Relative phase shift keying. Digital Phase Modulation: BPSK, QPSK, DQPSK

Binary phase shift keying.

AT digital systems mobile communications most commonly used is binary or M-ary phase shift keying. The simplest of them VRBC

(Bipolar Phase Shift Keying) binary phase shift keying, implemented at M=2, when the modulated signal takes only two values xi(t) or X2 (t)

Expressions (10.32) describing both states can be represented as

where um (t)- random binary signal with levels 1 or -1 in baseband at bit rate fb = 1/Tb, a (p(t) takes the values ​​0 or to.

Expressions (10.33) can be interpreted as the result of multiplication b(t) with harmonic signal, which is equivalent to conventional suppressed carrier amplitude modulation (BPSK AM-PN).

Obviously, BPSK detection is possible only with the use of a coherent phase detector, which is a voltage multiplier of the input signal with the voltage from the reference oscillator, which has a frequency and phase equal to the frequency and phase of the input, respectively.

The phase portrait of the carrier with binary phase shift keying has the form (Fig. 10.24)

Rice. 10.24

Rice. 10.25

Binary phase shift keying refers to antipodal modulation, when two time dependences coincide with each other, provided that the phase assignment law (10.33) is opposite.

Formation of the VREC signal, using the sequence rectangular pulses BVN, leads to an abrupt change in phase by ± i when the state of the digital sequence changes 0*-> 1 with an infinitely wide bandwidth of the digital path (Fig. 10.25). Nyquist or Gaussian LPF switched on at the input of the modulator reduce the bandwidth of the main path (BB), as well as the finiteness of the bandwidth of the radio path of the transmitter, which leads to a delay in transients. Their influence causes not only a delay in the moment of switching the phase at the output of the modulator, but also a change in the amplitude of the envelope, due to the action of the amplitude-phase conversion mechanism (Fig. 10.26).

Rice. 10.26

At the point in time Tb digital sequence (b to ) at the input of the modulator takes the value of logical zero. The signal at the output of the modulator should change the phase instantly, but due to the narrowband path, the active signal gradually decreases, and the signal with the new phase at the moment G = Th will slowly increase. Since their phases are opposite, and the frequencies are the same, they are subtracted, and at some point in time I-t+t 3 become equal, the envelope of the resulting process (thick dashed line) reaches zero and begins to gradually increase. Thus, in the process of forming the VR8K signal, like the ORBC, the change in the amplitude at the output of the modulator reaches 100%, which sharply reduces the noise immunity of the signal.

For coherent detection of the received signal, it is necessary to have receiving side exactly known value initial phase, which is not always possible. At the same time, during transmission, the phase of the signal can randomly take on opposite values ​​under the influence of interference, which leads to an error during reception in the time interval before the next phase failure.

The configuration of the coherent demodulator of the binary phase shift keying signal (BPSK) is shown in fig. 10.27

received signal g$) different from signal X(1 ), formed in a balanced modulator, which is due to the effect of interference in the radio channel and radio path. With coherent demodulation P(1 ) is mixed with a reference oscillator having the same frequency and phase. A circuit that provides such parameters to the reference oscillator signal is called a carrier recovery circuit (CHR).

The demodulator input is a modulated signal

where II is the amplitude of the received signal, and cp, (7) is the phase transmitted signal, which takes the values ​​0° or 180°. Multiplying the reference oscillator voltage (sooo?) in the simplest case with unit amplitude (heterodyne) and zero phase, as well as the received signal, we get

carrier oscillation and selects the constant component

the value of which is determined by the phase of the received signal (more precisely, the phase difference between the received signal and the reference oscillator). With #>, (7 ) = 0°, b, (1) = +1, which corresponds to a binary unit, and when (R (7) =180°, b, - (7) = - 1 (binary zero). So the demodulator restores the original binary sequence without returning to zero. Solver monitors the magnitude of the DC component of the received signal, comparing it with the threshold value of the binary ADC, exactly in middle(Fig. 10.3) the duration of the message, thus providing the best value of the signal-to-noise ratio and, accordingly, the minimum value of the error probability.

The main requirements for choosing the type of modulation are minimizing the width of the signal spectrum and choosing a sequence of elementary signals of various information sequences with maximum difference from each other (for the selected measure format) with the possibility of simple detection. On fig. 10.28 shows the normalized spectral signal power densities for various types modulation. Characteristics are given only for the upper sideband when modulated with an unfiltered signal.

