CN109379319B - Complex color shift keying constellation diagram design method for optical OFDM system - Google Patents

Abstract

The invention relates to a method for designing a complex color shift keying constellation diagram for an optical OFDM system, which can calculate to obtain an optimal amplification coefficient according to an optimization target of a Bit Error Rate (BER), and enables a constellation diagram after rotation amplification to have maximum Minimum Euclidean Distance (MED) and optimal Error Rate performance, thereby having obvious BER performance improvement compared with the original scheme. Meanwhile, the method uses the symmetry of the rotated constellation diagram to separately de-map the polar bit and the CSK bit, thereby reducing the de-mapping complexity to a half of the traditional scheme.

Description

Complex color shift keying constellation diagram design method for optical OFDM system

Technical Field

The invention relates to the technical field of visible light communication, in particular to a novel Complex Color Shift Keying (CCSK) constellation diagram design method based on rotary amplification. The CCSK constellation diagram designed by the method can be used for improving the error rate performance of an Optical Orthogonal Frequency Division Multiplexing (O-OFDM) communication system adopting the CCSK, and can effectively reduce the computational complexity of the existing CCSK system.

Background

In recent years, Visible Light Communication (VLC) has attracted much attention from researchers because of its advantages such as multiplexing, energy saving, environmental protection, and high-speed transmission, and has become a powerful complement to conventional wireless Communication [1 ]. VLC systems typically use Intensity Modulation/Direct Detection (IM/DD) techniques [1], with a signal being a positive real number [2 ].

In order to take advantage of the high speed and no intersymbol interference of Orthogonal Frequency Division Multiplexing (OFDM) technology, Optical OFDM (O-OFDM) schemes have received much research attention [3 ]. In view of the positive real characteristic limitation of VLC transmission signals, researchers have devised specific O-OFDM schemes, such as asymmetric Clipped O-OFDM (ACO-OFDM) [4], DC-biased O-OFDM (Direct Current biased O-OFDM, DCO-OFDM) [5], Unipolar OFDM (Unipolar OFDM, U-OFDM) [6], and so on. On the other hand, Color Shift Keying (CSK) is another high-speed transmission scheme under VLC, which provides multiplexing in the Color domain to increase system capacity [7 ]. The current CSK scheme has different modes such as traditional three-color lamp (TriLED, TLED) transmission [7] and extended four-color lamp (quad LED, QLED) [8] transmission.

The traditional O-OFDM scheme can modulate complex domain signals but cannot utilize color domain resources; while the conventional CSK can make full use of the color domain, it does not support modulation in the complex domain. In order to combine the advantages of both O-OFDM and CSK schemes, researchers have proposed a hybrid scheme, i.e., an O-OFDM scheme based on polar-Modulation (PM) and Complex Color Shift Keying (CCSK), that achieves higher transmission rates than both schemes [9 ]. However, the scheme does not reach the upper limit of theoretical performance, and the constellation diagram still has a large optimization space. Accordingly, the invention provides a novel CCSK constellation diagram design method, which is called a CCSK OFDM scheme based on Rotating Polar Modulation (RPM), and is called RPM-CCSK-OFDM for short. The novel CCSK constellation diagram designed by the scheme has the following two significant advantages:

firstly, the method can calculate the optimal amplification coefficient according to the optimization target of the Minimum Bit Error Rate (BER), so that the constellation diagram after rotation amplification has the maximum Minimum Euclidean Distance (MED) and the optimal Error Rate performance, thereby having significant BER performance improvement compared with the original scheme.

Secondly, the method uses the symmetry of the rotated constellation diagram to separately de-map the polar bit and the CSK bit, thereby reducing the de-mapping complexity to a half of the traditional scheme.

Disclosure of Invention

The method provided by the invention aims to solve the technical defect that the performance of an O-OFDM method based on polar modulation and complex color shift keying provided by the prior art is lower, and provides a complex color shift keying constellation diagram design method for an optical OFDM system. Meanwhile, the invention uses the symmetry of the rotated constellation diagram to separately de-map the polar bit and the CSK bit, thereby reducing the de-mapping complexity to a half of the existing scheme.

