CN104038457A - Soft output sphere decoding method in coding MIMO system based on initial sphere radius - Google Patents

Abstract

The invention discloses a soft output sphere decoding method in a coding MIMO system based on an initial sphere radius. The soft output sphere decoding method comprises the following steps that (1) in the coding MIMO system, a transmitting terminal is provided with NT antennas, a receiving terminal is provided with NR receiving antennas, NR >=MT, the transmitting terminal transmits a symbolic vector (please see the formula in the specification) to the receiving terminal, the receiving terminal carries out searching through a single-tree searching method, and a single-tree searching range is set through an initial sphere radius strategy; (2) the receiving vector received by the receiving terminal can be expressed as a complex value model (please see the formula in the specification), the receiving terminal obtains log-likelihood ratio information according to the complex value model (please see the formula in the specification), and a decoder decodes the symbolic vector (please see the formula in the specification) sent by the transmitting terminal according to the log-likelihood ratio information. According to the method, the symbolic vector sent by the transmitting terminal is decoded, and meanwhile the searching complexity can be lowered.

Description

Soft output spherical decoding method based on original ball radius in coded mimo systems

Technical field

The invention belongs to wireless communication technology field, relate to the soft output spherical decoding method based on original ball radius in a kind of coded mimo systems.

Background technology

In actual coded mimo systems, in order to realize the raising of spectrum efficiency, the detection algorithm of receiver is a major challenge.The detection algorithm that has occurred in recent years a lot of research high-performance low complex degrees, as the tree search algorithm of a class based on QR decomposition.According to the way of search of tree search, tree search can be divided into depth-first, breadth-first, and preferential three classes of measure value, and wherein first two mode can better be utilized tree structure thereby receive more concerns.In the detection of coded mimo systems, depth-first algorithm is used the Sphere Decoding Algorithm of soft output, and the algorithm of breadth-first is used the K-best algorithm of soft output.Because K the survival sequence of K-best algorithm use of soft output carried out log-likelihood ratio (log-likelihood ratio, LLR) calculating, therefore there will be all K survival sequences in the duplicate situation of value of a certain bit, cannot calculate soft information in this case.In order to address this problem, have two kinds of methods to estimate the LLR:1 of these bits) use the difference calculating between measure value maximum and minimum value in K survival sequence, be called differential technique K-best; 2) using a predefined value is that amplitude limit value replaces, and is called the fixedly K-best method of amplitude limit value.The Sphere Decoding Algorithm of soft output mainly comprises single tree search algorithm (Single Tree Search, STS), list Sphere Decoding Algorithm (List Sphere Decoder, LSD) and repeat tree search (Repeated Tree Search, RTS).Wherein STS a kind ofly can obtain the soft output Sphere Decoding Algorithm of different tradeoffs in performance and complexity by adjusting parameter, and it has better performance complexity tradeoff compared with LSD and RTS algorithm, and convenient, flexible configuration.But its cost that reaches while approaching optimum performance is high complexity overhead, and this expense is along with increasing with the increase of order of modulation of antenna number significantly raises.

Much research shows, the restriction by the radius of a ball to Sphere Decoding Algorithm search volume, and the complexity of globular decoding can be cut down by very big degree.Sum up the method for several main selection original ball radiuses at present, roughly have following three classes: 1) radius of a ball is determined by the output of linear detector, for example, use the measure value of MMSE solution as original ball radius.Because the accuracy of MMSE is subject to the impact of antenna number and channel conditions very large, therefore very inaccurate.2) original ball radius is since 0, and follows search procedure constantly to increase.Obviously, this method need to be searched for more layer when number of antennas is a lot of, and therefore, to also practical not in the many situations of number of antennas, above two kinds of methods are all difficult to reduce as much as possible complexity in the situation that of guaranteed performance.

Summary of the invention

The object of the invention is to overcome the shortcoming of above-mentioned prior art, the soft output spherical decoding method based on original ball radius in a kind of coded mimo systems is provided, in the symbolic vector that the method can send at decoding transmitting terminal, reduce the complexity of search.

