CN102571178A - Beam forming method used in equivalent isotropic radiated power limited systems - Google Patents

Abstract

The invention discloses a beam forming method used in equivalent isotropic radiated power (EIRP) limited systems, and aims to solve the problem of lower bit error rate performance of the conventional beam forming method. The method disclosed by the invention comprises the steps that firstly, antennas are selected and an initial beam vector is compressed; and then, a numerical solution is obtained through an exhaustive search method to approximate to the optimal solution. Compared with the prior art, the beam forming method disclosed by the invention has the advantages that through selecting and using more antennas, the information of a channel matrix is utilized more fully, power consumption in a radiation peak direction can be further reduced; and compared with a soft clipping scheme, the beam forming method disclosed by the invention has the advantages that greater received power is obtained and the bit error rate performance of a system is better. The beam forming method disclosed by the invention is suitable for the EIRP systems, and 60GHz systems as well except ultra-wideband systems.

Description

Beam-forming method in a kind of EIRP constrained system

Technical field

The invention belongs to wireless communication technology field, be specifically related to the beam-forming method in equivalent isotropically radiated power (EIRP, the Equivalent Isotropic Radiated Power) constrained system.

Background technology

In ultra broadband (UWB) system of multi-antenna multi-carrier-wave; Because the pair of orthogonal frequency division multiplexing (OFDM of US Federal Communication Committee (FCC); Orthogonal Frequency Division Multiple) power spectral density of multi-carrier ultra-broadband system has strict demand; Promptly for each subcarrier, equivalent isotropically radiated power is not more than-41.3dBm/MHz.Because traditional desirable transmit beam-forming algorithm (intrinsic beam shaping); The spatial beams that utilizes channel characteristic value to produce to have directivity; And the total transmitting power of only simple restriction can not make the power of all directions satisfy the EIRP restriction; So intrinsic beam shaping algorithm is not suitable for the EIRP constrained system.To the problems referred to above; Existed some to be applicable to the transmit beam-forming scheme of the limited radio ultra wide band system of EIRP; These methods have further limited transmitting power, and comprising compression intrinsic beam shaping scheme and antenna selecting plan, but these schemes make the bit error rate performance of system very poor.On the basis of compression intrinsic beam shaping, propose compression algorithm and the quick and smooth iterative algorithm designs the transmit beam-forming vector, further improved received signal to noise ratio.Compare with compression intrinsic beam shaping scheme, these algorithms have reduced the error rate and have improved systematic function.For multicarrier UWB system, under the EIPR confined condition, continuing to improve systematic function is challenging task.

Mathematical analysis about EIRP restriction: to the transmitting terminal antenna is that the situation of uniform straight line array (ULA) is analyzed.Suppose λ _cBe the transmission carrier wavelength, antenna distance is d _T, array element number is n _T, array element is arranged along the z direction of principal axis, with the angle of z axle be θ.n _TUnit's line array is as shown in Figure 1, the antenna array vector $a\left(\mathrm{\&theta;}\right)={(1,e^{\mathrm{j\&Omega;}},{\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">e</figure-callout>}}^j,...,e^{j(n_T-1)\mathrm{\&Omega;}})}^H,$ Ω=(2 π/λ wherein _c) d _TCos (θ).Suppose d _T>=λ _c/ 2, θ ∈ (π, π], obvious Ω ∈ (π, π].Suppose that the receiving terminal antenna number is n _R=1, transmitting terminal known channel state information, i.e. k number of sub-carrier upper signal channel matrix information H _k, the reception signal y on the k number of sub-carrier _k=H _kw _kx _k+ n _k, x _kBe that average is 0, variance is 1 multiple scalar symbol, n _kBe the multiple Gaussian noise vector of circulation symmetry, n _kEach element independent same distribution, and average is 0, variance is N ₀w _kRepresent the beam vector on the k number of sub-carrier.

EIRP restriction normalization on the k number of sub-carrier can be expressed as: $P_{\mathrm{EIRP}}\left(\mathrm{\&theta;}\right)=\underset{\mathrm{\&theta;}\&Element;\left(\mathrm{0,2}\mathrm{\&pi;}\right]}{\max }\left|{a\left(\mathrm{\&theta;}\right)}^H{w}_k\right|\&le;1,$ Following formula can also be expressed as P (Ω)=| a (Ω) ^Hw _k| ²≤1.P (Ω) can regard w as _kFourier inversion mould square.For uniform straight line array, can regard beam vector w as _kCarry out the IDFT sampling.n _T* 1 dimension beam vector w _kThe IDFT that corresponding length is K is r _k=(r ₁, r ₂..., r _K) ^T, wherein

