CN114389928A - Amplitude-limited Gaussian-like distribution constellation diagram design method - Google Patents

Amplitude-limited Gaussian-like distribution constellation diagram design method Download PDF

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CN114389928A
CN114389928A CN202111590859.9A CN202111590859A CN114389928A CN 114389928 A CN114389928 A CN 114389928A CN 202111590859 A CN202111590859 A CN 202111590859A CN 114389928 A CN114389928 A CN 114389928A
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张昌明
余显斌
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Zhejiang Lab
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L27/00—Modulated-carrier systems
    • H04L27/32—Carrier systems characterised by combinations of two or more of the types covered by groups H04L27/02, H04L27/10, H04L27/18 or H04L27/26
    • H04L27/34—Amplitude- and phase-modulated carrier systems, e.g. quadrature-amplitude modulated carrier systems
    • H04L27/3405—Modifications of the signal space to increase the efficiency of transmission, e.g. reduction of the bit error rate, bandwidth, or average power
    • H04L27/3411—Modifications of the signal space to increase the efficiency of transmission, e.g. reduction of the bit error rate, bandwidth, or average power reducing the peak to average power ratio or the mean power of the constellation; Arrangements for increasing the shape gain of a signal set

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Abstract

The invention discloses a method for designing an amplitude-limited Gaussian-like distribution constellation diagram, and belongs to the technical field of wireless communication modulation. The method comprises the following steps: for a constellation diagram with M constellation points, each constellation point is represented by M-log2M bit mapping, setting
Figure DDA0003429863910000011
The magnitude of the wave is a function of the amplitude,
Figure DDA0003429863910000012
a phase value of where m1+m2M; selecting a factor lambda larger than 1 and setting the ith amplitude value to
Figure DDA0003429863910000013
The j phase value of
Figure DDA0003429863910000014
Wherein φ is an arbitrary value; for m bits of information, the first m1Bits are mapped to amplitude values by gray, then m2Mapping the bits into phase values through Gray to obtain an initial result of the constellation diagram; comprehensively considering PAPR and channel capacity, adjusting the value of the factor lambda and calculating the peak value of the peak value m1+m2Adjusting m on the premise of m1And m2And (4) optimizing the constellation diagram design result. The invention can make the constellation point distribution approximate to Gaussian distribution, and simultaneously restrain the amplitude value within a certain threshold, and reduces the PAPR on the premise of ensuring higher communication capacity.

Description

Amplitude-limited Gaussian-like distribution constellation diagram design method
Technical Field
The invention relates to the technical field of wireless communication modulation, in particular to a design method of an amplitude-limited Gaussian distribution-like constellation diagram.
Background
With the rapid development of 5G/6G technology, various new business and intelligent application technologies relying on wireless communication are emerging, and in this context, the demand of wireless transmission rate is explosively increased, and on average, doubles every 18 months. However, spectrum resources related to communication capacity are in increasing shortage, and therefore, it is a necessary means for future wireless communication applications to improve spectrum efficiency, i.e., transmission rate per unit bandwidth. The signal modulation mode is a core element for determining the spectrum efficiency, and increasing the modulation order based on the conventional constellation diagram can improve the spectrum efficiency, but can significantly reduce the quality of signal transmission, so how to design a signal constellation diagram with high communication efficiency and high reliability is a hot point problem which is continuously concerned by academia and industry.
The traditional high-order Modulation mode is mainly Quadrature Amplitude Modulation (QAM), and the design of a constellation diagram pursues the maximization of the minimum euclidean distance, so that each constellation point is evenly distributed on a two-dimensional complex plane within a certain signal power range, and the minimization of Symbol Error Rate (SER) under the hard decision condition can be realized. In addition, gray mapping can be conveniently implemented by QAM constellation points, and is regarded as an optimal mapping manner in the industry, which makes only one Bit difference between adjacent constellation points, so that under the condition of a certain SER and without considering Error correction, gray mapping can implement minimization of Bit Error Rate (BER). Constellation points of Amplitude-Phase Shift Keying (APSK) are also uniformly distributed on a two-dimensional plane, which is also convenient for realizing the minimum euclidean distance maximization among the constellation points under the condition of limited power, but the existing uniform APSK has no simple and general gray mapping mode, and the performance of the existing uniform APSK is difficult to guarantee, so that the application in an actual communication system is less.