The width of the main lobe of the modulated M8K signal is ±3/4 7)n, and for the BP8K signal it is ± Th. However, the bandwidth of the M8K signal, even with optimal value modulation index ?3 \u003d 1/2 turns out to be greater than for signals with ORBC modulation (Fig. 10.28). The spectrum of the unfiltered MBK signal is 1/3 narrower than the spectrum of the RTBR and decays faster (in proportion to f " 4) than the signal with binary phase shift keying f" 2 .

The behavior of the spectrum outside the main lobe is very important when using a non-linear mode of operation of the power amplifier (PA) of the transmitter, close to the saturation mode. So the spectrum of a signal with GMSK modulation is concentrated near the carrier with an intense power drop during detuning. This allows, when generating a radio signal with GMSK modulation at the output of the PA transmitter, not to turn on the RF filter.

For BPSK signal, the value of the first next maximum of the spectral density is less than the value corresponding to the frequency of the carrier wave. A well-known rule of thumb for this type of manipulation is: 99% of the transmitted power is in the data rate band. The use of digital sequence filtering can significantly reduce the bandwidth of the transmitted signal. However, the carrier level should be chosen to avoid inter-symbol interference and spectrum enrichment. Theoretically, the spectral efficiency of BPSK modulation is one (bit/s)/Hz; in practice, a value of 1.4 (bit/s)/Hz can only be achieved by applying Nyquist pre-filtering.

Coherent detection of phase-shift keying signals requires the generation of a reference signal, for which carrier recovery schemes based on doubling or quadrupling the received signal are often used. This results in a phase ambiguity in the recovered carrier. In signals with BPSK modulation, the lack of information about the initial phase of the signal leads to the appearance of " reverse work» of the demodulator, when the demodulated signal is the inverse of the transmitted signal, which leads to the appearance of 100% errors.

phase keying

Combination of multilayer transmission methods with phase shift keying

Despite the higher information transfer rate achieved due to the increased information capacity of the symbol, multilayer transmission in pure form does not apply. It has already been noted above that interference and noise in the channel, as well as restrictions on the signal level in amplifiers, primarily affect the amplitude. For this reason, the considered method has not found application. At the same time, in combination with other methods (in particular, with frequency shifting), it gives a high effect and good noise immunity. The combination of multilevel transmission with phase modulation has received the greatest distribution. (Modulation is the process of changing the parameters of the carrier frequency (amplitude, frequency, phase); manipulation is the process of influencing the parameters of the carrier frequency with a digital signal.) This made it possible to dramatically expand the bandwidth in the subscriber area. Below we consider one of these methods - phase manipulation.

Phase keying transforms information by influencing the phase frequency signal. For example, in the simplest case of transmitting individual bits (Fig. 29), when going from 0 to 1, the phase changes by 180 °. In the situation shown in Fig. 29, a, one corresponds to a positive period at the beginning of the cycle, and zero - negative.

Rice. 29. Examples of phase keying for the cases: a) 2-PM b) 4-PM

With the 4-PM phase shift keying method (Fig. 29, 6), the phase shift is 45 °, while it is encoded as follows:

for 11 - shift +45° (π/4);

for 10 - shift +135° (З π /4);

at 00 - shift +225° (-З π /4);

at 01 - shift 315 ° (-π / 4).

The phase is determined by measuring the value of the cosine signal at the beginning of the period.

On the left in the figures are shown pie charts sinusoidal signal (in Fig. 29, b, the signal shows cosine values, and therefore is shifted by 90 °). The change in the value of the sinusoidal signal is compared with the value displayed on the circle. In this case, with the change in time, an imaginary vector (a radius placed in the center of the circle) rotates counterclockwise. The dot on the circle shows the value of the sinusoidal signal at that moment in time. The bottom point on the circle corresponds to the minimum negative value amplitude and is associated with a discrete unit, and the highest point corresponds to the maximum value and is identified with a discrete zero. For a diagram showing a fourfold phase shift, 4 points are marked.

Unlike amplitude modulation, phase shift keying is less affected by the transmission level (amplitude) and frequency. It is most adapted to the transmission of multilevel signals, which, as follows from the previous section, allow you to increase the information transfer rate without increasing the line rate in the channel. At the same time, it is strongly influenced by the inductive and capacitive parameters of the cable. For example, the already mentioned load coils, improving the parameters of a normal signal, introduce an artificial inductance, which, in turn, affects the signals compressed using phase keying.

