4314:
4272:
4255:
4227:
4186:
4154:communications channel
4119:
4040:(unlike, for example,
3994:
3934:
3911:
3826:
3784:
3767:
3749:
3680:
3597:
3188:
3064:
3063:{\displaystyle M>4}
3028:
3013:dual-polarization QPSK
3000:
2970:
2969:{\displaystyle \pi /4}
2935:
2934:{\displaystyle \pi /4}
2906:
2905:{\displaystyle \pi /4}
2881:
2827:
2803:
2777:
2708:
2686:
2515:
2315:
2234:
2226:
2197:
2106:
1905:
1730:differentially encoded
1676:
1650:
1570:
1479:
1423:
1377:
1331:
1239:
1143:
972:
824:differentially encoded
816:communications channel
799:
683:Frequency-shift keying
677:Amplitude-shift keying
598:process which conveys
7243:on September 15, 2015
7185:IEEE Std 802.11g-2003
7082:Communication Systems
6792:
6695:AWGN channel capacity
6688:
6649:numerical integration
6634:
6565:
6426:
6266:
6242:
6210:
6208:{\displaystyle P_{s}}
6174:
6172:{\displaystyle P_{b}}
6130:
6088:
6086:{\displaystyle T_{s}}
6056:
6054:{\displaystyle T_{b}}
6023:
5975:
5973:{\displaystyle E_{b}}
5933:
5891:
5842:
5840:{\displaystyle b_{k}}
5811:
5778:
5776:{\displaystyle e_{k}}
5751:
5718:
5685:
5683:{\displaystyle e_{k}}
5658:
5619:
5554:
5515:
5513:{\displaystyle b_{k}}
5488:
5486:{\displaystyle e_{k}}
5461:
5442:
5417:
5354:
5318:
5316:{\displaystyle e_{k}}
5291:
5289:{\displaystyle b_{k}}
5264:
5235:
5181:
5136:
5053:
5000:
4739:
4706:
4704:{\displaystyle r_{k}}
4679:
4646:
4644:{\displaystyle r_{k}}
4619:
4599:
4570:
4487:
4485:{\displaystyle n_{k}}
4456:
4429:
4427:{\displaystyle r_{k}}
4402:
4374:
4358:
4315:
4273:
4239:
4228:
4187:
4141:Differential encoding
4120:
4007:of the errors as the
3995:
3935:
3912:
3827:
3785:
3765:
3750:
3681:
3598:
3189:
3065:
3026:
2997:
2971:
2936:
2907:
2879:
2824:
2801:
2775:
2705:
2687:
2526:signal-to-noise ratio
2516:
2316:
2232:
2219:
2198:
2107:
1906:
1705:of the signal, or to
1670:
1651:
1571:
1480:
1424:
1378:
1332:
1240:
1144:
973:
826:prior to modulation.
796:Constellation diagram
794:
735:constellation diagram
27:Type of data encoding
7279:on December 22, 2015
7112:on February 10, 2020
6804:modulation formats.
6707:
6655:
6597:with BPSK and at 2.4
6546:
6282:
6255:
6240:{\displaystyle Q(x)}
6222:
6192:
6156:
6102:
6070:
6038:
5989:
5957:
5900:
5858:
5824:
5787:
5760:
5727:
5694:
5667:
5634:
5566:
5527:
5497:
5470:
5429:
5366:
5327:
5300:
5273:
5244:
5148:
5068:
5016:
4751:
4715:
4688:
4655:
4628:
4608:
4582:
4499:
4469:
4438:
4411:
4391:
4339:
4282:
4262:
4211:
4176:
4051:
3951:
3924:
3839:
3794:
3774:
3690:
3610:
3204:
3077:
3048:
2952:
2917:
2888:
2535:
2334:
2253:
2238:Probability of error
2125:
1927:
1748:
1580:
1508:
1497:(BER) of BPSK under
1441:
1387:
1344:
1266:
1155:
992:
848:
755:constellation points
7080:Haykin, S. (2001).
6819:Differential coding
6526:biometric passports
6497:Mbit/s. The 6 and 9
6329:
5010:complex conjugation
4994:
4904:
4784:
4597:{\displaystyle k-1}
4226:{\displaystyle t=0}
4147:differential coding
4029:Spectral efficiency
3466:
3448:
3328:
3132:
2948:On the other hand,
2768:Offset QPSK (OQPSK)
7308:. Addison Wesley.
6787:
6683:
6641:mutual information
6637:
6560:
6520:standards such as
6421:
6315:
6261:
6237:
6205:
6169:
6125:
6083:
6051:
6018:
5970:
5928:
5886:
5837:
5806:
5773:
5746:
5713:
5680:
5653:
5614:
5549:
5510:
5483:
5464:
5437:
5412:
5349:
5313:
5286:
5259:
5238:
5176:
5131:
5048:
4995:
4974:
4884:
4764:
4734:
4711:is projected onto
4701:
4674:
4641:
4614:
4594:
4565:
4482:
4451:
4424:
4397:
4377:
4353:
4333:Bluetooth standard
4310:
4268:
4256:
4223:
4182:
4115:
3990:
3930:
3907:
3822:
3780:
3768:
3755:are each Gaussian
3745:
3676:
3593:
3591:
3452:
3434:
3314:
3184:
3099:
3060:
3029:
3001:
2981:cellular telephone
2966:
2931:
2902:
2882:
2828:
2804:
2778:
2709:
2682:
2511:
2509:
2311:
2235:
2227:
2223:non-return-to-zero
2193:
2184:
2102:
2100:
1901:
1677:
1646:
1566:
1475:
1419:
1373:
1327:
1235:
1139:
968:
800:
672:represented data:
666:digital modulation
596:digital modulation
588:Phase-shift keying
341:Digital modulation
203:possibly contains
7410:Data transmission
7361:. Prentice-Hall.
7029:978-1-4673-0081-0
6914:. pp. 5 pp.