Modern communication systems usually employ channel coding with good error correction performance, such as Low Density Parity Check (LDPC) code, which performs error correction based on soft decision information, and at this time, even if gray mapping is employed, SER minimization under hard decision does not correspond to minimization of final BER, and at this time, whether an error occurs or a factor determining the BER size is more mainly communication capacity. The Shannon theorem gives an upper limit of the communication capacity, which requires that the modulation signal is approximately gaussian distributed, and the modulation signal based on the uniform constellation diagram has a large difference from the gaussian distribution, so the theoretical capacity has a certain distance from the Shannon capacity, for example, 4096QAM has a performance loss of 1.32dB relative to the Shannon capacity when the spectral efficiency is 8bps/Hz, i.e. the coding efficiency is 2/3. Therefore, a non-uniform constellation diagram approaching to the Gaussian distribution characteristic is adopted for signal modulation, and larger communication capacity can be obtained theoretically.
In order to implement gray mapping while constellation pattern is approaching to gaussian distribution, a non-uniform gray APSK modulation technique is proposed in the art, which is disclosed in the following documents: liu, q.xie, k.peng, and z.yang, "APSK condensation with Gray Mapping," IEEE commu., vol.15, No.12, pp.1271-1273, dec.2011. Consider a constellation diagram with M constellation points, which can represent M-log bits2M, wherein M1Bit modulation amplitude, m2＝m-m1The mapping mode of the phase, the amplitude and the phase of the bit modulation is Gray mapping. The difference between the non-uniform gray APSK and uniform APSK constellations mainly includes two aspects: (1) the number of constellation points at each amplitude in the non-uniform Gray APSK constellation is the same, i.e.
Figure BDA0003429863890000021
The species amplitudes are all correspondingly the same
Figure BDA0003429863890000022
The phase is seeded, and the number of position constellation points is more when the amplitude of the uniform APSK is larger; (2) the amplitude distribution in the non-uniform Gray APSK constellation is in a Gaussian distribution characteristic, and the size difference between adjacent amplitudes in the uniform APSK constellation is the same.
Although the existing gray APSK modulation can obtain the communication capacity gain, the signal amplitude is theoretically distributed between zero and infinity under the gaussian distribution condition, and for the limit condition that the number of constellation points is infinite, the Peak-to-Average Power Ratio (PAPR) of the signal is infinite, and even if the number of actual constellation points is limited, the PAPR is obviously increased relative to the classical QAM modulation. In order to avoid the power amplifier entering a nonlinear working region, a larger PAPR requires a lower Signal transmission power, which results in a loss of Signal-to-Noise Ratio (SNR), so that the communication capacity gain of the existing gray APSK may not correspond to a significant system gain in an actual system, and particularly for a nonlinear sensitive communication system, the system gain is even negative compared to that of a classical QAM modulation method.
Disclosure of Invention
In order to solve the problems, the invention provides a method for designing an amplitude-limited Gaussian distribution-like constellation diagram, which approaches Gaussian distribution on the premise of amplitude limitation by improving the distribution mode of constellation points, so that the communication capacity gain advantage of the conventional Gray APSK modulation can be maintained, and the PAPR can be obviously reduced.
The invention adopts the following technical scheme:
a method for designing an amplitude-limited Gaussian-like distribution constellation diagram comprises the following steps:
1) for a constellation diagram with M constellation points, each constellation point is represented by M-log2M bit mapping, setting
Figure BDA0003429863890000023
The magnitude of the wave is a function of the amplitude,
Figure BDA0003429863890000024
a phase value of where m1+m2M; at each amplitude there is M2Each constellation point corresponds to M2A phase value of M at each phase value1Each constellation point corresponds to M1An amplitude value;
2) selecting a factor lambda larger than 1, initializing each amplitude value and phase value, wherein the ith amplitude value is:
Figure BDA0003429863890000025
the j-th phase value is:
Figure BDA0003429863890000026
where φ is an arbitrary value, M1Number of amplitude values, M2The number of phase values;
3) for m bits of information, the first m1Bits are mapped to amplitude values by gray, then m2Mapping the bits into phase values through Gray to obtain an initial result of the constellation diagram;
4) comprehensively considering signal power peak-to-average ratio (PAPR) and channel capacity, adjusting the value of the factor lambda, and calculating the peak-to-average ratio m1+m2Adjusting m on the premise of m1And m2And (4) optimizing the constellation diagram design result.
