Quadrature modulation (QAM). Signal constellation

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Description

The radio signal is represented in the form of a two-dimensional dot plot on a complex plane, the points on which are all possible symbols, represented in geometric form. More abstractly, the diagram shows all the values that can be sampled by a given manipulation scheme, as points on a complex plane. The constellations generated by RF measurements can be used to determine the type of manipulation, the type of interference, and the level of distortion.

When the transmitted symbol is represented as a complex number and when the sine and cosine signal of the carrier frequency is modulated with real and imaginary parts, respectively, the symbol can be transmitted by two carriers with the same frequency. Such carriers are often called quadrature... Coherent detector ( ) is able to demodulate both carriers independently. The principle of using two independently modulated carriers is at the heart of Quadrature Modulation. In simple phase shift keying, the phase of the modulating symbol becomes the phase of the carrier signal.

When symbols are represented as complex numbers, they can be represented as points on the complex plane. The real and imaginary axes are often called in phase or I-axis and quadrature(quadrature) or Q-axis. By plotting points from multiple symbols, a constellation can be obtained. Points in a diagram are often referred to as signal points(or constellation points). They represent many modulating symbols, that is modulating alphabet.

Trellis coded modulation

When using block or convolutional coding, the noise immunity of radio communication is increased by expanding the frequency band and complicating the radio equipment without increasing the signal-to-noise ratio (SNR). To maintain noise immunity at the same SNR, it is possible to reduce the used frequency band and simplify the radio equipment using trellis-coded modulation (TCM), which was first developed in 1982 by Ungerbock. At the heart of TCM is a collaborative process of coding and modulation.

If a combined encoder / modulator is used, the general structure of which is shown in the figure, then the b0 bit allows you to select one of the two constellations that were obtained during the first division. Further, the selection is determined depending on the b1 and b2 bits.

Application

Consider detection based on the maximum likelihood method. When receiving a radio signal, the demodulator evaluates the received symbol, which is distorted during transmission or reception (for example, due to additive white Gaussian noise, fading, multipath propagation, attenuation, interference and imperfection of radio equipment). The demodulator selects the best fit to the transmitted signal, i.e. the nearest point of the constellation in terms of the Euclidean metric). Thus, if the signal distortion is strong enough, then a point different from the transmitted one can be selected, and the demodulator will give an incorrect result. Thus, the distance between the two nearest points of the constellation determines the noise immunity of manipulation.

For the purpose of analyzing the received signals, the constellation can simplify the detection of certain types of signal distortion. for instance

Gaussian noise appears as blurry constellation points
Incoherent single frequency interference looks like circles instead of a constellation point
Phase distortion is seen as signal points distributed around a circle.
Attenuation of the signal leads to the fact that the points located at the corners are closer to the center than they should be.

Signal constellations give a picture similar to eye diagram for one-dimensional signals. Eye diagrams are used to determine jitter in one modulation dimension.

Literature

Prokis J. Digital communication. - Per. from English // Ed. D. D. Klovsky. - M .: Radio and communication, 2000. - 800 p. - ISBN 5-256-01434-X
Sklyar B. Digital communication. Theoretical foundations and practical application. - Per. from English - M .: Publishing house "Williams", 2003. - 1104 p. -

By representing the transmitted symbol in the form of a complex number and by modulating the cosine and sine signals of the carrier frequency, respectively, the real and imaginary parts, the symbol can be transmitted by two carriers with the same frequency. Such carriers are often called quadrature... Coherent detector ( ) is able to demodulate both carriers independently. The principle of using two independently modulated carriers is at the heart of Quadrature Modulation. In simple phase shift keying, the phase of the modulating symbol becomes the phase of the carrier signal.

Trellis coded modulation

When using block or convolutional coding, the noise immunity of radio communication is increased by expanding the frequency band and complicating the radio equipment without increasing the signal-to-noise ratio (SNR). To maintain noise immunity at the same SNR, it is possible to reduce the used bandwidth and simplify the radio equipment using trellis coded modulation (TCM), which was first developed in 1982 by Ungerbock. At the heart of TCM is a collaborative process of coding and modulation.

