CN105516044A - Orthogonal frequency division multiplexing (OFDM) system peak-to-average power ratio (PAPR) suppression method based on differential evolution - Google Patents

Abstract

The invention discloses an orthogonal frequency division multiplexing (OFDM) system peak-to-average power ratio (PAPR) suppression method based on differential evolution, and belongs to the field of communication transmission. The method comprises the following steps: modeling a repeated clipping and filtering process into an optimization problem; under a constraint condition meeting the PAPR and an error vector magnitude (EVM), obtaining a local optimal solution of a system through the differential evolution; and finding an output signal of which the PAPR and the EVM satisfy system conditions at the same time and the time complexity is lowest. In consideration of actual application demands of an actual system, a global optimal solution does not need to be solved through a convex optimization method, namely, a minimum PAPR or a minimum EVM does not need to be solved, and only the system demands need to the met. On the basis of ensuring low time complexity and hardware overhead, the aim is to find a feasible solution, namely, the local optimal solution. Specifically, the aim is to find a PAPR value which meets system requirements within the shortest time, and meet the constraint of the EVM, namely, ensure that a bit error rate of the system is within a specified threshold.

Description

A kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm

Technical field

The invention belongs to field of communication transmission, be specifically related to a kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm.

Background technology

Abbreviation and Key Term definition

Peak-to-average force ratio: Peak-to-averagepowerratio (PAPR)

OFDM: OrthogonalFrequencyDivisionMultiplexing (OFDM)

Error vector magnitude: Errorvectormagnitude (EVM)

Differential evolution algorithm: Differentialevolution (DE)

The error rate: Biterrorrate (BER)

The iteration amplitude limit optimized and filtering: Optimizedclippingandfiltering (OICF)

Quadrature amplitude modulation: Quadratureamplitudemodulation (QAM)

Quadrature Phase Shift Keying: Quadraturephase-shiftkeying (QPSK)

Complementary Cumulative Distribution Function: Complementarycumulativedistributionfunction (CCDF)

OFDM technology has been recognized as the core transmission technology in 4G mobile communication system.OFDM technology adopts multiple mutually orthogonal subcarrier to carry out the transmission of business, in whole transmission band, the frequency spectrum of each subcarrier is overlapped, so not only greatly can improve the spectrum utilization efficiency of system, and data flow at a high speed can be assigned on each relatively low subcarrier of transmission rate by serial to parallel conversion and transmit, the expansion of transmission data on the time domain internal symbol cycle makes OFDM technology be highly resistant to the decline caused of radio channel multi-path time delay like this.In fact, present OFDM technology is widely used in terrestrial wireless broadcast system He in wireless broadband access system, as DVB-T (digital video broadcast terrestrial), WLAN (wireless local area network), DTMB (terrestrial DTV multimedia broadcasting) etc.

OFDM belongs to multi-carrier communication technology, because the time-domain signal exported through OFDM modulation is formed by multiple sub-carrier signal addition, separate between these subcarriers, when the signal phase on these carrier waves is identical, the instantaneous power much larger than average power signal can be produced, so just there will be larger papr.Higher peak-to-average force ratio can reduce the operating efficiency of transmitter internal power amplifier, increases the complexity of A/D and D/A converter realization; If the signal with relatively high power has exceeded the dynamic range of amplifier, then after amplifier, signal can produce distortion, destroys the orthogonality between subcarrier, causes whole system hydraulic performance decline.

High peak-to-average power ratio problem in recent years for ofdm signal has carried out large quantifier elimination, proposes the method for many suppression peak-to-average force ratios, comprises margining amplitude technique, selected mapping method, partial transmission sequence, precoding, constellation extension method etc.But these methods are often very loaded down with trivial details, when particularly number of subcarriers is very large, a lot of method is all impracticable.Limit filtration is the simplest and actual most popular method.Then directly limit filtration is carried out to signal inband signaling can be caused to decay and the radiation of out of band signal, need to adopt successive ignition filtering the PAPR of system could be reduced to below the threshold value of specifying simultaneously.Wang, Y2011 publishes thesis optimizediterativeclippingandfilteringforpaperreductiono fOFDMsignals on IEEEtransactionsoncommunications, propose the method Optimized Iterative and the filtering that adopt convex optimization, thus reduce the number of times of iteration.But go to solve convex optimization problem with interior point method in literary composition, make the computation complexity of problem reach O (N ³).So high computation complexity, makes this method really cannot promote the use of in the system of reality.