For achieving the above object, the soft output spherical decoding method based on original ball radius in coded mimo systems of the present invention comprises the following steps:

1), in coded mimo systems, establish transmitting terminal and be equipped with N _troot antenna, receiving terminal is equipped with N _rroot reception antenna, wherein, N _r>=N _t, transmitting terminal sends symbolic vector to receiving terminal, receiving terminal adopts the method for single tree search to search for, and by original ball radius strategy, the scope that single tree is searched for is set, wherein, the described method that the scope of single tree search is set by original ball radius strategy is: according to noise profile, when radius being initialized as to the probability during for (1-ε) that covers at least one leaf node on receiving terminal, the original ball radius of searching for is for the first time ρ _1-ε, when not searching reception vector for the first time, search for for the second time, wherein the original ball radius of search is set to ρ for the second time _{1-0.1 ε}, when not searching reception vector for the second time, search for for the third time, wherein the original ball radius of search is set to ρ for the third time _{1-0.01 ε}, when search does not for the third time have search not obtain receiving vector, the original ball radius of the 4th search is set to ρ _{1-0.001 ε}, while also not searching after searching for according to the 4th time, the original ball radius of the 5th search is set to ρ ₁;

2) the reception vector that receiving terminal receives can be expressed as complex value model wherein, for N _r* N _tthe channel matrix of dimension, for independent identically distributed multiple Gaussian noise, receiving terminal is according to described complex value model obtain log-likelihood ratio information, then described log-likelihood ratio information is forwarded in decoder, the symbolic vector that decoder sends transmitting terminal according to described log-likelihood ratio information carry out decoding.

If wherein, T representation vector or transpose of a matrix operation, complex value model be converted into real-valued model y=Hs+n;

According to log-likelihood ratio information table described in Max-Log approximation method, be shown

$L\left(x_{j,b}\right)\≈\underset{s\∈X_{j,b}^{\left(0\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2-\underset{s\∈X_{j,b}^{\left(1\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2$

Wherein, x _{j, b}for s is through b bit of j root antenna corresponding to gray mappings, with be respectively b bit of j root antenna and be 0 and be 1 set;

Order

$\left\{\begin{array}{c}{s}^{\mathrm{ML}}=\arg \underset{s\∈X_{j,b}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\\ {\mathrm{\λ}}^{\mathrm{ML}}={\left|\right|y-{\mathrm{Hs}}^{\mathrm{ML}}\left|\right|}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}=\underset{s\∈X_{j,b}^{\mathrm{\Θ}(j,b)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\end{array}\right.$

Wherein, s ^mLfor the maximum likelihood solution detecting; λ ^mLfor estimating of maximum likelihood solution, described maximum likelihood solution estimate λ ^mLfor receiving signal phasor and channel matrix H and maximum likelihood solution s ^mLeuclidean distance between product; for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution; set for all alternative hypothesis of b the bit of j root antenna of maximum likelihood solution;

Log-likelihood ratio information L (x _{j, b}) can be expressed as:

$L\left(x_{j,b}\right)=\left\{\begin{array}{cc}{\mathrm{\λ}}^{\mathrm{ML}}-{\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}},& x_{l,b}^{\mathrm{ML}}=0\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}-{\mathrm{\λ}}^{\mathrm{ML}},& x_{j,b}^{\mathrm{ML}}=1\end{array}\right.$

L (x wherein _{j, b}) be the log-likelihood ratio information of definition, λ ^mLfor estimating of maximum likelihood solution, for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution.

While searching for for the first time, λ ^mLwith be respectively:

$\left\{\begin{array}{c}{\mathrm{\λ}}^{\mathrm{ML}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2+L_{\max }(\∀j,b)\end{array}\right..$

The present invention has following beneficial effect:

Soft output spherical decoding method based on original ball radius in coded mimo systems of the present invention is when the symbolic vector of decoding transmitting terminal transmitting, receiving terminal adopts the method for single tree search to search for, the scope of single tree search is set by original ball radius strategy simultaneously, adopt the mode of five search to search for, and the radius of search constantly expands each time, the radius of the 5th search is extended to infinity, simultaneously in any search procedure, receive while receiving vector, just stop, thereby when guaranteeing symbolic vector decoding, reduce the complexity of search.Simulation result shows, in the situation that having complexity constraints, the present invention's tradition STS algorithm can reach with lower complexity identical performance, and its performance and complexity compromise are also better than the K-best of differential technique and the fixing K-best of amplitude limit value.