Further, initializing m according to the value of m1And m2Wherein when m is an odd number, m is initialized1(m-1)/2 and m2(m + 1)/2; when m is even number, initializing m1M/2-1 and m2＝m/2+1。
Further, in step 4), when the constellation design result is optimized by comprehensively considering the PAPR and the channel capacity, any m is given1And m2Value pairing, different amplification factors lambda correspond to a constellation diagram, and an objective function related to lambda can be established
Figure BDA0003429863890000031
The optimal magnification factor is the result of minimizing the scalar function, namely:
Figure BDA0003429863890000032
where ζ is the weighting factor, λoptIs the optimal amplification factor;
Figure BDA0003429863890000033
the minimum requirement of the target spectrum efficiency on the signal-to-noise ratio (SNR) can be obtained by calculation according to the communication capacity; PAPRdBAnd (lambda) is the signal power peak-to-average ratio.
Further, the PAPR of the signal power is calculated as:
Figure BDA0003429863890000034
further, for each m1And m2Value matching and selection of respective optimal amplification factor lambdaoptThen, the optimal amplification factor with the minimum objective function is selected from all the optimal amplification factors as the global optimal amplification factor, and the corresponding m is1And m2Value pair as optimal m1And m2The value of (a).
The invention has the beneficial effects that: according to the amplitude-limited Gaussian distribution-like constellation diagram design method, when the constellation point distribution is close to the Gaussian distribution, the amplitude value is constrained within a certain threshold, namely the probability density within the amplitude value is the result of the Gaussian distribution amplified in the same proportion, and the probability density outside the amplitude value is 0. Compared with a classical QAM modulation mode, the technical scheme of the invention can obtain the communication capacity gain caused by Gaussian shaping of constellation point distribution while retaining the superiority of Gray mapping; compared with the existing Gray APSK technology, the PAPR can be obviously reduced, and further more excellent system gain can be obtained.
Drawings
FIG. 1 is a graph of the probability density of the solution of the invention compared to a Gaussian distribution at an arbitrary amplitude a position;
FIG. 2 is a graph comparing the probability density of the amplitude distribution of the inventive arrangements with a Gaussian distribution;
FIG. 3 is a graph comparing the probability of the cumulative distribution of the amplitude of the Gaussian distribution in accordance with the inventive arrangements;
FIG. 4 is a flow chart of an embodiment of the present invention;
FIG. 5 is a diagram comparing the scheme of the present invention with the prior Gray APSK constellation diagram under the condition of 4096 constellation points;
FIG. 6 is a PAPR comparison graph of the present invention scheme with classical QAM and existing Gray APSK;
FIG. 7 is a diagram comparing the communication capacity of the present invention scheme with classical QAM and existing Gray APSK under 4096 constellation points;
FIG. 8 is a comparison graph of system cost indexes of the scheme of the present invention and classical QAM and existing Gray APSK;
fig. 9 is a diagram of the lowest system cost index when mapping the number of bits for different amplitudes according to the present invention.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The invention provides a method for designing an amplitude-limited Gaussian distribution constellation diagram, which can overcome the defect of high PAPR (peak-to-average power ratio) of the conventional Gray APSK modulation while obtaining theoretical communication capacity gain compared with the classical QAM modulation. The theory of the invention is explained by the infinite continuous distribution limit condition of the constellation point number, and then the constellation diagram is designed under the discrete distribution condition of the finite actual constellation point number.
For a two-dimensional Gaussian distribution, under the condition of signal power normalization, the probability density of the constellation point distributed at any position with the amplitude of a is
Figure BDA0003429863890000041
Wherein a is more than or equal to 0, and the amplitude value is the sum of the probability densities of all the positions of a, namely the amplitude probability density is
Figure BDA0003429863890000042
Here the maximum value of amplitude or power is theoretically infinite and the power average is 1, so the PAPR is infinite.