(j=1,2 ..., K).Therefore, r _kWith w _kCan represent: $r_k={\mathrm{\&Theta;}}_{K\&times;n_T}{w}_k.$ Wherein,

${\mathrm{\&Theta;}}_{K\&times;n_T}={\left[\begin{array}{cccc}1& 1& ...& 1\\ 1& {\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">e</figure-callout>}}^j& <figure-callout\; id="2"\; label="...\; e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">.</figure-callout>..& e^j{2\mathrm{\&pi;}(n_t-1)/K}^{}\\ M& M& O& M\\ 1& {\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">e</figure-callout>}}^j& <figure-callout\; id="2"\; label="...\; e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">.</figure-callout>..& e^j{2\mathrm{\&pi;}(n_T-1)(K-1)/K}^{}\end{array}\right]}_{K\&times;n_T}$

Be w _kCorresponding IDFT matrix, K are represented the direction in space number got, satisfy the EIRP restriction on the fixed direction getting.The value of K can change, and the K value is big more, calculates accurately more, but K must be a finite value.

By $P_{\mathrm{EIRP}}\left(\mathrm{\&theta;}\right)=\underset{\mathrm{\&theta;}\&Element;\left(\mathrm{0,2}\mathrm{\&pi;}\right]}{\mathrm{Max}}\left|{a\left(\mathrm{\&theta;}\right)}^H{w}_k\right|\&le;1$ Can obtain ${\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_k\left|\right|}_{\&infin;}^2\&le;1.$ In order to improve the error rate of system performance, need to consider how to design transmission beam vector w _kImprove received signal to noise ratio or the signal to noise ratio of receiving terminal is maximized.Problem equivalent in:

$\underset{w_k}{\max }\mathrm{\&rho;}\left(w_k\right){\left|\right|H_k{w}_k\left|\right|}_2^2$

$\mathrm{subjectto}{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_k\left|\right|}_{\&infin;}^2\&le;1$

Because

Only when getting equal sign, ρ (w _k) just can obtain optimal solution.Therefore,

$\mathrm{\&rho;}\left(w_k\right)=\frac{{\left|\right|H_k{w}_k\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_k\left|\right|}_{\&infin;}^2}.$

It line options (Antenna Selection) comprises emitting antenna selecting and reception antenna selection, and the antenna selection criterion mainly comprises the channel capacity maximal criterion and receives error rate minimum criteria.Here the emitting antenna selecting method of considering has been selected unique antenna, and adopts reception error rate minimum criteria, even received signal to noise ratio (SNR) maximization.It line options: unit vector e _lRepresent that l element is 1, beam vector in the antenna selecting method

Wherein

Its method is described in detail can list of references: Cheran M.Vithanage; Steve C.J.Parker; Magnus Sandell.Antenna selection with phase precoding for high performance UWB communication with legacy WiMedia multi-band OFDM devices.Proc.IEEE Int.Conf.Communications; Pp.3938-3942, May 2008.Its advantage is: when number of transmit antennas was two, emitting antenna selecting can make error rate of system minimum under the EIRP restrictive condition; When number of transmit antennas surpassed two, antenna selecting method was easy to realize that complexity is very low.Its shortcoming is: when number of transmit antennas surpasses two, and the error rate of system poor-performing.

Compression intrinsic beam shaping algorithm (Scaled Eigen-beamforming) compresses the intrinsic beam vector, makes transmitting power satisfy the EIRP requirement for restriction.Can obtain intrinsic beam vector w by the intrinsic beam shaping _EB, w _EBBe

Eigenvalue of maximum characteristic of correspondence vector, the beam vector after the compression

Can be applicable to the EIRP constrained system.This method is described in detail can list of references: C.M.Vithanage; Y.Wang; And J.P.Coon.Transmit beamforming methods for improved received signal-to-noise ratio in equivalent isotropic radiated power-constrained systems.IET Communications; Vol.3, PP.38-47,2009.Its shortcoming only is simply limit transmit power and causes received signal to noise ratio very little, thereby makes the bit error rate performance of system very poor.

Level and smooth reduction scheme (Soft clipping method) is similar to the principle that suppresses peak-to-average power ratio (PAPR) in the time domain OFDM system is applied to the spatial domain.This method is confirmed the radiation peak direction of compression intrinsic beam vector earlier; Change the peak value direction then, be similar to the limit filtration technology of ofdm system, the beam vector of frequency domain is transformed to time domain; Again the amplitude of corresponding time domain vector is necessarily handled; The peak-to-average power ratio that is amplitude carries out level and smooth iteration compression to satisfy particular requirement, and the time domain vector after the iterative processing transforms to frequency domain again, and frequency domain vectors is made the beam vector of the scheme of can smoothly being cut down after correspondence is handled again to radiation direction.Level and smooth reduction scheme is passed through repeatedly the power loss that iteration reduces the radiation peak direction, thereby obtains higher received power, improves systematic function.Relevant the method is narrated in more detail; Can list of references: C.M.Vithanage; Y.Wang, and J.P.Coon.Transmit beamforming methods for improved received signal-to-noise ratio in equivalent isotropic radiated power-constrained systems.IET Communications, Vol.3; PP.38-47,2009.