The shape of the modulated signal during phase keying is determined by the formula:

where \u003d 2π / p - the value by which the phases of adjacent signals differ; tn - symmetrical n-level signal in the form of pulses direct current without returning to zero, and the values ​​of the levels are ±1, ±3, etc.

The last expression is easily reduced to the form:

The formula makes it possible to reduce the process of phase manipulation to a combination of amplitude modulation of two signal sequences.

The representation of a sinusoidal oscillation as a linear combination of sinusoidal and cosine oscillations with a zero initial phase is commonly called a quadrature representation.

Functions sovf ietf for each cycle of signal transmission are constant, ᴛ.ᴇ. play the role of coefficients that take values ​​in accordance with the signal level. The functions and play the role of carrier frequencies shifted by 90°. When two amplitude-modulated signals are added, one phase-modulated function is obtained. Cosine waveforms are commonly referred to as ʼʼin-phaseʼʼ or ʼʼB-signalsʼʼ, while sinusoidal signals are called ʼʼquadratureʼʼ or ʼʼK-signalsʼʼ.

The block diagram of a phase modulator (PM) built according to this principle is shown in fig. thirty.

Rice. 30 Generalized phase modulator circuit: MB(t) - B-signal; Mk(t) - K-signal

Phase manipulation - concept and types. Classification and features of the category "Phase manipulation" 2017, 2018.

We said that these signals are obtained as a special case of frequency modulation with a digital modulating signal in the form of a sequence of pulses corresponding to zeros and ones of the binary stream. Since the pulses of the modulating signal change sign when the information bit changes, we got frequency shift keying.
Drawing an analogy, we can consider signals with phase shift keying (PSK), if we apply as a modulating signal to a phase modulator digital signal. In this article we will talk about binary phase shift key BPSK. This type modulation has found very wide application due to the high noise immunity and simplicity of the modulator and demodulator. In the domestic literature, BPSK modulation is referred to as PSK-2.

Binary Phase Shift Keying Signals

Consider a signal in the form of a sequence of pulses digital information, as shown in Figure 1.

Figure 1: Unipolar and bipolar digital signal

The upper graph shows a unipolar digital signal, in which the informational logical zero corresponds to , and the lower graph shows a bipolar digital signal, in which the informational logical zero corresponds to .
We apply a digital signal as a modulating signal to the phase modulator, as shown in Figure 2 with a phase deviation equal to rad.

Figure 2: Shaping a BPSK signal based on a phase modulator

Since it takes only values ​​equal to 0 and 1, then the in-phase and quadrature components of the complex envelope of the BPSK signal are equal to:
and the block diagram of the modulator can be simplified, as shown in Figure 3.

Figure 3: Simplified structural scheme BPSK modulator

The attentive reader will notice that this scheme is exactly the same as the previously discussed AM scheme with carrier suppression (DSB), with a modulating signal. Explanatory plots of the BPSK shaper are shown in Figure 4.

Figure 4: Explanatory plots of the BPSK modulator

Information is transmitted at a bit/s rate, the duration of one pulse of digital information is . The original modulating signal is multiplied by the carrier wave (in the figure) and we get a phase-shift keyed signal with a phase jump of rad. We observed the same phase jump when forming a DSB signal. Thus BPSK modulation is a degenerate type of phase shift keying that is the same as balanced amplitude modulation with a bipolar digital modulating signal.

Spectrum and vector diagram of a BPSK signal

Since the BPSK signal can be represented as a DSB signal, its spectrum is the spectrum of a digital bipolar modulating signal transferred to the carrier frequency. Figure 5 shows the spectrum of the BPSK signal at the information rate and carrier frequency . Figure 5 clearly shows that the spectrum of the BPSK signal has a main lobe and slowly decreasing side lobes. Figure 6 shows the main relationships between the BPSK spectrum and the parameters of the original modulating signal.

So the main lobe of the BPSK spectrum has a width equal to twice the information transfer rate and is symmetrical with respect to the carrier frequency. The level of the maximum (first) sidelobe of the spectrum is -13 dB. It can also be said that the width of the side lobes is equal to .
Consider a vector diagram of a BPSK signal. According to expression (1), the in-phase component of the complex envelope of the BPSK signal is , and the quadrature component is . In this case, it takes on the values ​​, then the vector diagram of the BPSK signal is shown in Figure 7.