In order to realize the purpose, the technical scheme is as follows:

the design method of the complex color shift keying constellation diagram for the optical OFDM system comprises the following steps:

arranging an input bit data sequence into

The matrix Q of (a), wherein,

number of bits, m, carried by 1 RPM-CCSK symbol _s2 denotes that each symbol additionally carries 2 polarity bits, m_CSK＝log₂M represents the number of bits carried by a CSK symbol, M represents the CSK modulation order, N_uRepresenting the number of RPM-CCSK symbols allowed to be input in one OFDM symbol;

each column of the matrix Q is equally divided into two parts, bits of the two parts are respectively modulated into a real part and an imaginary part of a final result through two PM-CCSK modules and two RPM modules, and the specific modulation process is as follows:

after entering into PM-CCSK module, the input bit of real part or imaginary part will be divided into 1 polarity bit and m_CSKOne CSK bit; the CSK bit is modulated into a CSK symbol, and the polarity bit represents the positive and negative of the symbol; the outputs of the two paths of PM-CCSK modules are respectively X_reAnd X_im；

X_reAnd X_imAre respectively sent into two paths of RPM modules, and are paired with X_reAnd X_imSequentially carrying out constellation diagram rotation, amplification and normalization operations to obtain an RPM-CCSK constellation diagram mapping result;

finally, combining the mapped real part and imaginary part to obtain (n)_c×N_u) A complex symbol matrix X;

the specific processes of the constellation diagram rotation, amplification and normalization operations are as follows:

(1) constellation rotation

The constellation diagram of PM-CCSK is divided into two sub-constellations with different polarity bits, and then the sub-constellation diagram with the polarity bit of 0 is along the axis p_r＝p_g＝p_bRotated by 180 degrees, wherein (p)_r,p_g,p_b) Color gamut coordinates representing constellation points;

(2) amplifying operation

Fixing the positive centers of the two sub-constellation diagrams, stretching and amplifying the two sub-constellation diagrams on respective planes, and setting an amplification coefficient to be xi;

(3) normalization

And dividing the coordinates of all the constellation points on the amplified constellation diagram by a normalization factor to enable the coordinates to meet the limitation of unit electric power.

It is noted that, to avoid repeated calculation, the operations such as rotation, amplification, normalization, etc. should be completed off-line, and the formed new RPM-CCSK constellation mapping relationship of each order is stored in the corresponding constellation mapping table for direct system invocation.

Preferably, the optimal solution method of the amplification factor ξ is as follows:

let BER expression of RPM-CCSK be

Wherein gamma is_bRepresenting the bit signal-to-noise ratio, gamma_b＝E_b/N₀，E_bRepresenting the energy per bit, N₀Representing the single-sided power spectral density of additive white gaussian noise, Q (-) represents the Q function,

representing the average electric energy of the CSK symbols in the RPM-CCSK; d represents the distance between any sub-constellation plane and the decision plane, d_HRepresenting the average Hamming distance between every two adjacent points in the CSK constellation diagram; f. of_M(. -) represents the symbol error rate of the CSK modulation of order M;

then the optimization objective function defining the above equation is:

solving a partial differential equation for the optimization objective function by using functions diff and solve of MATLAB to obtain xi_optAnd gamma_bFor all γ_bAnd selecting the same convergence value as an optimal amplification factor in the relation curve to design a constellation diagram.

Compared with the prior art, the invention has the beneficial effects that:

(1) in the existing PM-CCSK constellation, MED (denoted as d) inside two sub-CSK constellations_CSK) Smaller, and the distance (denoted as d) between the planes of the two sub-constellation diagrams₀) The larger the size, the more the error code is concentrated in the two sub-constellation maps, and the less the polarity bit for selecting the two sub-constellation maps. This phenomenon is more pronounced in higher-order CCSK systems. Therefore, in order to improve the MED of the constellation diagram and balance the error probability of each bit, the RPM-CCSK design method adopts the constellation diagram amplification method to reduce d₀Increase d at the expense of_CSKAnd by calculating the optimal amplification factor xi_optTo achieve optimal error rate performance.