Application

Consider detection based on the maximum likelihood method. When receiving a radio signal, the demodulator evaluates the received symbol, which is distorted during transmission or reception (for example, due to additive white Gaussian noise, fading, multipath propagation, attenuation, interference and imperfections of radio equipment). The demodulator selects the best fit to the transmitted signal, i.e. the nearest point of the constellation in terms of the Euclidean metric). If the signal distortion is strong enough, then a point different from the transmitted one can be selected, and the demodulator will give an incorrect result. Thus, the distance between the two nearest points of the constellation determines the noise immunity of manipulation.

For the purpose of analyzing the received signals, the constellation can simplify the detection of certain types of signal distortion. For instance,

Gaussian noise appears as blurry constellation points
Incoherent single frequency interference looks like circles instead of a constellation point
Phase distortions are seen as signal points distributed around a circle.
Attenuation of the signal leads to the fact that the points located at the corners are closer to the center than they should be.

Signal constellations give a picture similar to eye diagram for one-dimensional signals. Eye diagrams are used to determine jitter in one modulation dimension.

Write a review on the article "Signal Constellation"

Literature

Prokis, J. Digital communication = Digital Communications / Klovsky D. D .. - M .: Radio and communication, 2000. - 800 p. - ISBN 5-256-01434-X.
Sklyar B. Digital communication. Theoretical Foundations and Practical Application = Digital Communications: Fundamentals and Applications. - 2nd ed. - M .: Williams, 2007 .-- 1104 p. - ISBN 0-13-084788-7.