The technical scheme of prior art one

A kind of method for reducing signal PAR based on self-adapting EVM.Application number: 200810045331.1.

Program profile (Fig. 1): be a kind of method for reducing signal PAR based on self-adapting EVM, comprises step compositions such as the PAPR value of compute sign, the maximum EVM values of compute sign.The present invention utilizes or obtains the larger suppression to peak-to-average force ratio for EVM average can be better, and synchronous signal occurs that the possibility of high peak-to-average power ratio is less, the feature that more often peak-to-average force ratio is all lower, and the input signal larger to peak-to-average force ratio adopts higher EVM to process; And the input signal less to peak-to-average force ratio adopts less EVM to process, thus under ensure that the average EVM of all symbols is no more than the condition of thresholding EVM of system requirements, suppress those symbols occurring high peak-to-average power ratio as much as possible.The method realizes simply only needing to carry out simple amplitude limit and the inside and outside noise limit of band; Obvious to the inhibition of PAPR; Without the need to doing any change to receiver; Without the need to transmitting any secondary information; Applied widely, the system EVM and PAPR being had simultaneously to requirement can be used for, as OFDM or CDMA etc.

The shortcoming of prior art one

1. need to set up EVM and optimum CR relation table when system is set up by a large amount of emulated datas.

2. each iterative process, needs inquiry by EVM and the optimum CR relation table of emulation foundation, is found the amplitude limit rate of current iteration optimum, is carried out amplitude limit in advance.

3. need to set up out-of-band noise amplitude limit spectrum mask in advance.

4. need the process adopting successive ignition amplitude limit and the inside and outside noise level limit of band, just can reach the object reducing papr.

The technical scheme (Fig. 2) of prior art two

Traditional iteration and limit filtration method can cause out of band signal frequency spectrum again to increase and band attenuation.Signal band attenuation can cause error rate of system to increase.This technology uses the design of convex optimization method to replace each iterative process frequency domain response rectangular filter.Can the EVM of minimum signal by the Optimal Filter of design, ensure that the PAPR of signal is lower than specified value simultaneously.

The shortcoming of prior art two

1. still need the process of iteration.

2. program interior point method solves the convex optimization problem of Optimal Filter design.Convex optimization problem tries to achieve globally optimal solution, and the time complexity of system is too high, reaches O (N ³), cannot realize in the real system needing process in real time.

Summary of the invention

The present invention is directed to the deficiencies in the prior art, provide a kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm, the problem that the time complexity in order to resolution system is too high, specific as follows:

In 1.OFDM system, a multi-carrier signal is the superposition of multiple independent sub-carriers signal.Suppose that the data symbol that each subcarrier transmits is discrete-time signal x (n), usually can be represented as:

Wherein X (k) is 16-QAM modulation signal, and L is oversample factor.It has been generally acknowledged that, oversample factor L=4 can obtain the effect being similar to continuous time signal.

For discrete-time signal x (n), PAPR value can be defined as:

e [] represents mathematic expectaion.

2. iteration amplitude limit and filtering

The basic thought of margining amplitude technique is predetermined amplitude limit thresholding, and part ofdm signal envelope being exceeded to thresholding directly eliminates.Margining amplitude technique effectively can suppress signal peak-to-average ratio, and, because its complexity is low, successful and become one of peak-to-average force ratio Restrain measurement be most widely used at present.

Traditional slicing algorithm is exactly the peak value of input signal when exceeding a certain thresholding, just signal is limited on setting thresholding, if lower than this threshold value, then directly passes through.

Signal after amplitude limit can be expressed as:

But, because amplitude limit is a non-linear process, inband distortion and out-of-band radiation can be caused equally, thus reduce bit error rate performance and the spectrum efficiency of whole system.Amplitude limit post filtering can reduce out of band spectrum interference, but this will cause peak regeneration again.In order to avoid this situation, the mode process of Fig. 3 can be adopted.

First with longer IFFT, input data vector is transformed into time domain from frequency domain over-sampling.For given oversample factor J, between frequency domain data vector, add N (J-1) individual 0 to expand original data vector, after IFFT conversion, realize the interpolation of time-domain signal.Then amplitude limit is carried out to the signal after interpolation.Because amplitude limit is non-linear process, therefore it can bring in-band noise and the outer interference of band.In order to cancellation band is disturbed, filtering must be carried out to the signal after amplitude limit outward.Although filtering can cause peak regeneration, more much smaller than the signal peak before amplitude limit.In order to be reduced the peak value of signal as far as possible further by limit filtration process repeatedly.