Figure 7: Vector diagram of a BPSK signal

The complex envelope vector can take one of two values ​​(when transmitting an information zero) and when transmitting an information unit.

Relative (differential) binary phase shift keying (DBPSK)

When transmitting information using BPSK, it is required to use tracking systems to demodulate the signal. In this case, incoherent receiving devices are often used, which are not phase-matched with the master oscillator on the transmitting side, and, accordingly, cannot track random turn phase as a result of propagation that goes beyond the interval . For example, consider Figure 8.

Figure 8: Explanations for non-coherent BPSK reception

The original BPSK vector diagram (in the case of PSK signals, the vector diagram is often called a constellation) is shown in Figures 8a and 8d. Red indicates the value corresponding to informational zero, and blue one. As a result of propagation, the signal will acquire a random initial phase and the constellation will turn to some angle. Figure 8b shows the case when the rotation of the constellation lies in the range from to rad. In this case, with non-coherent reception, the entire constellation will be rotated as shown by the arrows in Figure 8b. Then after turning the constellation will take starting position and the information will be demodulated correctly. Figure 8e shows the case when the rotation of the constellation lies in the range from to rad. In this case, when receiving, the constellation will also be rotated for a horizontal position, but as follows from Figure 8e, the information zeros and ones will be mixed up.
In order to eliminate the confusion of information symbols, relative keying is used, or as it is also called differential BPSK (DBPSK). The essence of relative manipulation is that it is not the bit of information itself that is encoded, but its change. The structure of a data transmission system using DBPSK is shown in Figure 9.

Figure 9: Structure of a data communication system using DBPSK

The original bit stream is differentially encoded, then modulated with BPSK and demodulated at the receiving end by a non-coherent BPSK demodulator. The demodulated stream passes through the differential decoder and receives the received stream .
Consider the differential encoder shown in Figure 10.

Figure 10: Differential encoder

The summation is done modulo two, which corresponds to the logical XOR (exclusive OR). The designation means a delay of one bit of information. An example of differential coding is shown in Figure 11.

Figure 11: Example of differential bitstream encoding

The original bit stream is 011100101, we got 010111001 at the output of the differential encoder. The first bit (in the example above, the first 0 is not encoded), then the first bit is added modulo two of the previous bit at the output of the encoder and the current bit at the input. For differential decoding, it is necessary to do the reverse procedure according to the scheme shown in Figure 12 (the structure of the differential decoder is shown in Figure 9).

Figure 12: Example of differential bitstream decoding

As can be seen from the encoded bitstream 010111001, we received the original 011100101. Now consider a differential decoder if we invert all bits of the encoded stream on the receiving side, i.e. instead of 010111001 we will take 101000110. This is clearly shown in Figure 13.

Figure 13: Example of differential decoding with received stream inversion

It clearly follows from Figure 13 that when all bits of information are mixed up at the output of a differential decoder, the information is not distorted (with the exception of the first bit, shown in red), and this is the undoubted advantage of DBPSK, which can significantly simplify transmitting and receiving devices. But it is also necessary to say about the disadvantages of differential coding. The main disadvantage of DBPSK compared to BPSK is the lower noise immunity, since the reception errors propagate during the decoding stage.
Consider an example. Let the original stream be 011100101, the encoded stream be 010111001. Let the fourth bit of the encoded stream be received with an error, then the input of the decoder will be 010101001. And as a result of decoding, two whole bits will be decoded with an error (see Figure 14).

Figure 14: Reception error propagation in DBPSK decoding

Thus, we considered signals with binary phase shift keying (BPSK) and showed that BPSK is a special case of PSK with an input signal in the form of a stream of bipolar pulses, which is degenerate and reduces to a DSB signal. We have considered the BPSK spectrum and its spectral characteristics: main lobe width, side lobe level. The concept of relative or differential binary phase shift keying DBPSK was also introduced, which allows eliminating symbol inversion during incoherent reception at the decoding stage, but degrades the noise immunity of DBPSK compared to BPSK due to error propagation at the decoding stage.

Digital phase modulation is a versatile and widely used technique wireless transmission digital data.

In the previous article, we saw that we can use discrete changes in the amplitude or frequency of a carrier as a way to represent ones and zeros. Not surprisingly, we can also represent digital data with a phase; this method is called phase shift keying (PSK).