(2) Two sub-constellations in the constellation change from being symmetrical about the origin to being symmetrical about the decision plane p_r+p_g+p_bMirror symmetry is 0. The mirror symmetry has the advantage of reducing the computational complexity of the constellation diagram mapping. In the existing PM-CCSK scheme, maximum likelihood joint demapping is used for the polarity bits and the CSK bits, so that the complexity is high, and the complexity grows exponentially with the increase of the number of bits. However, in the RPM-CCSK scheme, since the two sub-constellations are mirror symmetric about the decision plane, the polarity bits and CSK can be de-mapped independently. Specifically, the method comprises the following steps:

first, demapping of the polarity bit can be simply determined according to the positional relationship between the received symbol and the decision plane, that is, if the received symbol is above the decision plane, the polarity bit is "1"; otherwise, it is "0".

Secondly, the demapping of the CSK bits still uses the maximum likelihood method, but the demapping complexity of the CSK bits is only half of that of the PM-CCSK scheme due to the reduction of the polarity bits.

Therefore, in general, compared with the existing PM-CCSK constellation, the RPM-CCSK constellation using the novel design method not only effectively reduces the error rate, but also reduces the demapping complexity by 50%.

Drawings

FIG. 1: and the schematic diagram of the RPM-CCSK-OFDM system adopting a novel constellation diagram.

FIG. 2: exemplary map of RPM-4CCSK constellation mapping process (taking the constellation of fig. 3(b) as an example).

FIG. 3: constellation design versus example diagram.

FIG. 4: amplification factor xi_optSchematic diagram of optimization of (1).

FIG. 5: BER performance of RPM scheme versus PM scheme, conventional QAM scheme. (in the figure: A: ACO-OFDM; D: DCO-OFDM; U: U-OFDM)

Detailed Description

The drawings are for illustrative purposes only and are not to be construed as limiting the patent;

the invention is further illustrated below with reference to the figures and examples.

Example 1

The novel constellation diagram method designed by the method can be used for various common O-OFDM schemes, such as ACO-OFDM, DCO-OFDM, U-OFDM and the like; it can also be used in CSK schemes with different LED configurations, such as TLED and QLED. For convenience of description, a system model in the TLED configuration will be taken as an example to illustrate a usage scheme of the novel RPM-CCSK constellation in the O-OFDM system. For simplicity, the "constellation" referred to below refers to the "CCSK constellation".

First, a CCSK-OFDM system model suitable for RPM-CCSK constellation diagram is introduced, such asAs shown in fig. 1. Two modules of RPM-CCSK constellation mapping and RPM-CCSK constellation mapping in the diagram adopt a novel CCSK constellation designed by the invention. As shown in fig. 1, bit information is first mapped to n by RPM-CCSK constellation_cA complex symbol of which n_cIs the number of configured LED lamps. In the TLED scheme, take n_c3. Then n is added_cThe complex symbols are respectively sent into n_cAmong the independent O-OFDM modulators, the O-OFDM modulator can use the common O-OFDM scheme such as ACO-OFDM, DCO-OFDM, U-OFDM, etc. And then, cyclic prefixes are respectively added to the three paths of output signals, and then the signals are respectively sent to the visible light channels through the red, green and blue LEDs. At a receiving end, the three optical signals are converted into three electrical signals through the optical filters with corresponding colors and the photodiodes, then the cyclic prefix is removed, the three electrical signals pass through the O-OFDM demodulator, and finally constellation mapping is realized through a maximum likelihood method, so that the transmitted bit information is restored.

The constellation mapping of RPM-CCSK in fig. 1 employs the novel scheme described in the summary of the invention. In order to more vividly and specifically show the mapping mode of the novel constellation diagram, the invention provides an example of the RPM-4CCSK mapping, as shown in FIG. 2, wherein all variables follow the definition in the summary of the invention. The processing of RPM-CCSK constellations of other orders may be analogized.

In order to more clearly illustrate the difference between the prior PM-CCSK scheme and the RPM-CCSK scheme proposed by the present invention, fig. 3 is a 4CCSK constellation diagram as an example, and a comparison between the two schemes is given. As is apparent from fig. 3, the rotation amplification makes the constellation mirror symmetric and increases the MED.

Next, the invention determines the optimal amplification factor xi_optThe value of (c). The optimization goal here is to minimize the system Bit Error Rate (BER), so a mathematical representation of BER needs to be derived first.