In successive ignition limit filtration process, the radiation outside signal band attenuation and band can be caused, so definition Error Vector Magnitude EVM retrains signal.

For an OFDM symbol, its EVM can be defined as:

If signal transmission meets EVM constraint, then receiving terminal correctly can recover data.

3. PAPR optimization problem is defined as a constrained optimization problem.

When meeting EVM and PAPR constraint, in subrange, searching a feasible solution, make the complexity of time and hardware minimum, thus can real system be successfully applied to.

4. new problem definition:

Problem: the PAPR optimization problem that hardware is feasible

Input: original time-domain signal x (n), the threshold value PA of PAPR setting _max, the threshold value E of EVM setting _max.

Export: after optimizing, prepare time-domain signal x ' (n) launched.

Constraints:

1)

2)

3)X _outband(k)＝0

This problem proposes motivation and is: can the angle of real time execution from real system, minimum for target with the time cost of real system process.Under the prerequisite of PAPR and EVM index request meeting system reality, seek minimum system processing time.

For overcoming the above problems, the technical solution used in the present invention is as follows, a kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm, iteration limit filtration process model building is repeatedly become optimization problem, under the constraints meeting PAPR and EVM, obtain the locally optimal solution of system with differential evolution algorithm, find PAPR and EVM to meet system condition and the minimum output signal of time complexity simultaneously.

As preferably, specifically comprise the following steps:

S1. the population of differential evolution algorithm is that N ties up noise vector, and initial noisc vector adopts random method to produce from whole spatial noise, and definition N ties up noise vector e (n)

E (n)=x (n)-x ' (n), e (n) is due to the skew that various method is introduced in filtering, and x (n) is original time-domain signal, and x ' (n) is filtered signal;

X′(K)＝DFT[x′(n)]

＝DFT(x(n)-e(n)) _LN

＝X(k)-DFT(e(n)) _LN

＝X(k)-E(k).

E (k) is the frequency domain vectors that e (n) is corresponding, is also the error vector of primary signal and filtered signal, searches for a most suitable E (k), thus meet the requirement of system, claims E (k) for PAPR reduction vector;

Population can be expressed as first generation initial noisc:

wherein, the upper bound and the lower bound of noise vector respectively, rand _j(0,1) is equally distributed stochastic variable, and rand _j(0,1) ∈ [0,1], the scope of noise vector is chosen as-0.5 to 0.5;

S2. by the equation of following differential variation, obtaining variation individuality is:

wherein for the individuality that makes a variation, F is zoom factor, represents that difference vector is to individual influence degree of future generation;

S3. interlace operation, crossing operation operation can be expressed as follows:

Wherein, rand () is the uniform random number between [0,1]; J=1,2 ... m represents a jth variable, and m is the dimension of variable;

S4. for determining trial vector U _i,twhether can become the member in the next generation, DEPR according to greedy criterion by the object vector E in trial vector and current population _i,tcompare, all individualities in the next generation are all better than the correspondence individuality of current population or at least equally good;

The value that in above-mentioned formula, target function obtains for calculating signal PAPR, can calculate according to following formula:

S5. the new individuality not meeting boundary constraint is regenerated trial vector according to S2, then carry out interlace operation, until the new individuality produced meets boundary constraint;

The constraints of EVM is as follows:

If S6. population meets end condition, namely produce an acceptable solution or reach maximum iteration time, then exporting, otherwise forward S2 to;

S7. the selection of controling parameters: population scale is got between 20 to 500; Between F zoom factor F value 0.5 to 1; Between cross-over control parameter value 0 to 1;

S8. set population scale as Np, then the complexity producing initial population is at random O (Np), and the complexity of variation is O (Np), and the complexity of intersection is O (Np), and the complexity of selection is O (Np*Nlog2N),

O(Np)+O(Np)+O(Np)+O(Np*Nlog ₂N)＝O(Np*Nlog ₂N)

Total complexity is:

Beneficial effect of the present invention is as follows: the demand applied in real time from the system of reality, does not need to ask for global optimum by the method for convex optimization, does not namely need to try to achieve minimum PAPR, or minimum EVM, and only demand fulfillment system needs.The present invention, on the basis ensureing low time complexity and hardware spending, is devoted to find feasible solution, that is to say locally optimal solution.Namely find with the fastest time PAPR value meeting system requirements, and meet the constraint of EVM, namely ensure that the error rate of system is within the thresholding formulated.Do not need successive ignition.Do not need to provide EVM and optimum CR relation table by a large amount of emulation experiment in advance.