Binary phase shift keying

The simplest type of PSK is called binary phase shift keying (BPSK), where "binary" refers to the use of two phase shifts (one for logic one and one for logic zero).

We can intuitively recognize that the system will be more reliable if the separation between these two phases is large - of course, it will be difficult for the receiver to distinguish between a symbol with a phase shift of 90° and a symbol with a phase shift of 91°. We have a 360° phase range to work with, so maximum difference between the phases of a logical unit and a logical zero is 180 °. But we know that switching a sine wave 180° is the same as inverting it; thus, we can think of BPSK as simply inverting the carrier signal in response to one logical state and leaving it in its original state in response to another logical state.

To take the next step, we remember that multiplying a sinusoid by a negative unit is the same as inverting it. This results in the possibility of implementing BPSK using the following basic hardware configuration:

Basic scheme for obtaining a BPSK signal

However, this scheme can easily lead to high slope transitions in the carrier waveform: if a logic state transition occurs when the carrier signal is in its maximum value, the carrier signal voltage should quickly go to the minimum value.

High slope in the BPSK waveform when the logic state of the baseband signal changes

Such high slope events are undesirable because they create energy at high frequency components that can interfere with other RF signals. In addition, amplifiers have a limited ability to produce sudden changes in output voltage.

If we improve the above implementation with two additional features, then we can provide smooth transitions between characters. First, we need to make sure that the period of a digital bit is equal to one or more full cycles of the carrier signal. Second, we need to synchronize the digital transitions with the carrier signal. With these improvements, we could design the system so that a 180° phase change occurs when the carrier signal is at (or close to) the zero crossing.

QPSK

BPSK transmits one bit per character, which is what we are used to. Everything we discussed about digital modulation, assumed that the carrier signal changes depending on whether the digital voltage logic high or low, and the receiver recreates the digital data, interpreting each character as a 0 or 1.

Before discussing quadrature phase shift keying (QPSK), we need to introduce the following important concept: there is no reason why one symbol can only carry one bit. It is true that the world of digital electronics is built around circuits where the voltage is at one extreme or the other, so that the voltage is always one digital bit. But the radio signal is not digital; rather, we use analog signals for digital data transmission, and it is perfectly acceptable to develop a system in which analog signals are encoded and interpreted in such a way that one character represents two (or more) bits.

The advantage of QPSK is more high speed data transfer: if we keep the same symbol duration, we can double the data transfer rate from the transmitter to the receiver. The disadvantage is the complexity of the system. (You might think that QPSK is more susceptible to bit errors than BPSK because the separation between possible values it has less. This is a reasonable guess, but if you look at their math, it turns out that the error probabilities are actually very similar.)

Options

QPSK modulation is of course effective method modulation. But it can be improved.

Phase jumps

Standard QPSK modulation ensures that symbol transitions occur with high slope; since the phase jumps can be ±90°, we cannot use the approach described for the 180° phase jumps generated by BPSK modulation.

This problem can be mitigated by using one of the two QPSK options. Offset QPSK (OQPSK), which involves adding a delay to one of the two digital data streams used in the modulation process, reduces the maximum phase jump to 90°. Another option is π/4-QPSK, which reduces the maximum phase jump to 135°. Thus, OQPSK has the advantage of reducing phase discontinuities, but π/4-QPSK wins because it is compatible with differential coding (discussed below).

Another way to deal with breaks between characters is to implement additional processing signals, which creates smoother transitions between characters. This approach is included in a modulation scheme called minimum shift keying (MSK) and an enhancement to MSK known as Gaussian MSK (GMSK).

Differential coding

Another difficulty is that the demodulation of PSK signals is more difficult than FSK signals. Frequency is "absolute" in the sense that changes in frequency can always be interpreted by analyzing signal changes over time. Phase, however, is relative in the sense that it does not have a universal reference point - the transmitter generates phase changes with respect to one point in time, while the receiver can interpret phase changes with respect to another point in time.

The practical manifestation of this is that if there are differences between the phases (or frequencies) of the oscillators used for modulation and demodulation, the PSK becomes unreliable. And we have to assume that there will be phase differences (unless the receiver includes a carrier recovery circuit).

Differential QPSK (DQPSK, differential QPSK) is a variant that is compatible with non-coherent receivers (i.e., receivers that do not synchronize the demodulation generator with the modulation generator). Differential QPSK encodes data by creating a certain phase offset from the previous symbol so that the demodulation circuit analyzes the phase of the symbol using a reference point that is common to both the receiver and the transmitter.