Accompanying drawing explanation

Fig. 1 is the schematic diagram of original ball radius strategy of the present invention;

Fig. 2 is that the complexity of the present invention upper bound is with the change curve of original ball radius parameter ε;

Fig. 3 is the impact of original ball radius parameter ε of the present invention on average access nodes;

Fig. 4 is the impact of original ball radius parameter ε of the present invention on BLER;

Fig. 5 is the performance complexity tradeoff that the present invention uses original ball radius strategy;

Fig. 6 is the impact of Probpruning technology of the present invention on BLER;

Fig. 7 is the present invention and existing methodical BLER comparison;

Fig. 8 is the present invention and existing methodical real addition comparison;

Fig. 9 is the present invention and existing methodical real multiplications comparison.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in further detail:

Soft output spherical decoding method based on original ball radius in coded mimo systems of the present invention comprises the following steps:

1), in coded mimo systems, establish transmitting terminal and be equipped with N _troot antenna, receiving terminal is equipped with N _rroot reception antenna, wherein, N _r>=N _t, transmitting terminal sends symbolic vector to receiving terminal, receiving terminal adopts the method for single tree search to search for, and by original ball radius strategy, the scope that single tree is searched for is set, wherein, the described method that the scope of single tree search is set by original ball radius strategy is: according to noise profile, when radius being initialized as to the probability during for (1-ε) that covers at least one leaf node on receiving terminal, the original ball radius of searching for is for the first time ρ _1-ε, when not searching reception vector for the first time, search for for the second time, wherein the original ball radius of search is set to ρ for the second time _{1-0.1 ε}, when not searching reception vector for the second time, search for for the third time, wherein the original ball radius of search is set to ρ for the third time _{1-0.01 ε}, when search does not for the third time have search not obtain receiving vector, the original ball radius of the 4th search is set to ρ _{1-0.001 ε}, while also not searching after searching for according to the 4th time, the original ball radius of the 5th search is set to ρ ₁;

2) the reception vector that receiving terminal receives can be expressed as complex value model wherein, for N _r* N _tthe channel matrix of dimension, for independent identically distributed multiple Gaussian noise, receiving terminal is according to described complex value model obtain log-likelihood ratio information, then described log-likelihood ratio information is forwarded in decoder, the symbolic vector that decoder sends transmitting terminal according to described log-likelihood ratio information carry out decoding.

If wherein, T representation vector or transpose of a matrix operation, complex value model be converted into real-valued model y=Hs+n;

According to log-likelihood ratio information table described in Max-Log approximation method, be shown

$L\left(x_{j,b}\right)\≈\underset{s\∈X_{j,b}^{\left(0\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2-\underset{s\∈X_{j,b}^{\left(1\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2$

Wherein, x _{j, b}for s is through b bit of j root antenna corresponding to gray mappings, with be respectively b bit of j root antenna and be 0 and be 1 set;

Order

$\left\{\begin{array}{c}{s}^{\mathrm{ML}}=\arg \underset{s\∈X_{j,b}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\\ {\mathrm{\λ}}^{\mathrm{ML}}={\left|\right|y-{\mathrm{Hs}}^{\mathrm{ML}}\left|\right|}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}=\underset{s\∈X_{j,b}^{\mathrm{\Θ}(j,b)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\end{array}\right.$

Wherein, s ^mLfor the maximum likelihood solution detecting; λ ^mLfor estimating of maximum likelihood solution, described maximum likelihood solution estimate λ ^mLfor receiving signal phasor and channel matrix H and maximum likelihood solution s ^mLeuclidean distance between product; for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution; set for all alternative hypothesis of b the bit of j root antenna of maximum likelihood solution;

Log-likelihood ratio information L (x _{j, b}) can be expressed as:

$L\left(x_{j,b}\right)=\left\{\begin{array}{cc}{\mathrm{\λ}}^{\mathrm{ML}}-{\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}},& x_{l,b}^{\mathrm{ML}}=0\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}-{\mathrm{\λ}}^{\mathrm{ML}},& x_{j,b}^{\mathrm{ML}}=1\end{array}\right.$

L (x wherein _{j, b}) be the log-likelihood ratio information of definition, λ ^mLfor estimating of maximum likelihood solution, for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution.

While searching for for the first time, λ ^mLwith be respectively:

$\left\{\begin{array}{c}{\mathrm{\λ}}^{\mathrm{ML}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2+L_{\max }(\∀j,b)\end{array}\right..$

Fig. 1 is the situation of change of radius in search procedure of the present invention, and wherein, the circle of two dotted line signs is not for adding single tree search of original ball radius limit, and its hunting zone is separated and determined by SIC.The circle of two solid line is the hunting zone that has original ball radius limit.Can see, hunting zone can be effectively dwindled in arranging of original ball radius.