Specific as follows:

1. the program realizes simple, only needs given input signal, and system directly produces by computing the system met the demands and exports;

2. without the need to doing any change to receiver;

3. without the need to transmitting any side information;

4. the inhibition of couple PAPR is obvious;

5. the Signal-to-Noise exported is higher, can meet the requirement of the set error rate, or the requirement of EVM that system is specified;

6. the scope of application is wide, can have the system of requirement to EVM and PAPR simultaneously, as: OFDM or CDMA.

7. with the comparing of the iterations of the interior point method adopted in prior art two, the effect that method of the present invention only needs iteration namely can reach interior point method iteration and can reach for 30 times for 10 times.

8. the comparison of the error rate of system.Because the constraint of the EVM adopted herein, so the error rate is identical with the method for original transmission side information.Lower than the error rate that conventional iterative amplitude limit method produces.Because the iteration amplitude limit of conventional method and filtering can cause the band attenuation of signal and the radiation outside band, the error rate is caused to increase.

9. use this method, 10 ^-2when probability, reach 6.3dB, the method many reductions 0.5dB in this method comparable technologies scheme two.

Accompanying drawing explanation

Fig. 1 is the schematic flow sheet of prior art one;

Fig. 2 is the schematic flow sheet of prior art two;

Fig. 3 is ofdm system transmitting terminal amplitude limit and over-sampling filtering principle figure;

Fig. 4 is S7.1 population test result schematic diagram;

Fig. 5 is optimal solution schematic diagram required by S7.2;

Fig. 6 is that parameter (crossover probability) regulates, and obtains optimal solution effect schematic diagram;

Fig. 7 is the result schematic diagram that the different CR value of S7.3 obtains;

Fig. 8 be with the iterations of the interior point method adopted in prior art two compare schematic diagram;

Fig. 9 is and conventional method, the interior point method adopted in technology two, the comparison schematic diagram of the iteration limit filtration method in technology one.

Figure 10.That reduces aspect of performance with the interior point method adopted in prior art two and conventional method at peak-to-average power ratio compares schematic diagram.

Embodiment

For making object of the present invention, technical scheme and advantage clearly understand, to develop simultaneously embodiment referring to accompanying drawing, the present invention is described in further details.

A kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm (DEPR),

S1. initial population is produced at random

The population of DEPR algorithm is that N ties up noise vector, and initial noisc vector adopts random method to produce from whole spatial noise.

First definition N is needed to tie up noise vector e (n)

e(n)＝x(n)-x′(n),

E (n) is due to skew that various method is introduced in filtering.Noise can produce identical object, but compares primary signal, can effectively reduce the search volume of signal.X (n) is original time-domain signal, and x ' (n) is filtered signal.

X′(K)＝DFT[x′(n)]

＝DFT(x(n)-e(n)) _LN

＝X(k)-DFT(e(n)) _LN

＝X(k)-E(k).

Discrete Fourier transform (DiscreteFourierTransform, be abbreviated as DFT), LN is the length of DFT, LN point DFT computing, E (k) is the frequency domain vectors that e (n) is corresponding, is also the error vector of primary signal and filtered signal.It directly determines the size of EVM.In the present invention, search for a most suitable E (k), thus meet the requirement of system, claim E (k) for PAPR reduction vector.

Initial noisc vector is produced by random method according to the upper and lower bound of spatial noise.Population can be expressed as first generation initial noisc:

Wherein, the upper bound and the lower bound of noise vector respectively.Rand _j(0,1) is equally distributed stochastic variable, and rand _j(0,1) ∈ [0,1].The scope of noise vector is chosen as (-0.5-0.5).

S2. mutation operation, by the equation of following differential variation, obtaining variation individuality is:

Wherein, for the individuality that makes a variation, F is zoom factor, represents that difference vector is to individual influence degree of future generation.F value size has material impact to algorithm performance.Value is excessive, and algorithm the convergence speed is slack-off; Value is too small, and population diversity reduces, and algorithm is easily absorbed in local optimum.