Summary

• Binary phase shift keying (BPSK) is a simple modulation technique that can transmit one bit per symbol.
• Quadrature phase shift keying (QPSK) is more complex, but it doubles the data rate (or achieves the same data rate with half the bandwidth).
• Quadrature phase-shift keying (OQPSK), π/4-QPSK, frequency modulation minimum phase shift (MSK) are modulation schemes that mitigate the effects of high-slope carrier signal voltage changes between symbols.
• Differential QPSK (DQPSK) uses the phase difference between adjacent symbols to avoid problems due to lack of phase synchronization between the transmitter and receiver.

AMn · FMn KAM FSK GMSK
OFDM COFDM TCM AIM DM PCM ΣΔ PWM PWM PIM FHSS DSSS CSS

Phase manipulation(FMN, eng. phase-shift keying (PSK)) - one of the types of phase modulation, in which the phase of the carrier oscillation changes abruptly depending on the information message.

Description

The phase-shift keyed signal has the following form:

$s\_m(t)=g(t)\backslash cos,$

where $g(t)$ determines the signal envelope; $\backslash varphi\_m(t)$ is the modulating signal. $\backslash varphi\_m(t)$ can take $M$ discrete values. $f\_c$- carrier frequency ; $t$- time.

If a $M=2$, then the phase manipulation is called binary phase shift keying(BPSK, B-Binary - 1 bit per 1 phase change) if $M=4$ - quadrature phase shift keying(QPSK, Q-Quadro - 2 bits per phase change), $M=8$(8-PSK - 3 bits per 1 phase change), etc. Thus, the number of bits $n$, transmitted by one phase jump, is the power to which two is raised when determining the number of phases required for transmission $n$- ordinal binary number.

phase-shift keyed signal $s\_i(t)$ can be viewed as a linear combination of two orthonormal signals $y\_1$ and $y\_2$ :

$S\_m(t)=S\_1\; Y\_1+S\_2\; Y\_2,$

$Y\_1(t)=\backslash sqrt(\backslash frac(2)(E\_g))S\_1(t)\backslash cos,$ $Y\_2(t)=-\backslash sqrt(\backslash frac(2)(E\_g))S\_2(t)\backslash sin.$

So the signal $S\_m(t)$ can be considered a two-dimensional vector $$. If the values $S\_1(m,\backslash ;M)$ set aside on the horizontal axis, and the values $S\_2(m,\backslash ;M)$- vertically, then points with coordinates $S\_1(m,\backslash ;M)$ and $S\_2(m,\backslash ;M)$ will form the spatial diagrams shown in the figures.

BPSK Gray Coded.svg

Binary Phase Shift Keying (BPSK)

QPSK Gray Coded.svg

Quadrature Phase Shift Keying (QPSK)

8PSK Gray Coded.svg

Octal Phase Shift Keying (8-PSK)

Binary phase shift keying

Coherent detection

Bit Error Probability(English) BER-Bit error rate ) for binary PSK in a channel with additive white Gaussian noise (AWGN) can be calculated by the formula:

$P\_b=Q\backslash left(\backslash sqrt(\backslash frac(2E\_b)(N\_0))\backslash right),$

$Q(x)=\backslash frac(1)(\backslash sqrt(2\backslash pi))\backslash int\backslash limits\_x^\backslash infty\; e^(-\backslash frac(t^2)(2))\backslash ,dt.$

Since there is 1 bit per symbol, the error probability per symbol is calculated using the same formula.

In the presence of an arbitrary phase change introduced by the communication channel, the demodulator is unable to determine which constellation point corresponds to 1s and 0s. As a result, data is often differentially encoded prior to modulation.

Incoherent detection

In the case of non-coherent detection, differential binary phase shift keying is used.

Implementation

Binary data is often transferred from following signals:

$s\_0(t)=\backslash sqrt(\backslash frac(2E\_b)(T\_b))\backslash cos(2\backslash pi\; f\_c\; t)$ for binary "0"; $s\_1(t)=\backslash sqrt(\backslash frac(2E\_b)(T\_b))\backslash cos(2\backslash pi\; f\_c\; t+\backslash pi)=-\backslash sqrt(\backslash frac(2E\_b)(T\_b))\backslash cos(2\backslash pi\; f\_c\; t\; )$ for binary "1",

where $f\_c$ is the frequency of the carrier wave.