Excerpt characterizing the Signal Constellation

“The thing is, your mom wasn’t here,” Stella whispered softly. - We met your mother where you "failed" here. They are very worried about you, because they cannot find you, so we offered to help. But, as you can see, we were not careful enough, and got ourselves into the same terrible situation ...
- How long have you been here? Do you know what they will do with us? - Trying to speak confidently, I asked quietly.
- We recently ... He always brings new people, and sometimes small animals, and then they disappear, and he brings new ones.
I looked at Stella with horror:
- This is a very real, real world, and a completely real danger! .. This is not the innocent beauty that we created! .. What are we going to do?
- Leave. - Again the baby repeated stubbornly.
- We can try, right? Yes, and grandmother will not leave us if it is really dangerous. Apparently, we can still get out on our own, if she does not come. Don't worry, she won't leave us.
I would have her confidence! .. Although usually I was far from shy, but this situation made me very nervous, since not only we were here, but also those for whom we came to this horror. And how to get out of this nightmare - I, unfortunately, did not know.
- There is no time here, but it usually comes at the same interval, approximately as there were days on earth. - Suddenly the boy answered my thoughts.
- Have you already been today? - clearly delighted, asked Stella.
The boy nodded.
- Well, let's go? - she looked at me attentively and I realized that she was asking to "put on" my "protection" on them.
Stella was the first to stick her red head out ...
- Nobody! - she was delighted. - Wow, what a horror it is! ..
I, of course, could not bear it and climbed after her. There really was a real "nightmare"! .. Near our strange "place of confinement", in a completely incomprehensible way, hung upside down in "bundles", were hanging human beings ... They were hung by their legs, and created, as it were, an inverted bouquet ...
We came closer - none of the people showed signs of life ...
- They are completely "pumped out"! - Stella was horrified. - They have not even a drop of vitality left! .. That's it, let's get away !!!
We rushed, as much as we could, somewhere to the side, absolutely not knowing where we were running, just further away from all this blood-freezing horror ... even worse, horror ...
Suddenly it darkened sharply. Blue-black clouds rushed across the sky, as if driven by a strong wind, although there was no wind yet. In the depths of the black clouds blazing lightning blazed, the tops of the mountains blazed with a red glow ... Sometimes the swollen clouds ripped open against the evil peaks and dark brown water poured from them like a waterfall. This whole scary picture reminded, the most creepy of the creepy, nightmares ...
- Daddy, dear, I'm so scared! - screeched thinly, forgetting his former belligerence, the boy.
Suddenly one of the clouds "broke", and a blindingly bright light blazed out of it. And in this light, in a sparkling cocoon, the figure of a very thin youth, with a face sharp as a knife blade, approached. Everything around him shone and shone, from this light black clouds "melted", turning into dirty, black scraps.
- Blimey! - Stella shouted joyfully. - How does he do it ?!
- Do you know him? - I was incredibly surprised, but Stella shook her head.
The young man sank down next to us on the ground and asked with an affectionate smile:
- Why are you here? This is not your place.
- We know we were just trying to get to the top! - already in full twitter the joyful Stella. - Will you help us get back upstairs? .. We definitely need to get home quickly! And then the grandmothers are waiting for us there, and now they are also waiting, but different.
The young man, meanwhile, for some reason very carefully and seriously examined me. He had a strange, piercing look, which somehow made me uncomfortable.
- What are you doing here, girl? He asked softly. - How did you manage to get here?
- We were just walking. - I answered honestly. “And so they were looking for them. - Smiling at the "foundlings", she pointed at them with her hand.
“But you’re alive, aren’t you?” - the savior could not calm down.
- Yes, but I've been here more than once. - I answered calmly.
- Oh, just not here, but "above"! - laughing, my girlfriend corrected me. “We certainly wouldn't return here, would we?
- Yeah, I think this will be enough for a long time ... In any case - for me ... - I was already cringing from recent memories.
- You have to get out of here. - Again, gently, but more insistently said the young man. - Now.
A sparkling "path" stretched from him and ran straight into the glowing tunnel. We were literally drawn in, without even having time to take a single step, and after a moment we found ourselves in the same transparent world in which we found our round Leah and her mother.
- Mom, Mom, Dad is back! And Great too! .. - little Leah rolled head over heels towards us, tightly clutching the red dragon to her chest .. Her round face shone with the sun, and she herself, unable to keep her stormy happiness, rushed to dad and, hanging on him neck, squeaked with delight.
I was happy for this family that had found each other, and a little sad for all my dead “guests” who came to earth for help, who could no longer hug each other as joyfully, since they did not belong to the same worlds .. ...
- Oh, daddy, here you are! I thought you were gone! And you took it and found it! How good it is! - the shining little girl squeaked with happiness.
Suddenly a cloud flew into her happy face, and it became very sad ... And in a completely different voice, the baby turned to Stella:
- Dear girls, thank you for your dad! And for the little brother, of course! Are you going to leave now? Will you come back sometime? Here's your dragon, please! He was very good, and he fell in love with me very, very much ... - it seemed that right now poor Leah would burst into tears, so much she wanted to hold even a little bit more of this lovely marvelous dragon! .. And he was about to be taken away and there will be no more ...

Recall from Section 4.3 that the QAM signal can be expressed as

where and are the information-containing amplitudes of the quadrature carriers, and is the signal pulse. Vector representation of these signals

(5.2.73)

To determine the probability of error in QAM, we must specify the points of the signal constellation. Let's start with the KAM signal ensemble, which has points. Rice. 5.2.14 illustrates two such ensembles. The first (a) is a four-phase modulated signal, and the second (b) is a four-phase QAM signal with two amplitude levels, denoted by and, and four phase values. Since the error probability is determined by the minimum distance between a pair of signal points, we assume that for both signal constellations, and calculate the average transmitted power, based on the assumption that all signal points are equally probable. For a four-phase signal, we have

(5.2.74)

For a two-amplitude four-phase KAM, we will place points on circles of radius and. Since, we have

(5.2.75)

which coincides with the average power for a four-phase constellation. Therefore, for all practical applications, the error probability of the two signal ensembles is the same. In other words, there is no advantage of a two-amplitude QAM signal over four-phase modulation.