6836:equivalent of PSK
6778:
6776:
6752:
6714:
6501:Mbit/s modes use
6491:IEEE 802.11g-2003
6464:IEEE 802.11b-1999
6411:
6400:
6399:
6378:
6346:
6313:
6312:
6264:{\displaystyle x}
6113:
5255:
5124:
5092:
4920:
4862:
4617:{\displaystyle k}
4527:
4400:{\displaystyle k}
4271:{\displaystyle M}
4185:{\displaystyle M}
4110:
4102:
4092:
4083:
3975:
3933:{\displaystyle M}
3897:
3881:
3783:{\displaystyle M}
3728:
3659:
3646:
3587:
3535:
3467:
3377:
3348:
3266:
2989:constant envelope
2835:The license-free
2676:
2675:
2633:
2632:
2585:
2584:
2487:
2486:
2441:
2440:
2305:
2304:
2180:
2179:
2156:
2155:
2061:
2060:
1977:
1976:
1863:
1802:
1801:
1640:
1639:
1604:
1560:
1559:
1405:
1359:
1301:
1300:
1209:
1208:
1113:
1112:
1046:
1045:
902:
901:
761:spacing around a
583:
582:
298:Analog modulation
266:
265:
258:
248:
247:
240:
205:original research
186:
185:
178:
124:
123:
116:
68:
16:(Redirected from
7417:
7391:
7372:
7353:
7320:
7319:
7298:
7289:
7288:
7286:
7284:
7278:
7267:
7259:
7253:
7252:
7250:
7248:
7233:
7227:
7226:
7215:
7209:
7208:
7205:Application Note
7202:
7194:
7188:
7182:
7176:
7170:
7164:
7158:
7149:
7148:
7145:10.1109/25.16541
7128:
7122:
7121:
7119:
7117:
7108:. Archived from
7102:
7096:
7095:
7077:
7071:
7070:
7060:
7040:
7034:
7033:
7007:
7001:
7000:
6990:
6981:
6975:
6974:
6948:
6942:
6941:
6903:
6891:
6885:
6884:
6866:
6840:Polar modulation
6830:Phase modulation
6796:
6794:
6793:
6788:
6783:
6779:
6777:
6775:
6774:
6765:
6764:
6755:
6753:
6748:
6740:
6738:
6729:
6728:
6716:
6715:
6712:
6692:
6690:
6689:
6684:
6682:
6681:
6672:
6667:
6666:
6600:
6593:
6577:
6573:
6569:
6567:
6566:
6561:
6556:
6530:American Express
6508:
6500:
6496:
6484:
6480:
6476:
6469:
6430:
6428:
6427:
6422:
6409:
6405:
6401:
6395:
6391:
6379:
6371:
6359:
6358:
6357:
6356:
6347:
6339:
6328:
6323:
6314:
6305:
6301:
6270:
6268:
6267:
6262:
6246:
6244:
6243:
6238:
6214:
6212:
6211:
6206:
6204:
6203:
6178:
6176:
6175:
6170:
6168:
6167:
6134:
6132:
6131:
6126:
6124:
6123:
6114:
6106:
6092:
6090:
6089:
6084:
6082:
6081:
6060:
6058:
6057:
6052:
6050:
6049:
6027:
6025:
6024:
6019:
6017:
6016:
6001:
6000:
5979:
5977:
5976:
5971:
5969:
5968:
5937:
5935:
5934:
5929:
5927:
5926:
5917:
5912:
5911:
5895:
5893:
5892:
5887:
5885:
5884:
5875:
5870:
5869:
5846:
5844:
5843:
5838:
5836:
5835:
5815:
5813:
5812:
5807:
5805:
5804:
5782:
5780:
5779:
5774:
5772:
5771:
5755:
5753:
5752:
5747:
5739:
5738:
5722:
5720:
5719:
5714:
5712:
5711:
5689:
5687:
5686:
5681:
5679:
5678:
5662:
5660:
5659:
5654:
5646:
5645:
5623:
5621:
5620:
5615:
5610:
5609:
5591:
5590:
5578:
5577:
5558:
5556:
5555:
5550:
5539:
5538:
5519:
5517:
5516:
5511:
5509:
5508:
5492:
5490:
5489:
5484:
5482:
5481:
5446:
5444:
5443:
5438:
5436:
5421:
5419:
5418:
5413:
5411:
5410:
5398:
5397:
5379:
5378:
5358:
5356:
5355:
5350:
5339:
5338:
5322:
5320:
5319:
5314:
5312:
5311:
5295:
5293:
5292:
5287:
5285:
5284:
5268:
5266:
5265:
5260:
5258:
5257:
5256:
5253:
5185:
5183:
5182:
5177:
5175:
5174:
5165:
5160:
5159:
5140:
5138:
5137:
5132:
5127:
5126:
5125:
5123:
5122:
5113:
5112:
5103:
5093:
5085:
5080:
5079:
5057:
5055:
5054:
5049:
5047:
5046:
5028:
5027:
5004:
5002:
5001:
4996:
4993:
4988:
4973:
4972:
4960:
4959:
4950:
4949:
4948:
4947:
4921:
4919:
4918:
4909:
4903:
4898:
4883:
4882:
4881:
4880:
4863:
4861:
4860:
4851:
4846:
4845:
4844:
4840:
4839:
4838:
4820:
4819:
4797:
4796:
4783:
4778:
4763:
4762:
4743:
4741:
4740:
4735:
4733:
4732:
4710:
4708:
4707:
4702:
4700:
4699:
4683:
4681:
4680:
4675:
4673:
4672:
4650:
4648:
4647:
4642:
4640:
4639:
4623:
4621:
4620:
4615:
4603:
4601:
4600:
4595:
4574:
4572:
4571:
4566:
4561:
4560:
4548:
4547:
4546:
4545:
4528:
4526:
4525:
4516:
4511:
4510:
4491:
4489:
4488:
4483:
4481:
4480:
4460:
4458:
4457:
4452:
4450:
4449:
4433:
4431:
4430:
4425:
4423:
4422:
4406:
4404:
4403:
4398:
4362:
4360:
4359:
4354:
4349:
4319:
4317:
4316:
4311:
4309:
4308:
4299:
4294:
4293:
4277:
4275:
4274:
4269:
4232:
4230:
4229:
4224:
4191:
4189:
4188:
4183:
4124:
4122:
4121:
4116:
4111:
4108:
4103:
4100:
4098:
4093:
4090:
4084:
4079:
4072:
4071:
4061:
4009:Hamming distance
3999:
3997:
3996:
3991:
3986:
3985:
3976:
3968:
3963:
3962:
3939:
3937:
3936:
3931:
3916:
3914:
3913:
3908:
3903:
3899:
3898:
3890:
3882:
3880:
3879:
3867:
3851:
3850:
3831:
3829:
3828:
3823:
3821:
3820:
3811:
3806:
3805:
3789:
3787:
3786:
3781:
3757:random variables
3754:
3752:
3751:
3746:
3744:
3740:
3739:
3738:
3729:
3721:
3702:
3701:
3685:
3683:
3682:
3677:
3675:
3671:
3670:
3669:
3660:
3652:
3647:
3645:
3644:
3635:
3622:
3621:
3602:
3600:
3599:
3594:
3592:
3588:
3586:
3585:
3576:
3575:
3566:
3557:
3556:
3540:
3536:
3534:
3533:
3524:
3523:
3514:
3505:
3504:
3485:
3484:
3468:
3465:
3460:
3447:
3442:
3433:
3407:
3406:
3405:
3404:
3399:
3395:
3394:
3393:
3378:
3376:
3375:
3366:
3349:
3341:
3327:
3322:
3313:
3312:
3311:
3310:
3298:
3297:
3288:
3287:
3267:
3265:
3254:
3245:
3241:
3240:
3227:
3226:
3225:
3224:
3193:
3191:
3190:
3185:
3180:
3179:
3167:
3163:
3162:
3149:
3148:
3147:
3146:
3131:
3127:
3118:
3114:
3089:
3088:
3069:
3067:
3066:
3061:
3019:Higher-order PSK
2975:
2973:
2972:
2967:
2962:
2940:
2938:
2937:
2932:
2927:
2911:
2909:
2908:
2903:
2898:
2758:
2757:
2753:
2749:
2745:
2737:
2735:
2731:
2727:
2723:
2714:
2691:
2689:
2688:
2683:
2681:
2677:
2674:
2673:
2664:
2663:
2654:
2653:
2638:
2634:
2631:
2630:
2629:
2616:
2615:
2606:
2605:
2590:
2586:
2583:
2582:
2573:
2572:
2563:
2562:
2547:
2546:
2520:
2518:
2517:
2512:
2510:
2503:
2502:
2497:
2493:
2492:
2488:
2485:
2484:
2475:
2474:
2465:
2464:
2446:
2442:
2439:
2438:
2429:
2428:
2419:
2418:
2400:
2396:
2395:
2390:
2386:
2385:
2384:
2350:
2349:
2320:
2318:
2317:
2312:
2310:
2306:
2303:
2302:
2293:
2292:
2291:
2278:
2277:
2265:
2264:
2202:
2200:
2199:
2194:
2189:
2188:
2181:
2175:
2174:
2165:
2164:
2157:
2151:
2150:
2141:
2140:
2111:
2109:
2108:
2103:
2101:
2097:
2093:
2089:
2088:
2062:
2059:
2058:
2046:
2045:
2027:
2026:
2013:
2009:
2005:
2004:
1978:
1975:
1974:
1962:
1961:
1943:
1942:
1910:
1908:
1907:
1902:
1869:
1865:
1864:
1856:
1830:
1829:
1803:
1800:
1799:
1790:
1789:
1788:
1775:
1774:
1760:
1759:
1711:exactly the same
1691:to minimize the
1655:
1653:
1652:
1647:
1645:
1641:
1638:
1637:
1628:
1627:
1618:
1617:
1605:
1597:
1592:
1591:
1575:
1573:
1572:
1567:
1565:
1561:
1558:
1557:
1548:
1547:
1546:
1533:
1532:
1520:
1519:
1484:
1482:
1481:
1476:
1468:
1467:
1428:
1426:
1425:
1420:
1406:
1404:
1403:
1394:
1382:
1380:
1379:
1374:
1360:
1358:
1357:
1348:
1336:
1334:
1333:
1328:
1302:
1299:
1298:
1286:
1285:
1244:
1242:
1241:
1236:
1210:
1207:
1206:
1197:
1196:
1195:
1182:
1181:
1167:
1166:
1148:
1146:
1145:
1140:
1114:
1111:
1110:
1101:
1100:
1099:
1086:
1085:
1047:
1044:
1043:
1034:
1033:
1032:
1019:
1018:
1004:
1003:
977:
975:
974:
969:
903:
900:
899:
890:
889:
888:
875:
874:
860:
859:
810:
798:example for BPSK
575:
568:
561:
292:
275:
274:
261:
254:
243:
236:
232:
229:
223:
220:inline citations
196:
195:
188:
181:
174:
170:
167:
161:
156:this article by
147:inline citations
134:
133:
126:
119:
112:
108:
105:
99:
79:
78:
71:
60:
38:
37:
30:
21:
7425:
7424:
7420:
7419:
7418:
7416:
7415:
7414:
7395:
7394:
7388:
7369:
7350:
7342:. McGraw Hill.
7329:
7324:
7323:
7316:
7299:
7292:
7282:
7280:
7276:
7265:
7261:
7260:
7256:
7246:
7244:
7235:
7234:
7230:
7217:
7216:
7212:
7200:
7196:
7195:
7191:
7183:
7179:
7171:
7167:
7159:
7152:
7129:
7125:
7115:
7113:
7104:
7103:
7099:
7092:
7078:
7074:
7041:
7037:
7030:
7008:
7004:
6988:
6982:
6978:
6971:
6949:
6945:
6930:
6904:
6892:
6888:
6881:
6867:
6863:
6858:
6810:
6770:
6766:
6760:
6756:
6754:
6741:
6739:
6737:
6733:
6724:
6720:
6711:
6710:
6708:
6705:
6704:
6677:
6673:
6668:
6662:
6658:
6656:
6653:
6652:
6629:
6598:
6591:
6575:
6571:
6552:
6547:
6544:
6543:
6506:
6498:
6494:
6482:
6478:
6474:
6467:
6449:
6441:fading channels
6390:
6386:
6370:
6352:
6348:
6338:
6334:
6330:
6324:
6319:
6300:
6283:
6280:
6279:
6256:
6253:
6252:
6223:
6220:
6219:
6199:
6195:
6193:
6190:
6189:
6163:
6159:
6157:
6154:
6153:
6119:
6115:
6105:
6103:
6100:
6099:
6095:symbol duration
6077:
6073:
6071:
6068:
6067:
6045:
6041:
6039:
6036:
6035:
6012:
6008:
5996:
5992:
5990:
5987:
5986:
5964:
5960:
5958:
5955:
5954:
5948:
5922:
5918:
5913:
5907:
5903:
5901:
5898:
5897:
5880:
5876:
5871:
5865:
5861:
5859:
5856:
5855:
5831:
5827:
5825:
5822:
5821:
5794:
5790:
5788:
5785:
5784:
5767:
5763:
5761:
5758:
5757:
5734:
5730:
5728:
5725:
5724:
5701:
5697:
5695:
5692:
5691:
5674:
5670:
5668:
5665:
5664:
5641:
5637:
5635:
5632:
5631:
5599:
5595:
5586:
5582:
5573:
5569:
5567:
5564:
5563:
5534:
5530:
5528:
5525:
5524:
5504:
5500:
5498:
5495:
5494:
5477:
5473:
5471:
5468:
5467:
5435:
5430:
5427:
5426:
5406:
5402:
5387:
5383:
5374:
5370:
5367:
5364:
5363:
5334:
5330:
5328:
5325:
5324:
5307:
5303:
5301:
5298:
5297:
5280:
5276:
5274:
5271:
5270:
5252:
5251:
5247:
5245:
5242:
5241:
5230:
5202:continuous wave
5170:
5166:
5161:
5155:
5151:
5149:
5146:
5145:
5118:
5114:
5108:
5104:
5102:
5098:
5094:
5084:
5075:
5071:
5069:
5066:
5065:
5036:
5032:
5023:
5019:
5017:
5014:
5013:
4989:
4978:
4968:
4964:
4955:
4951:
4937:
4933:
4926:
4922:
4914:
4910:
4908:
4899:
4888:
4876:
4872:
4868:
4864:
4856:
4852:
4850:
4828:
4824:
4815:
4811:
4810:
4806:
4802:
4798:
4792:
4788:
4779:
4768:
4758:
4754:
4752:
4749:
4748:
4722:
4718:
4716:
4713:
4712:
4695:
4691:
4689:
4686:
4685:
4662:
4658:
4656:
4653:
4652:
4635:
4631:
4629:
4626:
4625:
4609:
4606:
4605:
4604:symbol and the
4583:
4580:
4579:
4556:
4552:
4541:
4537:
4533:
4529:
4521:
4517:
4515:
4506:
4502:
4500:
4497:
4496:
4476:
4472:
4470:
4467:
4466:
4465:(AWGN) term as
4445:
4441:
4439:
4436:
4435:
4418:
4414:
4412:
4409:
4408:
4392:
4389:
4388:
4369:
4345:
4340:
4337:
4336:
4304:
4300:
4295:
4289:
4285:
4283:
4280:
4279:
4263:
4260:
4259:
4252:
4245:
4212:
4209:
4208:
4177:
4174:
4173:
4149:
4143:
4138:
4107:
4099:
4094:
4089:
4067:
4063:
4062:
4060:
4052:
4049:
4048:
4031:
3981:
3977:
3967:
3958:
3954:
3952:
3949:
3948:
3925:
3922:
3921:
3889:
3875:
3871:
3866:
3865:
3861:
3846:
3842:
3840:
3837:
3836:
3816:
3812:
3807:
3801:
3797:
3795:
3792:
3791:
3775:
3772:
3771:
3734:
3730:
3720:
3713:
3709:
3697:
3693:
3691:
3688:
3687:
3665:
3661:
3651:
3640:
3636:
3634:
3633:
3629:
3617:
3613:
3611:
3608:
3607:
3590:
3589:
3581:
3577:
3571:
3567:
3565:
3558:
3552:
3548:
3545:
3544:
3529:
3525:
3519:
3515:
3513:
3509:
3497:
3493:
3486:
3480:
3476:
3473:
3472:
3461:
3456:
3443:
3438:
3432:
3425:
3419:
3418:
3400:
3389:
3385:
3371:
3367:
3365:
3355:
3351:
3350:
3340:
3336:
3332:
3323:
3318:
3306:
3302:
3293:
3289:
3283:
3279:
3272:
3268:
3258:
3253:
3246:
3236:
3232:
3228:
3220:
3216:
3215:
3211:
3207:
3205:
3202:
3201:
3175:
3171:
3158:
3154:
3150:
3142:
3138:
3137:
3133:
3123:
3119:
3110:
3103:
3084:
3080:
3078:
3075:
3074:
3049:
3046:
3045:
3042:
3021:
3006:
2958:
2953:
2950:
2949:
2923:
2918:
2915:
2914:
2894:
2889:
2886:
2885:
2874:
2859:ternary symbols
2833:
2770:
2765:
2755:
2751:
2747:
2743:
2741:
2733:
2729:
2725:
2721:
2720:
2712:
2699:
2669:
2665:
2659:
2655:
2652:
2648:
2625:
2621:
2617:
2611:
2607:
2604:
2600:
2578:
2574:
2568:
2564:
2561:
2557:
2542:
2538:
2536:
2533:
2532:
2508:
2507:
2498:
2480:
2476:
2470:
2466:
2463:
2459:
2455:
2451:
2450:
2434:
2430:
2424:
2420:
2417:
2413:
2398:
2397:
2391:
2380:
2376:
2369:
2365:
2364:
2351:
2345:
2341:
2337:
2335:
2332:
2331:
2298:
2294:
2287:
2283:
2279:
2276:
2272:
2260:
2256:
2254:
2251:
2250:
2240:
2183:
2182:
2170:
2166:
2163:
2158:
2146:
2142:
2139:
2129:
2128:
2126:
2123:
2122:
2099:
2098:
2084:
2080:
2073:
2069:
2054:
2050:
2044:
2037:
2022:
2018:
2015:
2014:
2000:
1996:
1989:
1985:
1970:
1966:
1960:
1953:
1938:
1934:
1930:
1928:
1925:
1924:
1919:basis functions
1855:
1825:
1821:
1814:
1810:
1795:
1791:
1784:
1780:
1776:
1773:
1755:
1751:
1749:
1746:
1745:
1738:
1681:quadriphase PSK
1665:
1633:
1629:
1623:
1619:
1616:
1612:
1596:
1587:
1583:
1581:
1578:
1577:
1553:
1549:
1542:
1538:
1534:
1531:
1527:
1515:
1511:
1509:
1506:
1505:
1491:
1463:
1459:
1442:
1439:
1438:
1399:
1395:
1393:
1388:
1385:
1384:
1353:
1349:
1347:
1345:
1342:
1341:
1294:
1290:
1284:
1267:
1264:
1263:
1202:
1198:
1191:
1187:
1183:
1180:
1162:
1158:
1156:
1153:
1152:
1106:
1102:
1095:
1091:
1087:
1084:
1039:
1035:
1028:
1024:
1020:
1017:
999:
995:
993:
990:
989:
895:
891:
884:
880:
876:
873:
855:
851:
849:
846:
845:
839:
808:
787:
767:complex numbers
715:By viewing the
704:By viewing the
662:
633:communication.