Quadrature phase shift keying

π/4-QPSK

Here are two separate constellations using Gray coding, which are rotated 45° relative to each other. Usually, even and odd bits are used to determine the points of the corresponding constellation. This reduces the maximum phase jump from 180° to 135°.

On the other hand, the use of π/4-QPSK results in simple demodulation and hence it is used in systems cellular communication with time division channels.

FSK of higher orders

FSK with an order greater than 8 is rarely used.

Differential PSK

When implementing PSK, the problem of constellation rotation may occur, for example, in continuous transmission without synchronization. For solutions similar problem coding based not on the position of the phase, but on its change can be used.

For example, for DBPSK, the phase changes by 180° for the transmission "1" and remains unchanged for the transmission "0".

see also

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Notes

Literature

• Prokis, J. Digital communications = Digital Communications / Klovsky D. D. - M .: Radio and communication, 2000. - 800 p. - ISBN 5-256-01434-X.
• Sklyar, Bernard. Digital communication. Theoretical basis and practical use= Digital Communications: Fundamentals and Applications. - 2nd ed. - M .: "Williams", 2007. - S. 1104. - ISBN 0-13-084788-7.
• Feer K. Wireless digital communication. Modulation and Spread Spectrum Methods = Wireless Digital Communications: Modulation and Spread Spectrum Applications. - M .: Radio and communication, 2000. - 552 p. - ISBN 5-256-01444-7.

Links

An excerpt characterizing Phase Keying

- How can I tell you, - Natasha answered, - I was in love with Boris, with a teacher, with Denisov, but this is not at all the same. I am calm, firm. I know that there are no people better than him, and I feel so calm, good now. Not at all like before...
Nikolai expressed his displeasure to Natasha that the wedding had been postponed for a year; but Natasha attacked her brother with bitterness, proving to him that it could not be otherwise, that it would be bad to enter the family against the will of her father, that she herself wanted it.
“You don’t understand at all,” she said. Nicholas fell silent and agreed with her.
Her brother was often surprised looking at her. It was not at all like she was a bride in love separated from her fiancé. She was even, calm, cheerful, completely as before. This surprised Nikolai and even made him look incredulously at Bolkonsky's matchmaking. He did not believe that her fate had already been decided, especially since he had not seen Prince Andrei with her. It always seemed to him that something was not right in this proposed marriage.
"Why the delay? Why didn't you get engaged?" he thought. Having talked once with his mother about his sister, he, to his surprise and partly to his pleasure, found that his mother, in the same way, in the depths of her soul, sometimes looked with distrust at this marriage.
“He writes,” she said, showing her son a letter from Prince Andrei with that hidden feeling of hostility that a mother always has against her daughter’s future marital happiness, “writes that she will not arrive before December. What kind of business could hold him back? That's right, a disease! Health is very weak. Don't tell Natasha. Don't look at how cheerful she is: this is the last girl's time, and I know what happens to her every time we receive his letters. But God willing, everything will be fine, - she concluded every time: - he is an excellent person.