The present invention discusses the setting of original ball radius by case theory.In case theory, partial sequence in each dimension is all regarded a lattice point as, lattice are divided into unlimited lattice and Finite Lattice, unlimited lattice refer to that grid point distribution is in the total space, be not limited to the scope of planisphere, the lattice point of Finite Lattice can only be distributed in planisphere region, according to the conclusion of unlimited case theory, the lattice point of k layer, its upper bound can approximate representation be:

$S_k\left(\mathrm{\ρ}\right)=\frac{V_k\left(\mathrm{\ρ}\right)}{\det \left(R_k^H{R}_k\right)}$

Wherein, $V_k\left(\mathrm{\ρ}\right)=\frac{{\mathrm{\π}}^k{\left({\mathrm{\ρ}}^2\right)}^k}{k!},$ ρ is the radius of a ball, supposes that maximum four search necessarily can search leaf node, ignores and within four times, also searches for the possibility less than leaf node.So, for k node layer, only consider that the complexity upper bound of front four search is

$\mathrm{UB}\left(k\right)\≈\frac{{\mathrm{\π}}^k}{\det \left(R_k^H{R}_k\right)k!}{\mathrm{\Σ}}_{i=1}^{4}P\left(i\right){\left({\mathrm{\ρ}}_{1-{0.1}^{i-1}\mathrm{\ϵ}}\right)}^{2k}$

Wherein, P (i)=0.1 ^{(i-1) (i-2)/2}ε ^i-1(1-0.1 ^i-1ε)., for coverage rate is 1-0.1 ^i-1the radius of a ball of ε, for choosing of original ball radius, denominator (is ignored in the upper bound of each layer of complexity that can draw according to above formula ), with reference to the smallest point in the complexity upper bound of 2, the 1,2 layers, figure near ε=0.4, along with the smallest point in the higher complexity of the number of plies upper bound is distributed near ε=0.3.And high-rise complexity magnitude much larger than low layer, becomes the major influence factors of complexity.Therefore, getting ε=0.3 can drop to complexity minimum as the parameter of controlling initial radium.

For the modulation system of U-QAM, corresponding real-valued constellation set is:

$\mathrm{\Ω}=\{-(\sqrt{U}-1),-(\sqrt{U}-3),...,-\mathrm{1,1,3},...,(\sqrt{U}-3),(\sqrt{U}-1)\}$

The line number m of table:

Table 1 has provided when modulation system is 16QAM, and SE enumerates the storage list of order.

Table 1

In order to verify performance of the present invention, algorithm of the present invention and other algorithm have been carried out to analogy.Mainly from BLER, super average access nodes, this three aspects: of real multiplications addition number of times carrys out comparison.The present invention considers the mimo system of coding, on the link of LTE-A and emulation platform, verifies, the main parameter of emulation is in Table 2.

Table 2

A. the checking of original ball radius strategy

Fig. 3 and Fig. 4 figure provide the average access nodes of different original ball radiuses under each amplitude limit value and the variation of BLER.Compositive complexity and performance two aspects, the most suitable interval of initialization radius of a ball parameter ε is [0.2,0.4], in this is interval, can obtain good performance complexity tradeoff.Fig. 5 provides performance and the complexity tradeoff of each algorithm, the STS that has original ball radius that the present invention is proposed and traditional STS, and the K-best of differential technique, and adopt the fixedly K-best of amplitude limit value to compare.There are the STS of original ball radius and the size that the other numeral of traditional STS curve is amplitude limit value, and two kinds of values that the other numeral of K-best method curve is corresponding K.As shown in the figure, under each amplitude limit value, after interpolation original ball radius, reduced complexity surpasses 50%, and performance approximately has the gain of 0.5～1dB.Compare with the K-best algorithm of other two kinds of soft outputs, the STS adding after original ball radius also can reach identical BLER performance with lower average access nodes, obtain better performance complexity tradeoff.

B. the checking of Probpruning

In order to verify the impact of Probpruning on performance and complexity.Fig. 6 provides the performance comparison of adding Probpruning front and back, wherein D _avg=3500, L _max=1.5, ε=0.3.Beta pruning probability used herein is 0.8.As shown in the figure, near the signal to noise ratio interval 6～8dB (BLER=0.01) being concerned about, performance loss is 0.1～0.2dB approximately.Such performance loss is negligible substantially, and increases with signal to noise ratio, and such performance loss is more and more less.Table 3 has provided the contribution of Probpruning aspect reduction complexity.

Table 3

At each signal to noise ratio place, probability of use beta pruning can surpass 50% by reduced complexity.

C. with other algorithm comparison

Fig. 7 is the comparison of the present invention and other several algorithm BLER performances.Obviously, amplitude limit value is larger, and the performance of the algorithm of carrying is better.As shown in the figure, the performance of three curves of the algorithm of carrying in BLER=0.01 vicinity is all better than other algorithms.Fig. 8 and Fig. 9 are respectively the comparisons of the present invention and other several algorithm real addition number of times and real multiplications number of times, and near BLER=0.01, the complexity of the algorithm of carrying is lower than other algorithms.Performance and complexity that carried algorithm is described are all better than other algorithm, have obtained better performance and complexity tradeoff.In addition, the parameter of the algorithm of carrying can be adjusted flexibly, convenient configuration.