Rice. 5.2.14. Two 4-point signal constellations

Rice. 5.2.15. Four 8-point constellations of QAM signals

Next, consider an eight-level QAM signal. In this case, there are many possible signal constellations. Consider the four signal constellations shown in Fig. 5.2.15. All of them are characterized by two amplitudes and have minimum distances between signal points. The coordinates for each signal point, normalized by, are given in the figure. Assuming that all signal points are equally probable, we obtain for the average transmitted signal power

where are the coordinates of the signal points, normalized to. Two signal ensembles (a) and (c) in Fig. 5.2.15 contain signal points that lie on the grid of a rectangle and have Signal ensemble (b) requires the transmitted average power and ensemble (d) requires Therefore, the fourth signal ensemble requires about 1 dB less power than the first two, and by 1, 6 dB less power than the third one in order to achieve the same error rate. This constellation is known as the best 8-point QAM constellation because it requires the least power for a given minimum distance between signal points.

There are many more possibilities for choosing QAM signal points in two-dimensional space. For example, we can select circular multilevel constellations for, as shown in Fig. 4.3.4. In this case, the signal points at a given amplitude are rotated in phase relative to the signal points of adjacent amplitude levels. This constellation of 16 QAM is a generalization of the optimal constellation of 8 QAM. However, the circular constellation 16 QAM is not the best 16-point QAM constellation in an AWGN channel.

The rectangular QAM constellation has a distinct advantage in terms of ease of generation, as two AM signals transmitted on quadrature phase carriers. Moreover, it is easily demodulated. Although it is not the best QAM α-positional constellation for, the average transmitted power required to achieve a given minimum distance is only slightly greater than the average power required for the best QAM constellation. Based on these considerations, the rectangular-positional signal constellation QAM is most often used in practice.

For rectangular signal constellations with where is even, the QAM signal constellation is equivalent to the sum of two AM signals on quadrature carriers, each having signal points. Since the signals in the quadrature components can be accurately separated in the demodulator, the error probability for QAM is easily determined from the AM error probability. More specifically, the probability of a correct solution for the -positional KAM system is

(5.2.77)

where is the error probability for a-positional AM with half the average power in each QAM equivalent quadrature signal. By slightly modifying the expression for the error probability in the -positional AM, we obtain

(5.2.78)

where is the average SNR per symbol. Therefore, the probability of error per symbol for -positional QAM is

(5.2.79)

We emphasize that this result is exact for, when is even. On the other hand, if odd, there is no equivalent -positional system AM. However, there is no problem here, since it is always easier to determine the probability of error for a rectangular ensemble of signals. If we use an optimal detector that bases its decisions on the use of distance metrics defined by (5.1.49), it is relatively easy to show that the probability of error per symbol has a dense upper bound

(5.2.80)

for all, where is the average SNR per bit.

Rice. 5.2.16. Error probability per symbol for QAM

For non-rectangular QAM constellations, we can derive an upper bound for the error probability using the combined bound. Obvious upper bound

where is the minimum Euclidean distance between signal points. This border can be loose when it is large. In this case, we can approximate by replacing with, where is the largest number of the nearest points that have a distance from any point of the constellation.

It is interesting to compare the quality characteristics of QAM and AM for a given volume of signals, since both types of signals are two-dimensional. Recall that for the -positional FM, the error probability per symbol is approximated as follows:

(5.2.81)

where is SNR per symbol. For -positional KAM we can use expression (5.2.78). Since the probability of error is determined by the -function argument, we can compare the arguments for the two signal formats. The ratio of the two arguments under discussion is equal. For example, it can be seen that the 32 QAM system has a 7 dB SNR gain relative to the 32 PM system.

Fig.4.5. Constellation and envelope phase transitions QPSK and O-QPSK.