The first time of his arrival, Nikolai was serious and even boring. He was tormented by the imminent need to intervene in these stupid household affairs for which his mother had called him. In order to get this burden off his shoulders as soon as possible, on the third day of his arrival, he angrily, without answering the question where he was going, went with frowning eyebrows to Mitenka's wing and demanded from him the accounts of everything. What these accounts of everything were, Nikolai knew even less than Mitenka, who had come in fear and bewilderment. The conversation and accounting of Mitenka did not last long. The headman, the elector and the zemstvo, who were waiting in the ante-room of the wing, heard with fear and pleasure at first how the young count’s voice, which seemed to rise ever higher, hummed and crackled, heard abusive and terrible words, pouring out one after another.
- Rogue! Ungrateful creature! ... I will chop up a dog ... not with my father ... robbed ... - etc.
Then these people, with no less pleasure and fear, saw how the young count, all red, with bloodshot eyes, pulled Mitenka by the collar, with great dexterity, with great dexterity, between his words, pushed him in the behind and shouted: “Get out! so that your spirit, bastard, is not here!
Mitenka flew headlong down the six steps and ran into the flower bed. (This flowerbed was a well-known area for saving criminals in Otradnoye. Mitenka himself, when he arrived drunk from the city, hid in this flowerbed, and many residents of Otradnoye, hiding from Mitenka, knew the saving power of this flowerbed.)
Mitenka's wife and sisters-in-law, with frightened faces, leaned out into the hallway from the door of the room, where a clean samovar was boiling and the clerk's high bed stood under a quilted blanket sewn from short pieces.
The young count, panting, paying no attention to them, walked past them with resolute steps and went into the house.
The countess, who immediately learned through the girls about what had happened in the wing, on the one hand, calmed down in the sense that now their condition should get better, on the other hand, she was worried about how her son would endure this. She tiptoed to his door several times, listening to him smoke pipe after pipe.
The next day the old count called his son aside and said to him with a timid smile:
- Do you know, you, my soul, got excited in vain! Mitenka told me everything.
"I knew, thought Nikolai, that I would never understand anything here in this stupid world."
- You were angry that he did not enter these 700 rubles. After all, he wrote them in transport, and you didn’t look at the other page.
- Daddy, he's a scoundrel and a thief, I know. And what he did, he did. And if you don't want me, I won't tell him anything.
- No, my soul (the count was also embarrassed. He felt that he was a bad manager of his wife's estate and was guilty before his children, but did not know how to fix it) - No, I ask you to take care of business, I'm old, I ...
- No, papa, you will forgive me if I did something unpleasant for you; I can do less than you.
“To hell with them, with these men and money, and transports along the page,” he thought. Even from a corner of six kush, I once understood, but from the page of transport - I don’t understand anything, ”he said to himself, and since then he has no longer intervened in business. Only once did the countess call her son to her, inform him that she had Anna Mikhailovna's bill for two thousand and asked Nikolai what he was thinking of doing with him.
“But how,” Nikolai answered. – You told me that it depends on me; I do not love Anna Mikhailovna and I do not love Boris, but they were friendly with us and poor. So that's how! - and he tore the bill, and with this act, with tears of joy, he made the old countess sob. After that, young Rostov, no longer intervening in any business, with passionate enthusiasm, took up the still new for him cases of dog hunting, which in large sizes was instituted by the old count.

There were already winters, morning frosts shackled the ground moistened with autumn rains, already the greenery had become narrower and bright green separated from the stripes of turning brown, knocked out by cattle, winter and light yellow spring stubble with red stripes of buckwheat. The peaks and forests, which at the end of August were still green islands between the black fields of winter and stubble, became golden and bright red islands in the midst of bright green winters. The hare was already halfway lost (molted), the fox broods began to disperse, and the young wolves were larger than the dog. It was the best hunting time. The dogs of the hot, young hunter Rostov not only entered the hunting body, but also knocked out so that in general council The hunters decided to give the dogs a rest for three days, and on September 16 to go on a trip, starting from the oak forest, where there was an untouched wolf brood.
This was the state of affairs on the 14th of September.
All that day the hunt was at home; it was frosty and poignant, but in the evening it began to rejuvenate and warmed up. On September 15, when young Rostov looked out the window in the morning in a dressing gown, he saw such a morning, better than which nothing could be better for hunting: as if the sky was melting and descending to the ground without wind. The only movement that was in the air was the quiet movement from top to bottom of descending microscopic drops of mist or mist. Transparent drops hung from the bare branches of the garden and fell on the newly fallen leaves. The ground in the garden, like a poppy, turned glossy wet black, and at a short distance merged with a dull and damp cover of fog. Nikolay went out onto the porch, wet with dirt, which smelled of withering forest and dogs. The black-spotted, broad-assed bitch Milka, with big black bulging eyes, saw her master, got up, stretched back and lay down like a brown, then unexpectedly jumped up and licked him right on the nose and mustache. Another greyhound dog, seeing the owner from the colored path, arching its back, quickly rushed to the porch and raising the rule (tail), began to rub against Nikolai's legs.
- Oh goy! - that inimitable hunting echo was heard at that time, which combines both the deepest bass and the thinnest tenor; and from around the corner came Danilo, a hunter and hunter, trimmed in Ukrainian brackets, a gray-haired, wrinkled hunter with a bent rapnik in his hand and with that expression of independence and contempt for everything in the world that only hunters have. He took off his Circassian hat in front of the master, and looked at him contemptuously. This contempt was not offensive to the master: Nikolai knew that this Danilo, who despised everything and stood above all, was still his man and hunter.