Let the mathematical expression of BER of the traditional M-order CSK modulation be denoted as P_b,MCSK. The Symbol Error Rate (SER) formula [10 ] of M-order CSK modulation is provided in the related literature]Can be expressed as P_s,MCSK＝f_M(γ) wherein P_s,MCSKRepresenting SER, f_M(y) represents a function with an argument y, which is related to the modulation order, and also to the distribution of constellation points,

pseudo-Signal-Noise Ratio (pSNR) defined as the electrical domain, where P_T＝p_r+p_g+p_bIs the total optical power, σ²Is the variance of Additive White Gaussian Noise (AWGN) in any of the three channels, red, green and blue (RGB). For M-4, 8,16 or even higher order f_MSpecific expressions of (. gamma.) are available in the literature [10 ]]Is found in (1).

On the other hand, the symbol average electric energy in the CSK constellation diagram is defined as E_sIt is and

in direct proportion. Order to

Wherein e_sIs as follows

The average electrical energy of the CSK symbols of time is shown in table 1. Defining the Signal-to-Noise Ratio (SNR) of the electric domain as gamma_sRepresenting the electrical domain signal-to-noise ratio of the receiving end if the 3-way photodiode of the receiving end is replaced by an equivalent electrical domain receiver, then

Substituting it into the previous SER function yields:

next, the existing SER needs to be converted into an expression for BER. To simplify the problem, it is assumed that only adjacent constellation points will be in error, and the probability of error from any constellation point to any adjacent constellation point is the same, which is assumed to be p. Then according to the related document [11], the SER and BER of the CSK system can be obtained as follows:

wherein the content of the first and second substances,

indicating the l-th symbol s in the system transmission constellation diagram_lIs erroneously detected as the kth symbol s_kProbability of (c)_i(i ═ 0,1, Λ, M-1) denotes mapping to s_iBit vector of d (c)_l,c_k) Represents c_lAnd c_kHamming distance between, set K_l＝{k|0≤k≤M-1,k≠l,s_kAnd s_lAdjacent }, n_lRepresents K_lThe number of elements in (c).

By substituting formula (1) or formula (2) into formula (3), it is possible to obtain:

wherein

Represents the average hamming distance between every two neighboring points in the CSK constellation, and the values are listed in table 1.

Next, the BER expression of the RPM-CCSK scheme needs to be derived. Although the output of RPM-CCSK is complex symbol, the real part and the imaginary part are mapped by using the same constellation diagram, and the BER of the real part (or the imaginary part) is only considered in the invention. As explained earlier, the real part modulation contains 1 polarity bit and m_CSKOne CSK bit. Assuming a BER of polar bits is

Because the two sub-constellation diagrams of the constellation diagram of the RPM-CCSK are parallel and mirror symmetric, two events of the polarity bit error and the CSK bit error are independent of each other, so the overall BER of the RPM-CCSK can be expressed as:

wherein

The error probability of CSK bit in RPM-CCSK can be calculated by analogy with equation (4). Without loss of generality, power normalization is considered, i.e.

The constellation of (a). At this time, the BER of the polarity bit may use Binary Phase Shift Keying (BPSK) bit error rate expression [12]And calculating to obtain:

wherein

Represents the distance between any sub-constellation plane and the decision plane, Q (-) represents the Q function,

the average electric energy of the CSK symbol in the RPM-CCSK scheme, which is different from the conventional CSK, is no longer a constant but a function related to the amplification coefficient ξ, and can be expressed as:

wherein d is_lRepresents the l-th point on the traditional CSK constellation diagram and the center of the constellation diagramc₀The distance of (a) to (b),

to represent

Average value of (a). The amplification operation in the design of RPM-CCSK constellation diagram not only brings about

Also another important change is caused by the change of (c), i.e. the SER function of CSK is represented by f_MIs changed into

Namely:

assume energy per bit as E_bDefining the bit signal-to-noise ratio gamma_b＝E_b/N₀Then for the real part symbol under consideration, γ is known_s＝(m_CSK+1)γ_b. Therefore, based on equations (4), (5), (6) and (8), the BER expression of the final RPM-CCSK can be obtained as:

based on equation (9), an optimization objective function can be defined as follows:

the partial differential equation can be solved for equation (10) using the functions diff and solve of MATLAB to obtain the results of fig. 4. As can be seen from FIG. 4(a), when γ is_bSmaller, xi_optWill follow gamma_bBut when γ changes_bWhen it is larger, xi_optWill tend to converge to several constant values

Corresponding to RPM-CCSK constellations of orders 4,8,16, respectively. FIG. 4(b) compares the values for different gamma_bUsing an adaptive optimum value xi_optFor all gamma_bAll using the same convergence value xi_optPerformance of the design method of (1). It can be seen that the same convergence value ξ is used_optDoes not cause obvious performance degradation, but can avoid the problem of different gamma_bAdaptively selecting different xi_optOf the system. Therefore, all γ can be addressed_bAll using the same convergence value xi_optTo design the constellation. For example, using fixed values

The constellation diagram in fig. 3(b) can be obtained.