The invention carries out certain constraint on the distribution range of the amplitude a, so that a is more than or equal to 0 and less than or equal to amax< + ∞whereasthe cumulative distribution probability of the amplitude values if the constellation points still obey a Gaussian distribution within the constraint range
Figure BDA0003429863890000043
Which is not in accordance with the actual situation. For this purpose, a is 0 ≦ amaxWithin the range, the distribution probability density is amplified according to a certain proportion, namely the probability density of the constellation point distributed at the position with any amplitude value as a is
Figure BDA0003429863890000044
Where λ is an amplification factor and λ > 1, and the probability density of the amplitude a is
Figure BDA0003429863890000045
The cumulative distribution probability of the amplitude from 0 to a is
Figure BDA0003429863890000046
The maximum amplitude can be obtained by setting the cumulative distribution probability to 1, and then the peak power can be obtained, as shown in formula (2).
Figure BDA0003429863890000047
On the other hand, the power mean value can be obtained by derivation calculation
Figure BDA0003429863890000048
Therefore, the PAPR of the scheme of the present invention in the limit case that the number of constellation points is infinite and is continuously distributed can be expressed as shown in equation (4).
Figure BDA0003429863890000051
Fig. 1 compares the probability density curves for arbitrary amplitude a positions for a gaussian distribution and the present invention at λ 1.1 and 1.2. It can be seen that the distribution probability density is 0 when the amplitude is greater than a certain threshold, and the probability density within the threshold is enlarged according to a certain proportion, and the overall distribution is still close to the Gaussian function envelope, namely, the distribution is similar to Gaussian distribution with limited amplitude. Fig. 2 and 3 compare the corresponding magnitude distribution probability density curve and cumulative distribution probability curve.
For the modulation and demodulation process of an actual communication system, the constellation points are limited in number and only can be discretely distributed at specific positions, the distribution probability shown in the formula (1) cannot be strictly met at any amplitude value, and the key point of the constellation diagram design is how to make the constellation point distribution approach the theoretical situation as much as possible. The constellation points and the phase values of the Gray APSK under each amplitude are the same, so that Gray code mapping can be performed, the invention still performs constellation diagram design based on the thought, and a corresponding implementation flow chart is given in FIG. 4.
Consider a constellation diagram with M constellation points, which can represent M ═ log bits2M is a positive integer, where the first M1Bit-determined amplitude, m2＝m-m1The bit determines the phase, and the mapping modes of the amplitude and the phase are gray mapping. Here, the amplitude value has one in common
Figure BDA0003429863890000052
All phase values being
Figure BDA0003429863890000053
At each amplitude there is M2A constellation point corresponding to the M2A phase value of M at each phase value1A constellation point corresponding to the M1An amplitude value. In general, m1Slightly less than m2The best result of the communication capacity can be obtained, and when m is an odd number, m can be set in advance1(m-1)/2 and m2When m is an even number, m may be set in advance to (m +1)/21M/2-1 and m2＝m/2+1。
For the ith amplitude (i ═ 1,2, …, M1) With an amplitude value of aiThe cumulative distribution probability of the left and right limit positions is respectively
Figure BDA0003429863890000054
And
Figure BDA0003429863890000055
while the amplitude aiThe cumulative distribution probability of (c) can be taken as the average of the two, i.e., p (a) ═ i-1/2)/M1. The amplitude values obtained by combining equation (1) are shown in equation (5), where the amplitude values can be scaled according to the power requirement, and the gaussian-like characteristics do not change.
Figure BDA0003429863890000056
For the j-th phase, the phase value can take the value shown in equation (6), where φ is any phase value.
Figure BDA0003429863890000057
After determining the set of good amplitude and phase values, for any m bits of information, the first m1The bits are gray mapped to corresponding amplitude values, m2And mapping the bits into corresponding phase values through Gray, thereby obtaining an initial design result of the constellation diagram.
In order to obtain the maximum system benefit, i.e., the minimum system cost, the optimal amplification factor λ may be selected according to the PAPR and the communication capacity. In one aspect, the constellation point PAPR can be expressed as
Figure BDA0003429863890000061
In an actual system, all the transmission symbols belong to elements in a constellation point set, and the transmission symbols can be obtained only through waveform shaping, so that the PAPR of the transmission signals is greater than that of the constellation points. For example, considering waveform shaping using a root-raised cosine filter with a roll-off coefficient of 0.2, the PAPR may be increased by approximately 6 dB. However, under different constellation diagrams, the same waveform shaping method has the same theoretical value for the PAPR boost amount, so the present invention directly adopts the constellation point PAPR shown in formula (7) to describe the PAPR under the corresponding modulation method.