579:
442:Spread spectrum
273:
262:
251:
250:
249:
244:
233:
227:
224:
209:
197:
193:
182:
171:
165:
162:
152:Please help to
151:
135:
131:
120:
109:
103:
100:
92:help improve it
89:
80:
76:
39:
35:
28:
23:
22:
15:
12:
11:
5:
7423:
7413:
7412:
7407:
7393:
7392:
7386:
7373:
7367:
7354:
7348:
7328:
7325:
7322:
7321:
7314:
7290:
7254:
7228:
7210:
7189:
7177:
7165:
7150:
7139:(3): 151–157.
7123:
7097:
7090:
7072:
7051:(3): 1839–62.
7035:
7028:
7002:
6976:
6969:
6943:
6928:
6886:
6879:
6860:
6859:
6857:
6854:
6853:
6852:
6847:
6842:
6837:
6827:
6821:
6816:
6809:
6806:
6798:
6797:
6786:
6782:
6773:
6769:
6763:
6759:
6751:
6747:
6744:
6736:
6732:
6727:
6723:
6719:
6680:
6676:
6671:
6665:
6661:
6628:
6625:
6559:
6555:
6551:
6448:
6445:
6433:
6432:
6420:
6417:
6414:
6408:
6404:
6398:
6394:
6389:
6385:
6382:
6377:
6374:
6369:
6366:
6363:
6355:
6351:
6345:
6342:
6337:
6333:
6327:
6322:
6318:
6311:
6308:
6304:
6299:
6296:
6293:
6290:
6287:
6260:
6236:
6233:
6230:
6227:
6217:
6216:
6202:
6198:
6187:
6166:
6162:
6151:
6122:
6118:
6112:
6109:
6097:
6080:
6076:
6065:
6048:
6044:
6033:
6015:
6011:
6007:
6004:
5999:
5995:
5984:
5967:
5963:
5947:
5944:
5925:
5921:
5916:
5910:
5906:
5883:
5879:
5874:
5868:
5864:
5834:
5830:
5803:
5800:
5797:
5793:
5770:
5766:
5745:
5742:
5737:
5733:
5710:
5707:
5704:
5700:
5677:
5673:
5652:
5649:
5644:
5640:
5625:
5624:
5613:
5608:
5605:
5602:
5598:
5594:
5589:
5585:
5581:
5576:
5572:
5548:
5545:
5542:
5537:
5533:
5507:
5503:
5480:
5476:
5434:
5423:
5422:
5409:
5405:
5401:
5396:
5393:
5390:
5386:
5382:
5377:
5373:
5348:
5345:
5342:
5337:
5333:
5310:
5306:
5283:
5279:
5250:
5229:
5226:
5173:
5169:
5164:
5158:
5154:
5142:
5141:
5130:
5121:
5117:
5111:
5107:
5101:
5097:
5091:
5088:
5083:
5078:
5074:
5045:
5042:
5039:
5035:
5031:
5026:
5022:
5006:
5005:
4992:
4987:
4984:
4981:
4977:
4971:
4967:
4963:
4958:
4954:
4946:
4943:
4940:
4936:
4932:
4929:
4925:
4917:
4913:
4907:
4902:
4897:
4894:
4891:
4887:
4879:
4875:
4871:
4867:
4859:
4855:
4849:
4843:
4837:
4834:
4831:
4827:
4823:
4818:
4814:
4809:
4805:
4801:
4795:
4791:
4787:
4782:
4777:
4774:
4771:
4767:
4761:
4757:
4731:
4728:
4725:
4721:
4698:
4694:
4684:. That is, if
4671:
4668:
4665:
4661:
4638:
4634:
4613:
4593:
4590:
4587:
4576:
4575:
4564:
4559:
4555:
4551:
4544:
4540:
4536:
4532:
4524:
4520:
4514:
4509:
4505:
4479:
4475:
4448:
4444:
4421:
4417:
4396:
4368:
4365:
4352:
4348:
4344:
4307:
4303:
4298:
4292:
4288:
4267:
4250:
4243:
4222:
4219:
4216:
4181:
4145:Main article:
4142:
4139:
4137:
4134:
4126:
4125:
4114:
4106:
4097:
4088:
4082:
4078:
4075:
4070:
4066:
4059:
4056:
4030:
4027:
4001:
4000:
3989:
3984:
3980:
3974:
3971:
3966:
3961:
3957:
3929:
3918:
3917:
3906:
3902:
3896:
3893:
3888:
3885:
3878:
3874:
3870:
3864:
3860:
3857:
3854:
3849:
3845:
3819:
3815:
3810:
3804:
3800:
3779:
3743:
3737:
3733:
3727:
3724:
3719:
3716:
3712:
3708:
3705:
3700:
3696:
3674:
3668:
3664:
3658:
3655:
3650:
3643:
3639:
3632:
3628:
3625:
3620:
3616:
3604:
3603:
3584:
3580:
3574:
3570:
3564:
3561:
3559:
3555:
3551:
3547:
3546:
3543:
3539:
3532:
3528:
3522:
3518:
3512:
3508:
3503:
3500:
3496:
3492:
3489:
3487:
3483:
3479:
3475:
3474:
3471:
3464:
3459:
3455:
3451:
3446:
3441:
3437:
3431:
3428:
3426:
3424:
3421:
3420:
3417:
3414:
3411:
3403:
3398:
3392:
3388:
3384:
3381:
3374:
3370:
3364:
3361:
3358:
3354:
3347:
3344:
3339:
3335:
3331:
3326:
3321:
3317:
3309:
3305:
3301:
3296:
3292:
3286:
3282:
3278:
3275:
3271:
3264:
3261:
3257:
3252:
3249:
3247:
3244:
3239:
3235:
3231:
3223:
3219:
3214:
3210:
3209:
3195:
3194:
3183:
3178:
3174:
3170:
3166:
3161:
3157:
3153:
3145:
3141:
3136:
3130:
3126:
3122:
3117:
3113:
3109:
3106:
3102:
3098:
3095:
3092:
3087:
3083:
3059:
3056:
3053:
3041:
3040:Bit error rate
3038:
3020:
3017:
3005:
3002:
2965:
2961:
2957:
2930:
2926:
2922:
2901:
2897:
2893:
2873:
2867:
2832:
2829:
2769:
2766:
2764:
2761:
2760:
2759:
2738:
2693:
2692:
2680:
2672:
2668:
2662:
2658:
2651:
2647:
2644:
2641:
2637:
2628:
2624:
2620:
2614:
2610:
2603:
2599:
2596:
2593:
2589:
2581:
2577:
2571:
2567:
2560:
2556:
2553:
2550:
2545:
2541:
2522:
2521:
2506:
2501:
2496:
2491:
2483:
2479:
2473:
2469:
2462:
2458:
2454:
2449:
2445:
2437:
2433:
2427:
2423:
2416:
2412:
2409:
2406:
2403:
2401:
2399:
2394:
2389:
2383:
2379:
2375:
2372:
2368:
2363:
2360:
2357:
2354:
2352:
2348:
2344:
2340:
2339:
2322:
2321:
2309:
2301:
2297:
2290:
2286:
2282:
2275:
2271:
2268:
2263:
2259:
2239:
2236:
2204:
2203:
2192:
2187:
2178:
2173:
2169:
2162:
2159:
2154:
2149:
2145:
2138:
2135:
2134:
2132:
2113:
2112:
2096:
2092:
2087:
2083:
2079:
2076:
2072:
2068:
2065:
2057:
2053:
2049:
2043:
2040:
2038:
2036:
2033:
2030:
2025:
2021:
2017:
2016:
2012:
2008:
2003:
1999:
1995:
1992:
1988:
1984:
1981:
1973:
1969:
1965:
1959:
1956:
1954:
1952:
1949:
1946:
1941:
1937:
1933:
1932:
1912:
1911:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1872:
1868:
1862:
1859:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1828:
1824:
1820:
1817:
1813:
1809:
1806:
1798:
1794:
1787:
1783:
1779:
1772:
1769:
1766:
1763:
1758:
1754:
1737:
1736:Implementation
1734:
1693:bit error rate
1683:, 4-PSK, or 4-
1664:
1661:
1657:
1656:
1644:
1636:
1632:
1626:
1622:
1615:
1611:
1608:
1603:
1600:
1595:
1590:
1586:
1564:
1556:
1552:
1545:
1541:
1537:
1530:
1526:
1523:
1518:
1514:
1495:bit error rate
1490:
1489:Bit error rate
1487:
1474:
1471:
1466:
1462:
1458:
1455:
1452:
1449:
1446:
1418:
1415:
1412:
1409:
1402:
1398:
1392:
1372:
1369:
1366:
1363:
1356:
1352:
1338:
1337:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1297:
1293:
1289:
1283:
1280:
1277:
1274:
1271:
1258:basis function
1247:
1246:
1245:for binary "1"
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1205:
1201:
1194:
1190:
1186:
1179:
1176:
1173:
1170:
1165:
1161:
1150:
1149:for binary "0"
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1109:
1105:
1098:
1094:
1090:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1042:
1038:
1031:
1027:
1023:
1016:
1013:
1010:
1007:
1002:
998:
979:
978:
967:
964:
961:
958:
955:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
898:
894:
887:
883:
879:
872:
869:
866:
863:
858:
854:
838:
837:Implementation
835:
786:
783:
731:
730:
713:
690:
689:
686:
680:
661:
658:
606:of a constant
581:
580:
578:
577:
570:
563:
555:
552:
551:
550:
549:
544:
539:
534:
529:
524:
519:
514:
509:
504:
499:
494:
489:
484:
476:
475:
471:
470:
469:
468:
463:
458:
453:
445:
444:
438:
437:
436:
435:
430:
422:
421:
415:
414:
413:
412:
407:
402:
397:
392:
387:
382:
377:
372:
367:
362:
357:
352:
344:
343:
337:
336:
335:
334:
329:
324:
319:
314:
309:
301:
300:
294:
293:
285:
284:
270:Pre-shared key
264:
263:
246:
245:
200:
198:
191:
184:
183:
138:
136:
129:
122:
121:
83:
81:
74:
69:
43:
42:
40:
33:
26:
9:
6:
4:
3:
2:
7422:
7411:
7408:
7406:
7403:
7402:
7400:
7389:
7387:0-471-62947-2
7383:
7379:
7374:
7370:
7368:0-13-081223-4
7364:
7360:
7355:
7351:
7349:0-07-113814-5
7345:
7341:
7340:
7334:
7333:
7332:
7317:
7315:0-201-10709-0
7311:
7307:
7303:
7302:Blahut, R. E.
7297:
7295:
7275:
7271:
7264:
7258:
7242:
7238:
7232:
7224:
7220:
7214:
7206:
7199:
7193:
7186:
7181:
7174:
7169:
7162:
7157:
7155:
7146:
7142:
7138:
7134:
7127:
7111:
7107:
7101:
7093:
7091:0-471-17869-1
7087:
7083:
7076:
7068:
7064:
7059:
7054:
7050:
7046:
7039:
7031:
7025:
7021:
7017:
7013:
7006:
6998:
6994:
6987:
6980:
6972:
6970:0-7803-6521-6
6966:
6962:
6958:
6954:
6947:
6939:
6935:
6931:
6929:0-7803-9414-3
6925:
6921:
6917:
6913:
6909:
6901:
6897:
6890:
6882:
6880:0-13-121929-4
6876:
6872:
6865:
6861:
6851:
6848:
6846:
6843:
6841:
6838:
6835:
6831:
6828:
6825:
6822:
6820:
6817:
6815:
6812:
6811:
6805:
6803:
6784:
6780:
6771:
6767:
6761:
6757:
6749:
6745:
6742:
6734:
6730:
6725:
6721:
6717:
6703:
6702:
6701:
6698:
6696:
6678:
6674:
6669:
6663:
6659:
6650:
6646:
6642:
6633:
6624:
6622:
6617:
6615:
6611:
6605:
6603:
6596:
6589:
6585:
6584:IEEE 802.15.4
6581:
6557:
6553:
6549:
6541:
6537:
6535:
6531:
6527:
6523:
6522:ISO/IEC 14443
6519:
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6504:
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5219:
5218:optical field
5215:
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5207:
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2019:
2010:
2006:
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1997:
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1963:
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988:
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986:
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962:
959:
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953:
949:
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934:
928:
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922:
919:
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904:
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885:
881:
877:
870:
864:
856:
852:
844:
843:
842:
834:
832:
827:
825:
821:
817:
812:
806:
797:
793:
789:
782:
780:
776:
772:
768:
764:
760:
756:
751:
748:
744:
740:
739:complex plane
736:
728:
724:
723:
718:
714:
711:
707:
703:
702:
701:
699:
695:
687:
684:
681:
678:
675:
674:
673:
671:
667:
657:
655:
649:
647:
643:
639:
638:binary digits
634:
632:
628:
624:
623:wireless LANs
620:
616:
612:
609:
605:
601:
597:
593:
589:
585:
576:
571:
569:
564:
562:
557:
556:
554:
553:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
515:
513:
510:
508:
505:
503:
500:
498:
495:
493:
490:
488:
485:
483:
480:
479:
478:
477:
473:
472:
467:
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462:
459:
457:
454:
452:
449:
448:
447:
446:
443:
440:
439:
434:
431:
429:
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425:
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411:
408:
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403:
401:
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388:
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376:
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371:
368:
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361:
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353:
351:
348:
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346:
345:
342:
339:
338:
333:
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328:
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323:
320:
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305:
304:
303:
302:
299:
296:
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291:
287:
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283:
280:
277:
276:
271:
260:
257:
242:
239:
231:
221:
217:
213:
207:
206:
201:This article
199:
190:
189:
180:
177:
169:
159:
155:
149:
148:
142:
137:
128:
127:
118:
115:
107:
97:
93:
87:
84:This article
82:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
7377:
7358:
7338:
7330:
7305:
7283:December 20,
7281:. Retrieved
7274:the original
7269:
7257:
7247:September 8,
7245:. Retrieved
7241:the original
7231:
7222:
7213:
7204:
7192:
7180:
7168:
7136:
7132:
7126:
7114:. Retrieved
7110:the original
7100:
7081:
7075:
7048:
7044:
7038:
7011:
7005:
6996:
6992:
6979:
6952:
6946:
6911:
6900:10150/604890
6889:
6870:
6864:
6799:
6699:
6638:
6618:
6606:
6604:with OQPSK.