Claims (3)

1. the soft output spherical decoding method based on original ball radius in coded mimo systems, is characterized in that, comprises the following steps:

1), in coded mimo systems, establish transmitting terminal and be equipped with N _troot antenna, receiving terminal is equipped with N _rroot reception antenna, wherein, N _r>=N _t, transmitting terminal sends symbolic vector to receiving terminal, receiving terminal adopts the method for single tree search to search for, and by original ball radius strategy, the scope that single tree is searched for is set, wherein, the described method that the scope of single tree search is set by original ball radius strategy is: according to noise profile, when radius being initialized as to the probability during for (1-ε) that covers at least one leaf node on receiving terminal, the original ball radius of searching for is for the first time ρ _1-ε, when not searching reception vector for the first time, search for for the second time, wherein the original ball radius of search is set to ρ for the second time _{1-0.1 ε}, when not searching reception vector for the second time, search for for the third time, wherein the original ball radius of search is set to ρ for the third time _{1-0.01 ε}, when search does not for the third time have search not obtain receiving vector, the original ball radius of the 4th search is set to ρ _{1-0.001 ε}, while also not searching after searching for according to the 4th time, the original ball radius of the 5th search is set to ρ ₁;

2) the reception vector representation that receiving terminal receives is complex value model wherein, for N _r* N _tthe channel matrix of dimension, for independent identically distributed multiple Gaussian noise, receiving terminal is according to described complex value model obtain log-likelihood ratio information, then described log-likelihood ratio information is forwarded in decoder, the symbolic vector that decoder sends transmitting terminal according to described log-likelihood ratio information carry out decoding.

2. the soft output spherical decoding method based on original ball radius in coded mimo systems according to claim 1, is characterized in that,

If wherein, T representation vector or transpose of a matrix operation, complex value model be converted into real-valued model y=Hs+n;

According to log-likelihood ratio information table described in Max-Log approximation method, be shown

$L\left(x_{j,b}\right)\≈\underset{s\∈X_{j,b}^{\left(0\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2-\underset{s\∈X_{j,b}^{\left(1\right)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2$

Wherein, x _{j, b}for s is through b bit of j root antenna corresponding to gray mappings, with be respectively b bit of j root antenna and be 0 and be 1 set;

Order

$\left\{\begin{array}{c}{s}^{\mathrm{ML}}=\arg \underset{s\∈X_{j,b}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\\ {\mathrm{\λ}}^{\mathrm{ML}}={\left|\right|y-{\mathrm{Hs}}^{\mathrm{ML}}\left|\right|}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}=\underset{s\∈X_{j,b}^{\mathrm{\Θ}(j,b)}}{\min }{\left|\right|y-\mathrm{Hs}\left|\right|}^2\end{array}\right.$

Wherein, s ^mLfor the maximum likelihood solution detecting; λ ^mLfor estimating of maximum likelihood solution, described maximum likelihood solution estimate λ ^mLfor receiving signal phasor and channel matrix H and maximum likelihood solution s ^mLeuclidean distance between product; for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution; set for all alternative hypothesis of b the bit of j root antenna of maximum likelihood solution;

Log-likelihood ratio information L (x _{j, b}) can be expressed as:

$L\left(x_{j,b}\right)=\left\{\begin{array}{cc}{\mathrm{\λ}}^{\mathrm{ML}}-{\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}},& x_{l,b}^{\mathrm{ML}}=0\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}-{\mathrm{\λ}}^{\mathrm{ML}},& x_{j,b}^{\mathrm{ML}}=1\end{array}\right.$

L (x wherein _{j, b}) be the log-likelihood ratio information of definition, λ ^mLfor estimating of maximum likelihood solution, for the minimum value in all alternative hypothesis measure values of b the bit of j root antenna of maximum likelihood solution.

3. the soft output spherical decoding method based on original ball radius in coded mimo systems according to claim 2, is characterized in that, while searching for for the first time, and λ ^mLwith be respectively:

$\left\{\begin{array}{c}{\mathrm{\λ}}^{\mathrm{ML}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2\\ {\mathrm{\λ}}_{j,b}^{\stackrel{\‾}{\mathrm{ML}}}={\mathrm{\ρ}}_{1-\mathrm{\ϵ}}^2+L_{\max }(\∀j,b)\end{array}\right..$