S3. interlace operation, interlace operation is for increasing the diversity of population.

Crossing operation operation can be expressed as follows

Wherein U trial vector, rand () is the uniform random number between [0,1]; J=1,2 ... m represents a jth variable (gene), and m is the dimension of variable.CR is the crossover probability factor, and CR is larger, right contribution larger, algorithm evolution speed is faster, is also more easily absorbed in local optimum; CR is less, right contribution larger, be more beneficial to and keep the diversity of population, improve the ability of searching optimum of algorithm.

S4. select

For determining trial vector U _i,twhether can become the member in the next generation, DEPR according to greedy criterion by the object vector E in trial vector and current population _i,tcompare.If target function will be minimized, so have and will win a position on the ground compared with the vector of Small object function in population of future generation.All individualities in the next generation are all better than the correspondence individuality of current population or at least equally good.

Wherein, E _{i, t+1}the individuality of new population, f is target function.E _i,tfor previous generation population at individual, U _i,tfor the trial vector produced, the value that in above-mentioned formula, target function obtains for calculating signal PAPR, can calculate according to following formula.

S5. the process of boundary condition

The constraint of demand fulfillment EVM of the present invention, it is necessary for guaranteeing to produce the feasible zone that new individual parameter value is positioned at problem.The new individuality not meeting boundary constraint is regenerated trial vector according to S2, then carries out interlace operation, until the new individuality produced meets boundary constraint.

The constraints of EVM is as follows:

add the frequency domain vectors of the signal after noise, X _kadd the frequency domain vectors of the signal before noise.

S6. termination of iterations

If population meets end condition (namely produce an acceptable solution or reach maximum iteration time), then export, otherwise forward S2 to.

S7. the selection of controling parameters

S7.1 population quantity

From computation complexity analysis, population scale is larger, and the possibility searching globally optimal solution is larger, but the time complexity needed for calculating can increase.But population scale increases sometimes, the precision of optimal solution can be made to reduce, therefore, need Rational choice population scale.

Through a large amount of emulation testing, when setting zoom factor F=0.5 and intersecting factor CR=0.9, population scale is got 1024 from 10, and test result is as Fig. 4.

Test result shows, when population scale is got between 20 to 2000, optimum results is better.

S7.2 zoom factor F

Setting population scale NP=100, intersection factor CR=0.9, required optimal solution as shown in Figure 5.

Time between zoom factor F value [0.51], the result that algorithm obtains is better.As F<0.5 or F>1, the solution that algorithm is tried to achieve of low quality.As can be seen from Figure 5, almost to all test functions when F=0.5, average optimal value is all ideal.

Zoom factor F is for controlling differential vector to the individual impact of variation.When F is larger, differential vector is comparatively large on the impact that variation is individual, can produce larger disturbance, thus is conducive to keeping population diversity.Otherwise when F is less, disturbance is less, zoom factor can play the effect of local fineization search.

S7.3 cross-over control parameter CR

Setting NP=100, F=0.5, the result that different CR values obtains as shown in Figure 7.

As can be seen from Figure 7, experimental subjects individual by variation individual with parent mutually to intersect between component and to produce.The value of CR is larger, then right contribution more, be conducive to opening up new space, thus accelerating ated test, but be tending towards same at later stage variation individuality.Be unfavorable for keeping diversity, thus be easy to precocious, poor stability; The value of CR is less, then right contribution more, so just reduce the ability that algorithm opens up new space, convergence rate is relatively slow, but is conducive to keeping population diversity, thus can have higher success rate.

S8. the complexity of algorithm

If population scale is Np, then the complexity producing initial population is at random O (Np), and the complexity of variation is O (Np), and the complexity of intersection is O (Np), and the complexity of selection is O (Np*Nlog2N).

Therefore, total complexity is

O(Np)+O(Np)+O(Np)+O(Np*Nlog ₂N)＝O(Np*Nlog ₂N)

In this article, get Population Size Np=100 and just can obtain a good optimum results.And the size of Np is well below the sub-carrier number N of reality, because N=1024. establishes G to be maximum iteration time, so DEPR algorithm finds the complexity of optimal solution to be O (GNlog ₂n), this time complexity is well below additive method.