6538:
6515:
6460:wireless LAN
6457:
6450:
6447:Applications
6434:
6218:
6184:
6063:bit duration
6029:
5949:
5853:
5849:
5817:
5629:
5626:
5522:
5465:
5424:
5239:
5222:
5191:
5188:
5143:
5060:
5007:
4577:
4386:
4381:
4378:
4367:Demodulation
4330:
4325:
4257:
4248:
4241:
4204:
4202:
4197:
4170:
4166:
4161:
4160:rather than
4157:
4150:
4127:
4037:
4032:
4020:
4013:
4005:Lee distance
4002:
3919:
3769:
3605:
3196:
3043:
3030:
3012:
3011:(DPQPSK) or
3008:
3007:
2985:
2947:
2943:
2883:
2869:
2856:
2848:
2843:
2840:-offset QPSK
2836:
2834:
2816:
2812:
2805:
2792:
2788:
2784:
2780:
2779:
2710:
2698:
2694:
2523:
2326:
2323:
2245:
2241:
2212:
2208:
2205:
2117:
2114:
1916:
1913:
1739:
1727:
1715:
1710:
1706:
1699:
1697:
1680:
1678:
1658:
1492:
1431:
1339:
1255:
1250:
1248:
980:
840:
833:modulation.
828:
813:
801:
788:
753:In PSK, the
752:
732:
722:differential
721:
716:
694:carrier wave
691:
663:
660:Introduction
653:
650:
635:
611:carrier wave
591:
587:
586:
584:
547:Multiplexing
487:Demodulation
389:
252:
234:
225:
202:
172:
166:October 2022
163:
144:
110:
104:October 2022
101:
85:
61:
54:
48:
47:Please help
44:
6181:probability
5946:Definitions
5723:differ and
5630:Therefore,
4172:one of the
4164:the phase.
4023:union bound
3942:Gray coding
2999:"blue" one.
2826:components.
2707:boundaries.
1723:electronics
1689:Gray coding
1673:Gray coding
983:binary data
820:Costas loop
805:demodulator
710:demodulator
696:(usually a
646:demodulator
492:Line coding
158:introducing
7399:Categories
7327:References
7223:Planet Fox
7058:1908.09501
6824:Modulation
6534:ExpressPay
6462:standard,
5455:addition.
5447:indicates
5214:photodiode
4194:comparator
282:modulation
228:March 2024
212:improve it
141:references
50:improve it
7380:. Wiley.
7270:Black Box
6746:π
6731:
6718:≃
6550:π
6540:Bluetooth
6416:≥
6384:
6336:−
6326:∞
6317:∫
6310:π
6185:bit-error
5799:−
5706:−
5604:−
5593:⊕
5544:±
5433:⊕
5400:⊕
5392:−
5100:−
5041:−
5034:ϕ
5030:−
5021:ϕ
4991:∗
4983:−
4942:−
4935:φ
4928:−
4901:∗
4893:−
4874:φ
4833:−
4826:φ
4822:−
4813:φ
4781:∗
4773:−
4727:−
4667:−
4589:−
4539:ϕ
4443:ϕ
4407:timeslot
4343:π
4105:⋅
4074:
4055:ρ
3965:≈
3892:π
3887:
3873:γ
3853:≈
3790:and high
3704:∼
3624:∼
3550:γ
3507:
3499:−
3478:θ
3387:θ
3383:
3369:γ
3360:−
3338:−
3325:∞
3316:∫
3304:θ
3300:
3281:γ
3274:−
3263:π
3234:θ
3218:θ
3173:θ
3156:θ
3140:θ
3121:π
3108:π
3105:−
3101:∫
3097:−
2983:systems.
2956:π
2921:π
2892:π
2646:
2598:
2549:≈
2448:−
2374:−
2362:−
2161:±
2137:±
2078:π
2067:
2020:ϕ
1994:π
1983:
1936:ϕ
1858:π
1847:−
1819:π
1808:
1703:bandwidth
1610:
1457:π
1448:
1408:ϕ
1391:−
1362:ϕ
1316:π
1307:
1270:ϕ
1224:π
1215:
1128:π
1119:
1082:−
1073:π
1061:π
1052:
938:−
929:π
917:π
908:
747:imaginary
725:schemes,
670:digitally
631:Bluetooth
608:frequency
216:verifying
56:talk page
7304:(1988).
7116:July 15,
6938:11020777
6834:analogue
6808:See also
6614:Broadcom
5818:inverted
5453:modulo-2
5204:, and a
5186:values.
2763:Variants
2713:11000110
781:of two.
698:sinusoid
474:See also
279:Passband
7225:. 2014.