In summary, the optimal amplification factor ξ is based on equation (10)_optThe RPM-CCSK constellation diagram can achieve the optimal BER performance, and meanwhile, the lower demapping complexity can be obtained.

Example 2

To more fully illustrate the benefits of the present invention, the following further describes the effectiveness and advancement of the present invention in conjunction with simulation analysis and results.

Consider N_uThe RPM-CCSK-OFDM system of 64 adopts the LED configuration of TLED, and uses the Color Band Combination One (CBC-1) CSK modulation method defined by the ieee802.15.7 standard. Without loss of generality, ACO-OFDM is chosen as the O-OFDM scheme.

First, compare the performance of RPM-CCSK-OFDM and the existing PM-CCSK-OFDM, as shown in FIG. 5 (a). Therefore, compared with the existing PM-CCSK-OFDM scheme, the RPM-CCSK-OFDM scheme with the novel constellation diagram has obvious performance improvement, and the performance improvement is more obvious along with the increase of the modulation order. On the other hand, assuming that the peak transmission rate is m to 6 bits/symbol, the BER performance of RPM-4CCSK-OFDM is compared with the BER performance of conventional 64QAM-OFDM at the same transmission rate using ACO-OFDM, DCO-OFDM (direct current offset selection 13dB), and U-OFDM schemes, respectively, as shown in fig. 5 (b). It can be seen that the RPM scheme can provide better BER performance than the conventional scheme at the same transmission rate.

In addition, the RPM scheme is also applicable to the CCSK-OFDM system configured with QLED devices, and the design method is the same as that described before for the CCSK-OFDM system using TLED, and the relevant parameters can be obtained through calculation, as shown in table 1.

Table 1: relevant parameters of CSK and CCSK constellations of each order

It should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.

Reference to the literature

[1].J.M.Kahn and J.R.Barry,“Wireless infrared communications,”in Proceedings of the IEEE,vol.85,no.2,pp.265-298,Feb.1997.

[2].T.Komine and M.Nakagawa,“Fundamental analysis for visible-light communication system using LED lights,”in IEEE Transactions on Consumer Electronics,vol.50,no.1,pp.100-107,Feb.2004.

[3].J.Armstrong,“OFDM for optical communications,”in Journal of Lightwave Technology,vol.27,no.3,pp.189-204,Feb.2009.

[4].S.D.Dissanayake and J.Armstrong,“Comparison of ACO-OFDM,DCO-OFDM and ADO-OFDM in IM/DD systems,”in Journal of Lightwave Technology,vol.31,no.7,pp.1063-1072,April 2013.

[5].O.Gonzalez,R.Perez-Jimenez,S.Rodriguez,J.Rabadan,and A.Ayala,“OFDM over indoor wireless optical channel,”in IEE Proceedings-Optoelectronics,vol.152,no.4,pp.199-204,Aug.2005.

[6].D.Tsonev,S.Sinanovic,and H.Haas,“Novel unipolar orthogonal frequency division multiplexing(u-ofdm)for optical wireless,”2012 IEEE 75th Vehicular Technology Conference(VTC Spring),Yokohama,2012,pp.1-5.

[7].IEEE Standard for Local and Metropolitan Area Networks—Part 15.7:Short-Range Wireless Optical Communication Using Visible Light,in IEEE Standard 802.15.7-2011,pp.1-309,Sept.2011.

[8].R.Singh,T.O’Farrell,and J.P.R.David,“An enhanced color shift keying modulation scheme for high-speed wireless visible light communications,”in Journal of Lightwave Technology,vol.32,no.14,pp.2582-2592,Jul.2014.

[9].Y.Chen,M.Jiang,L.Zhang and X.Chen,“Polarity modulated complex colour shift keying for OFDM-based visible light communication,”2017 IEEE/CIC International Conference on Communications in China(ICCC),Qingdao,2017,pp.1-5.