Those of ordinary skill in the art will appreciate that, embodiment described here is to help reader understanding's implementation method of the present invention, should be understood to that protection scope of the present invention is not limited to so special statement and embodiment.Those of ordinary skill in the art can make various other various concrete distortion and combination of not departing from essence of the present invention according to these technology enlightenment disclosed by the invention, and these distortion and combination are still in protection scope of the present invention.

Claims (2)

1. the ofdm system method for suppressing peak to average ratio based on differential evolution algorithm, it is characterized in that, iteration limit filtration process model building is repeatedly become optimization problem, under the constraints meeting PAPR and EVM, obtain the locally optimal solution of system with differential evolution algorithm, find PAPR and EVM to meet system condition and the minimum output signal of time complexity simultaneously.

2. a kind of ofdm system method for suppressing peak to average ratio based on differential evolution algorithm according to claim 1, is characterized in that, specifically comprise the following steps:

S1. the population of differential evolution algorithm is that N ties up noise vector, and initial noisc vector adopts random method to produce from whole spatial noise, and definition N ties up noise vector e (n)

E (n)=x (n)-x ' (n), e (n) is due to the skew that various method is introduced in filtering, and x (n) is original time-domain signal, and x ' (n) is filtered signal;

X′(K)＝DFT[x′(n)]

＝DFT(x(n)-e(n)) _LN

＝X(k)-DFT(e(n)) _LN

＝X(k)-E(k).

E (k) is the frequency domain vectors that e (n) is corresponding, is also the error vector of primary signal and filtered signal, searches for a most suitable E (k), thus meet the requirement of system, claims E (k) for PAPR reduction vector;

Population can be expressed as first generation initial noisc:

$E_{ij}^{\left(0\right)}=E_j^L+{\mathrm{rand}}_j(0,1)\&times;(E_j^U-E_j^L)$

I=1,2 ..., N; J=1,2 ..., D, wherein, the upper bound and the lower bound of noise vector respectively, rand _j(0,1) is equally distributed stochastic variable, and rand _j(0,1) ∈ [0,1], the scope of noise vector is chosen as-0.5 to 0.5;

S2. by the equation of following differential variation, obtaining variation individuality is:

wherein for the individuality that makes a variation, F is zoom factor, represents that difference vector is to individual influence degree of future generation;

S3. interlace operation, crossing operation operation can be expressed as follows:

$U_{ij}^{(t+1)}=\{\begin{array}{cc}{V}_{ij}^{(t+1)}& \begin{array}{cccc}{\mathrm{rand}}_j(0,1)>CR& or& j=j_{rand}& j=1,2...,D\end{array}\\ E_{ij}^{(t+1)}& {\mathrm{rand}}_j(0,1)>CR\end{array}.$

Wherein, rand () is the uniform random number between [0,1]; J=1,2 ... m represents a jth variable, and m is the dimension of variable;

S4. for determining trial vector U _i,twhether can become the member in the next generation, DEPR according to greedy criterion by the object vector E in trial vector and current population _i,tcompare, all individualities in the next generation are all better than the correspondence individuality of current population or at least equally good;

$E_{i,t+1}=\{\begin{array}{cc}{U}_{i,t}& iff\left(U_{ij}^{(t+1)}\right)\&le;f\left(E_{ij}^{\left(t\right)}\right)\\ E_{i,t}& otherwise\end{array}.$

The value that in above-mentioned formula, target function obtains for calculating signal PAPR, can calculate according to following formula:

$\frac{max|x^{\&prime;}\left(n\right){|}^2}{E\&lsqb;|x^{\&prime;}\left(n\right){|}^2\&rsqb;}\&le;{\mathrm{PAPR}}_{max};$

S5. the new individuality not meeting boundary constraint is regenerated trial vector according to S2, then carry out interlace operation, until the new individuality produced meets boundary constraint;

The constraints of EVM is as follows:

If S6. population meets end condition, namely produce an acceptable solution or reach maximum iteration time, then exporting, otherwise forward S2 to;

S7. the selection of controling parameters: population scale is got between 20 to 500; Between F zoom factor F value 0.5 to 1; Between cross-over control parameter value 0 to 1;

S8. set population scale as Np, the complexity then producing initial population is at random O (Np), the complexity of variation is O (Np), the complexity of intersecting is O (Np), the complexity selected is O (Np*Nlog2N), and total complexity is: O (Np)+O (Np)+O (Np)+O (Np*Nlog ₂n)=O (Np*Nlog ₂n).