6542:2 uses
5240:At the
4492:. Then
2872:/4-QPSK
2524:If the
759:angular
594:) is a
210:Please
154:improve
90:Please
7384:
7365:
7346:
7312:
7088:
7026:
6999:: 156.
6967:
6936:
6926:
6877:
6621:modems
6610:DVB-S2
6599:
6592:
6588:Zigbee
6576:
6572:
6507:
6499:
6495:
6483:
6479:
6475:
6468:
6410:
5449:binary
5425:where
4335:where
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4158:change
3197:where
3004:DPQPSK
2870:π
2838:shaped
2831:SOQPSK
1249:where
809:
763:circle
717:change
642:symbol
619:cosine
400:SC-FDE
143:, but
7277:(PDF)
7266:(PDF)
7201:(PDF)
7053:arXiv
6989:(PDF)
6934:S2CID
6856:Notes
6850:PSK63
6845:PSK31
6137:noise
5198:laser
4130:M-QAM
4042:M-FSK
2844:FQPSK
2793:SQPSK
2785:OQPSK
831:2-QAM
779:power
706:phase
685:(FSK)
679:(ASK)
604:phase
497:Modem
7382:ISBN
7363:ISBN
7344:ISBN
7310:ISBN
7285:2015
7249:2015
7118:2020
7086:ISBN
7024:ISBN
6965:ISBN
6924:ISBN
6875:ISBN
6639:The
6608:The
6518:RFID
6503:OFDM
6471:Mbit
6458:The
6381:erfc
6032:bits
5816:are
5783:and
5690:and
4651:and
4091:bits
3832:by:
3686:and
3606:and
3055:>
2978:TDMA
2808:bits
2643:erfc
2595:erfc
1700:same
1607:erfc
1493:The
775:QPSK
771:BPSK
745:and
743:real
727:some
629:and
627:RFID
617:and
615:sine
600:data
537:OFDM
466:THSS
461:FHSS
456:DSSS
370:MFSK
355:APSK
18:DPSK
7141:doi
7063:doi
7016:doi
6957:doi
6916:doi
6896:hdl
6802:QAM
6722:log
6647:by
6602:GHz
6595:MHz
6532:'s
6183:of
5982:bit
5663:if
5466:So
5451:or
5192:In
4382:not
4247:= 2
4162:set
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4044:):
3884:sin
3495:tan
3380:cos
3291:sin
2863:LEO
2852:MSK
2795:).
2064:sin
1980:cos
1805:cos
1685:QAM
1576:or
1445:cos
1304:cos
1212:cos
1116:cos
1049:cos
905:cos
592:PSK
542:FDM
532:ΔΣM
527:PWM
522:PDM
517:PCM
512:PAM
507:PoM
502:AnM
451:CSS
433:WDM
428:QAM
410:WDM
405:TCM
395:QAM
390:PSK
385:PPM
380:OOK
375:MSK
365:FSK
360:CPM
350:ASK
332:SSB
322:QAM
214:by
94:to
7401::
7293:^
7268:.
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7153:^
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7135:.
7061:.
7049:22
7047:.
7022:.
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6995:.
6991:.
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6932:.
6922:.
6910:.
6713:MI
6513:.
6275::
6148:Hz
6135:,
5820:,
5254:th
4233:.
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3759:.
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1899:4.
1485:.
966:1.
625:,
327:SM
317:PM
312:FM
307:AM
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7318:.
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7032:.
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6940:.
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6902:.
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6883:.
6785:.
6781:)
6772:0
6768:N
6762:s
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6750:e
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6726:2
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6670:/
6664:s
6660:E
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6554:/
6431:.
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6407:,
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6397:2
6393:x
6388:(
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6368:=
6365:t
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6295:)
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6289:(
6286:Q
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6235:)
6232:x
6229:(
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6146:/
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6142:(
6121:0
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6111:2
6108:1
6079:s
6075:T
6047:b
6043:T
6030:n
6014:b
6010:E
6006:n
6003:=
5998:s
5994:E
5966:b
5962:E
5924:0
5920:N
5915:/
5909:b
5905:E
5882:0
5878:N
5873:/
5867:b
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5833:k
5829:b
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5792:e
5769:k
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5744:0
5741:=
5736:k
5732:b
5709:1
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5699:e
5676:k
5672:e
5651:1
5648:=
5643:k
5639:b
5612:,
5607:1
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5588:k
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5580:=
5575:k
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5541:=
5536:k
5532:e
5506:k
5502:b
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5341:(
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5120:0
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5110:b
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5090:2
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5082:=
5077:b
5073:P
5044:1
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5025:k
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4980:k
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4347:/
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4297:/
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4218:=
4215:t
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4113:]
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4087:[
4081:2
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3502:1
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3430:=
3423:V
3416:,
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3402:2
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3251:=
3243:)
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3230:(
3222:r
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3152:(
3144:r
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3125:/
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3112:/
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3091:=
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2964:4
2960:/
2929:4
2925:/
2900:4
2896:/
2791:(
2783:(
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2661:b
2657:E
2650:(
2640:=
2636:)
2627:0
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2602:(
2592:=
2588:)
2580:0
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2500:2
2495:]
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2457:Q
2453:[
2444:)
2436:0
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2393:2
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2382:b
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2367:(
2359:1
2356:=
2347:s
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2308:)
2300:0
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2285:E
2281:2
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2270:Q
2267:=
2262:b
2258:P
2191:.
2186:)
2177:2
2172:s
2168:E
2153:2
2148:s
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2131:(
2095:)
2091:t
2086:c
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2071:(
2056:s
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2042:=
2035:)
2032:t
2029:(
2024:2
2011:)
2007:t
2002:c
1998:f
1991:2
1987:(
1972:s
1968:T
1964:2
1958:=
1951:)
1948:t
1945:(
1940:1
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1893:3
1890:,
1887:2
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1881:1
1878:=
1875:n
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1867:)
1861:4
1853:)
1850:1
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1841:2
1838:(
1835:+
1832:t
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1797:s
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1778:2
1771:=
1768:)
1765:t
1762:(
1757:n
1753:s
1643:)
1635:0
1631:N
1625:b
1621:E
1614:(
1602:2
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1589:e
1585:P
1563:)
1555:0
1551:N
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1540:E
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1529:(
1525:Q
1522:=
1517:b
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1461:f
1454:2
1451:(
1417:)
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1411:(
1401:b
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1108:b
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1070:+
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1001:0
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926:+
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