[10].L.Jia,J.Wang,W.Zhang,M.Chen and J.Wang,“Symbol error rate analysis for colour-shift keying modulation in visible light communication system with RGB light-emitting diodes,”in IET Optoelectronics,vol.9,no.5,pp.199-206,Oct.2015.

[11].L.Xiao and X.Dong,“A new approach to calculating the exact transition probability and bit error probability of arbitrary two-dimensional signaling,”IEEE Global Telecommunications Conference,2004.GLOBECOM’04,Dallas,TX,2004,pp.1239-1243,Vol.2.

[12].A.Goldsmith,Wireless Communications,Cambridge,U.K.:Cambridge Univ.Press,2005.

Claims (2)

1. The design method of the complex color shift keying constellation diagram for the optical OFDM system is characterized in that: the method comprises the following steps:

arranging an input bit data sequence into

The matrix Q of (a), wherein,

number of bits, m, carried by 1 RPM-CCSK symbol_s2 denotes that each symbol additionally carries 2 polarity bits, m_CSK＝log₂M represents the number of bits carried by a CSK symbol, M represents the CSK modulation order, N_uRepresenting the number of RPM-CCSK symbols allowed to be input in one OFDM symbol; RPM-CCSK represents complex color shift keying based on rotating polar modulation;

each column of the matrix Q is equally divided into two parts, bits of the two parts are respectively modulated into a real part and an imaginary part of a final result through two PM-CCSK modules and two RPM modules, and the specific modulation process is as follows:

after entering into PM-CCSK module, the input bit of real part or imaginary part will be divided into 1 polarity bit and m_CSKOne CSK bit; the CSK bit is modulated into a CSK symbol, and the polarity bit represents the positive and negative of the symbol; the outputs of the two paths of PM-CCSK modules are respectively X_reAnd X_im；

X_reAnd X_imAre respectively sent into two paths of RPM modules, and are paired with X_reAnd X_imSequentially carrying out constellation diagram rotation, amplification and normalization operations to obtain an RPM-CCSK constellation diagram mapping result;

finally, combining the mapped real part and imaginary part to obtain (n)_c×N_u) A complex symbol matrix X; n is_cIndicating the number of configured LED lamps;

the specific processes of the constellation diagram rotation, amplification and normalization operations are as follows:

(1) constellation rotation

The constellation diagram of PM-CCSK is divided into two sub-constellations with different polarity bits, and then the sub-constellation diagram with the polarity bit of 0 is along the axis p_r＝p_g＝p_bRotated by 180 degrees, wherein (p)_r,p_g,p_b) Color gamut coordinates representing constellation points;

(2) amplifying operation

Fixing the positive centers of the two sub-constellation diagrams, stretching and amplifying the two sub-constellation diagrams on respective planes, and setting an amplification coefficient to be xi;

(3) normalization

Dividing the coordinates of all constellation points on the amplified constellation diagram by a normalization factor to enable the coordinates to meet the limitation of unit electric power;

and the rotation, amplification and normalization operations are completed off line, and the formed new RPM-CCSK constellation map mapping relation of each order is stored in a corresponding constellation map mapping table for direct calling of a system.

2. The method of claim 1, wherein the complex color shift keying constellation design method for optical OFDM systems comprises: the optimization solving method of the amplification factor xi is as follows:

let BER expression of RPM-CCSK be

Wherein gamma is_bRepresenting the bit signal-to-noise ratio, gamma_b＝E_b/N₀，E_bRepresenting the energy per bit, N₀Representing the single-sided power spectral density of additive white gaussian noise, Q (-) represents the Q function,

representing the average electric energy of the CSK symbols in the RPM-CCSK; d represents the distance between any sub-constellation plane and the decision plane, d_HRepresenting the average Hamming distance between every two adjacent points in the CSK constellation diagram; f. of_M(. -) represents the symbol error rate of the CSK modulation of order M; n is_cIndicating the number of configured LED lamps;

then the optimization objective function defining the above equation is:

solving a partial differential equation for the optimization objective function by using functions diff and solve of MATLAB to obtain xi_optAnd gamma_bFor all γ_bAnd selecting the same convergence value as an optimal amplification factor in the relation curve to design a constellation diagram.

Images