On the other hand, in the case of specifying the constellation and the SNR, the communication capacity can be calculated by averaging the mutual information values, as shown in equation (8). Wherein χ is the constellation point value set, Ex,y[·]Representing that a received symbol y is expected for a transmitted symbol x, the value set of x is χ, y can be taken on the whole two-dimensional complex plane, p (y | x) is conditional probability, and under the condition of signal power normalization, p (y | x) ═ SNR · exp (-SNR · | x-y @ is2)/π。
Figure BDA0003429863890000062
Practical systems generally give a code rate, i.e. a clear communication spectral efficiency, so that the minimum requirement for SNR can be numerically calculated as equation (8), which can be expressed in dB
Figure BDA0003429863890000063
The higher the PAPR, the lower the allowed transmit power, and the greater the system cost;
Figure BDA0003429863890000064
the higher the SNR requirement, the greater the system cost. However, reducing the transmission power due to the increase in the PAPR also produces an advantageous effect in terms of system power consumption, so that the PAPR and the PAPR are combined
Figure BDA0003429863890000065
When the system cost index is integrated into a comprehensive system cost index, the PAPR can be endowed with a weight factor zeta smaller than 1, and thus after the PAPR is expressed by taking dB as a unit, the comprehensive cost index can be expressed as
Figure BDA0003429863890000066
Further, the optimal amplification factor λ is a value that minimizes the system cost index, as shown in equation (9).
Figure BDA0003429863890000067
Further, m can be changed1And m2To increase the system gain, i.e. at each m1And m2Under the condition of value pairing, respectively optimal amplification factors lambda are selectedoptRespectively obtaining the optimal system cost indexes under each pairing condition, and then comparing a pair of m with the minimum optimal system cost index1And m2And selecting values. Here, m1And m2The value pairs have m types, but the optimal solution is close to the default value, so that the method can be used for searching in a small range close to the default value.
Finally, according to the selected m1And m2Value pairing and corresponding optimal amplification factor lambdaoptAnd (5) and (6) to obtain a final constellation diagram design result.
To further illustrate the effect of the present invention with respect to the prior art, the following embodiments are specifically described, and without particularly describing, the number of constellation points is considered to be 4096, the number of amplitude and phase mapping bits, i.e., m1And m2Set by default.
Fig. 5 shows the constellation diagram design results in the prior art and the technical scheme of the present invention on the premise that the power normalization, i.e., the average power of the constellation points is 1, where the prior art is gray APSK modulation in which the constellation point positions are designed based on standard gaussian distribution, and the technical scheme of the present invention considers two cases of amplification factors, i.e., λ 1.1 and λ 1.2. Further, φ in equation (6) is set to π/128. As can be seen from fig. 5, the constellation diagram distribution trend of the scheme of the present invention is similar to that of the existing gray APSK modulation, and the constellation points at positions with larger amplitude values are distributed sparsely, which conforms to the characteristics of gaussian distribution. Here, although the inner-circle amplitude distribution is relatively sparse, the number of constellation points at each amplitude value is the same as the outer circle, and the average density of the constellation point distribution near the inner-circle position is actually greater than that near the outer-circle position. In contrast, the peak amplitude value of the technical scheme of the invention is significantly smaller than that of the existing scheme, and theoretically, the possibility of reducing the PAPR is provided.
Fig. 6 compares the PAPR of the constellation point for the case of modulation orders of 1024, 4096, 16384 and infinity. In addition to the three cases referred to in fig. 5, a classical QAM modulation is added for comparison. As can be seen from fig. 6, the PAPR of the existing gray APSK modulation is greatly improved compared to the classical QAM modulation, and the higher the modulation order, the larger the improvement amount, and when the number of constellation points is in the limit of infinity, the PAPR is also infinite. By adopting the technical scheme of the invention, compared with Gray APSK modulation, the PAPR can be obviously reduced, and when lambda is 1.2, the PAPR of the technical scheme of the invention under various modulation orders is even lower than that of classical QAM modulation.