19. Why the CHMMS signal can be formed according to the quadrature scheme of the offset FM-4?

ChMMS can be considered as a special case of a coherent ChMNF with the index ChM t = 0.5. According to (4.12) and (4.14), it can be written for b 1= ± 1 and ± Df =± 1 / (4T c):

where the phase increment of the carrier wave (envelope quadrature) on the interval T c equals ± p / 2(as with offset O-QPSK) and depends on the character signs b i ≡ ± 1 modulating signal u (t). Therefore, the FMMS modulator can be implemented according to the quadrature scheme in Fig. 4.13, which provides t = 0.5 with less error than a VCO-based circuit. The implementation diagram of the quadrature modulator (4.16) is shown in Fig. 4.13.

Figure 4.13. Implementation diagram of the QMMS quadrature modulator.

20. Why is the QAM signal sensitive to the linearity of the communication channel path and what elements of the path are decisive for the implementation of this linearity?

The width of the QAM spectrum is approximately the same as the spectrum of the M-ary FM signal. However, the QAM signal can provide a lower probability of bit error, but has a large crest factor and increased requirements for the linearity of the transmitter path and communication channel.

21. The spectrum of which signal (information or PSP) determines the width of the NLS spectrum: a) in a system with direct spreading of the spectrum; b) in a system with frequency hops; c) in a system with jumps in time?

a) Direct spreading of the spectrum is carried out by multiplying the information signal u inf. (t) per pseudo-random signal r (t) generated from the MSC during the entire communication session.

b) When the spectrum of the radio signal is expanded with jumps in frequency, the frequency of the carrier oscillation changes discretely in time, taking on a finite number of different values. The sequence of its values can be considered as the bandwidth, which is formed in accordance with some code.

c) The emission of a signal with this method is performed at short time intervals T psr whose position on the time axis is determined by a pseudo-random code. The time axis is divided into frames with M windows. In one frame, the subscriber transmits information only in one of the M windows, the number of which is determined by the code allocated to the subscriber. To transmit all the information in the window, the signal bandwidth is increased by M times, i.e. spreading factor (signal base) B = M.

22. Draw the QPSK complex envelope constellation at I and Q ± 1.

Note that by changing the I and Q values, one can obtain amplitude and phase modulation(with AM I and Q change proportionally) .

If I and Q take the values +1 or -1, then the amplitude of such a signal (4.8) is constant and equal to √2, and the phase φ takes on the values shown in the signal constellation in Fig. 4.5b (in the Gray code).

Fig.4.5. Constellation and envelope phase transitions QPSK and O - QPSK.

23. How is the quadrature of the complex envelope obtained in QPSK?

Figure 4.5a shows the quadrature principle

formation of this complex amplitude from the sequence

input rectangular modulating electrical pulses with duration 2T with with values +1 or -1.

At quadrature amplitude modulation(QAM, QAM - Quadrature Amplitude Modulation) both the phase and the amplitude of the signal change, which makes it possible to increase the number of encoded bits and, at the same time, significantly increase the noise immunity. Currently, modulation methods are used in which the number of information bits encoded on one baud interval can reach 8 ... 9, and the number of signal positions in the signal space - 256 ... 512.

The quadrature representation of signals is a convenient and fairly universal means of describing them. Quadrature representation consists in expressing the oscillation by a linear combination of two orthogonal components - sinusoidal and cosine:

S (t) = x (.t) sin (wt + (p) + y (t) cos (wt + (p),
where x (t) and y (t) - bipolar discrete quantities. This discrete modulation (keying) is performed on two carrier channels shifted by 90 ° relative to each other, i.e. which are in quadrature (hence the name of the representation and the method of generating signals).