Fig. 7 compares the communication capacity bounds under an Additive White Gaussian Noise (AWGN) channel, where the difference between the capacity limit value of the SNR and the Shannon limit is given when different spectral efficiencies are satisfied. As can be seen from fig. 7, compared with the classical QAM, the existing gray APSK and the technical solution of the present invention can obtain capacity gain when the spectral efficiency is in the conventional middle region, which is mainly benefited by the constellation point design method using gaussian distribution characteristics. Compared with the existing Gray APSK, the technical scheme of the invention may have smaller gain loss, but the gain loss is far lower than the improvement amount of the PAPR, so that the scheme of the invention can still obtain certain system gain.
In fig. 8, comparing the system cost indexes, where the weighting factor ζ is set to 0.3, it can be found that the cost indexes of the technical scheme of the present invention are all significantly lower than those of the classical QAM and the existing gray APSK, where λ takes a value of 1.15-1.20, the present invention can obtain the optimal cost index, and can obtain profits of 0.38 and 1.05dB compared with the classical QAM and the existing gray APSK, respectively. Further, FIG. 9 maps the number of bits at different amplitudes, i.e., different m1The lowest system cost index of the present invention is given, and it can be seen that it is near the default value, i.e., m1When m is 4, 5, or 6 (in this case 28, 7, 6, respectively), a smaller system cost index can be obtained globally. Therefore, for 4096 order modulation, m can be used when actually designing the constellation diagram1And m2The values are set to 5 and 7, respectively, and λ is set to 1.75.
The foregoing lists merely illustrate specific embodiments of the invention. It is obvious that the invention is not limited to the above embodiments, but that many variations are possible. All modifications which can be derived or suggested by a person skilled in the art from the disclosure of the present invention are to be considered within the scope of the invention.

Claims (5)

1. A method for designing an amplitude-limited Gaussian-like distribution constellation diagram is characterized by comprising the following steps:
1) for a constellation diagram with M constellation points, each constellation point is represented by M-log2M bit mapping, setting
Figure FDA0003429863880000011
The magnitude of the wave is a function of the amplitude,
Figure FDA0003429863880000012
a phase value of where m1+m2M; at each amplitude there is M2Each constellation point corresponds to M2A phase value of M at each phase value1Each constellation point corresponds to M1An amplitude value;
2) selecting an amplification factor lambda larger than 1, and initializing each amplitude value and phase value, wherein the ith amplitude value is:
Figure FDA0003429863880000013
the j-th phase value is:
Figure FDA0003429863880000014
where φ is an arbitrary value, M1Number of amplitude values, M2The number of phase values;
3) for m bit signalsThen, m in front of1Bits are mapped to amplitude values by gray, then m2Mapping the bits into phase values through Gray to obtain an initial result of the constellation diagram;
4) comprehensively considering signal power peak-to-average power ratio (PAPR) and channel capacity, adjusting the value of an amplification factor lambda, and calculating the peak-to-average power ratio (m)1+m2Adjusting m on the premise of m1And m2And (4) optimizing the constellation diagram design result.
2. The method of claim 1, wherein initializing m according to m is based on m value1And m2Wherein when m is an odd number, m is initialized1(m-1)/2 and m2(m + 1)/2; when m is even number, initializing m1M/2-1 and m2＝m/2+1。
3. The method according to claim 1, wherein in the step 4), when the constellation design result is optimized by comprehensively considering PAPR and channel capacity, any m is given1And m2Value pairing, different amplification factors lambda correspond to a constellation diagram, and an objective function related to lambda can be established
Figure FDA0003429863880000015
The optimal magnification factor is the result of minimizing the scalar function, namely:
Figure FDA0003429863880000016
where ζ is the weighting factor, λoptIs the optimal amplification factor;
Figure FDA0003429863880000017
the minimum requirement of the target spectrum efficiency on the signal-to-noise ratio (SNR) can be obtained by calculation according to the communication capacity; PAPRdBAnd (lambda) is the signal power peak-to-average ratio.
4. The method according to claim 3, wherein the PAPR is calculated as:
Figure FDA0003429863880000021
5. the method of claim 3, wherein the constellation design method for the amplitude-limited Gaussian-like distribution is performed for each m1And m2Value matching and selection of respective optimal amplification factor lambdaoptThen, the optimal amplification factor with the minimum objective function is selected from all the optimal amplification factors as the global optimal amplification factor, and the corresponding m is1And m2Value pair as optimal m1And m2The value of (a).
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