Let us explain the operation of the quadrature circuit (Fig. 6.2) using the example of the formation of four-phase FM signals (FM-4).
The original sequence of binary symbols of duration T divided into odd pulses using a shift register y, which are fed into the quadrature channel (coswt), and even - X, entering the common-mode channel (sinwt). Both sequences of pulses are fed to the inputs of the corresponding shapers of manipulating pulses, at the outputs of which sequences of bipolar pulses are formed x (t) and y (t). The manipulating impulses have an amplitude of C / d / ^ s and a duration of 2T. Impulses x (t) and y (t) are fed to the inputs of the channel multipliers, at the outputs of which two-phase (0, l) FM oscillations are formed. After summation, they form an FM-4 signal. In accordance with the method of generating the FM-4 signal, it is also called quadrature PM signal(QPSK - Quadrature PSK).

With a simultaneous change of symbols in both channels of the modulator (from 10 to 01, or from 00 to 11), the phase jump occurs in the DPSK signal by 180 ° (i).

Rice. 6.2.

Rice. 6.3.

Four-phase shifted FM(OQPSK - Offset QPSK)(Figure 6.3) avoids 180 ° phase jumps and therefore deep envelope modulation. Signal shaping in the quadrature circuit is the same as in the FM-4 modulator, except that the manipulation elements of the information sequence x (t) and y (t) shifted in time by the duration of one element T, as shown in fig. 6.3, b, c. The phase change with such a displacement of the modulating fluxes is determined by only one element of the sequence, and not two, as with FM-4. As a result, there are no 180 "phase jumps since each sequence element entering the I / Q modulator can cause a 0 °, + 90 °, or -90 ° phase change.

The expression given at the beginning of the section for describing the signal is characterized by the mutual independence of multilevel manipulating pulses x (t), y (t) in channels, i.e. a single level in one channel can correspond to a single or a zero level in the other channel. As a result, the output signal of the quadrature circuit changes not only in phase but also in amplitude. Since AMK is performed in each channel, this type of modulation is called amplitude modulation quadrature keying(QASK - Quadrature Amplitude Shift Keying) or simply quadrature amplitude modulation - QAM.

Using a geometric interpretation, each QAM signal can be represented by a vector in the signal space. Marking only the ends of the vectors, for QAM signals we obtain an image in the form of a signal point, the coordinates of which are determined by the values x (t) and y (t). The collection of signal points forms the so-called signal constellation (signal constellation).
In fig. 6.4 shows a block diagram of the U-signal constellation modulator for the case when - (0 and y (t) take values ± 1, ± 3 (4-level KAM).

Rice. 6.4.

Values ± 1, ± 3 define modulation levels and are relative. The constellation contains 16 signal points, each of which corresponds to four transmitted information bits.

The combination of levels ± 1, ± 3, ± 5 can form a constellation of 36 signal points. However, of these, the ITU-T protocols use only 16 points evenly distributed in the signal space.

There are several ways to practically implement a 4-level KAM, the most common of which is the so-called superposition modulation method(SPM - Supersposed Modulation). In the circuit that implements this method, two identical 4-phase modulators are used (Fig. 6.2). The block diagram of the SPM modulator and diagrams explaining its operation are shown in Fig. 6.5.

It is known from communication theory that with an equal number of points in the signal constellation, the spectrum of QAM signals is identical to the spectrum of PM signals. However, the noise immunity of FM and QAM systems is different. With a large number of points, the QAM system signals have better characteristics than the PM systems. The main reason for this is that the distance between signal points in the PM system is less than the distance between signal points in the QAM system.

In fig. 6.6 shows the signal constellations of the KAM-16 and FM-16 systems with the same signal power. Distance d between adjacent points of the signal constellation in the QAM system with L modulation levels is determined by the expression:
c? = v2 / (JL-l). Similarly for FM
d = 2sin (n / M), where M - number of phases.

Fig 6 5

And h of the above expressions it follows that with an increase in the value M and the same power level of the QAM system is preferable to the FM systems For example, for M = 16 (Ј = 4)

Fig 6 6

Quadrature modulation (QAM). Signal constellation

Description

Trellis coded modulation

Application

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Trellis coded modulation

Application

see also

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