In general, the bit error rate performance of above-mentioned three kinds of methods does not all reach requirement, and antenna selecting method is poor with compression intrinsic beam shaping algorithm bit error rate performance, though smoothly the scheme of cutting down has improved bit error rate performance, the amplitude of improving is little.

Summary of the invention

The objective of the invention is to have proposed the beam-forming method in a kind of EIRP constrained system, on the basis of level and smooth reduction scheme, further improve the error rate of system performance in order to solve the lower problem of existing beam-forming method bit error rate performance.

Technical scheme of the present invention is: the beam-forming method in a kind of EIRP constrained system, the idiographic flow sketch map is as shown in Figure 2, comprises the steps:

Step 1: day line options: the n from the EIRP constrained system _TSelect two corresponding antenna S of maximum and time big channel gain and T in the root antenna, wherein, n _T＞2, the weights of antenna S and T are respectively: p, λ p, all the other all n _TThe weights of-2 antennas are 0, n _TRoot antenna probability assignments is designated as

Wherein, λ is the weights coefficient of antenna T, and magnitude range is: 0＜λ≤1, then the initial beam vector of i root antenna 1≤i≤n _T, (H _k) _iBe the channel gain on the k number of sub-carrier on the i root antenna, ∠ representes to ask phase bit arithmetic, () _iI element got in expression;

Step 2: compression initial beam vector: be chosen in the compression initial beam vector on the k number of sub-carrier

Wherein, w _{0, k}Corresponding direction spin matrix

${\mathrm{\&Theta;}}_{K\&times;n_T}=\left[\begin{array}{cccc}1& 1& ...& 1\\ 1& {\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">e</figure-callout>}}^j& <figure-callout\; id="2"\; label="...\; e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">.</figure-callout>..& e^j{2\mathrm{\&pi;}(n_T-1)/K}^{}\\ M& M& O& M\\ 1& {\mathrm{<figure-callout\; id="2"\; label="e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">e</figure-callout>}}^j& <figure-callout\; id="2"\; label="...\; e\; j"\; filenames="HDA0000138991640000031.png,HDA0000138991640000041.png"\; state="\{\{state\}\}">.</figure-callout>..& e^j{2\mathrm{\&pi;}(n_T-1)(K-1)/K}^{}\end{array}\right],$ K representes the direction in space number got, || || _∞Represent infinite norm computing; Obtain the initial received power on the k number of sub-carrier || || ₂Represent 2 norm computings;

Step 3: loop iteration number of times J and M are set, loop initialization variable j=1,

λ _jBe the weights coefficient of the j time circulation time antenna T, w _{0, k}(λ _j) be the j time compression initial beam vector on the circulation time k number of sub-carrier, loop initialization variable m=1;

Step 4: calculate $r_{k,m}={\mathrm{\&Theta;}}_{n_T\&times;n_T}{w}_{m-1,k}\left({\mathrm{\&lambda;}}_j\right),$ Wherein, r _{K, m}Be n _T* 1 column vector, r _{K, m}The peak-to-average power ratio of amplitude does

r _i=(r _{K, m}) _iExpression r _{K, m}I element, 1≤i≤n _T, PAPR ₀=0, Δ=| PAPR _m-PAPR _M-1|;

If Δ＞ε, wherein, ε is a preset threshold value, ${\left(r_{k,m+1}\right)}_i=\left(\right|{\left(r_{k,m}\right)}_i|-\frac{{\left|{\left(r_{k,m}\right)}_i\right|}^3}{3})\mathrm{Exp}(j\&angle;{\left(r_{k,m}\right)}_i),$

Obtain

The peak value direction of space radiation by

Value is definite,

Calculate

With $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$ Otherwise, w _{M, k}(λ _j)=w _M-1, k (λ _j), $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$

Step 5:m=m+1, if m≤M, repeating step 4; Otherwise; obtains

and

if j＜J; J=j+1 then; M=1; Repeating step 4; Otherwise, forward step 6 to;

Step 6: calculate

wherein $\underset{x}{\arg \max }f\left(x\right):=\left\{x\right|\&ForAll;y:f\left(y\right)\&le;f\left(x\right)\};$

Step 7: if

Then beam vector does

Otherwise beam vector is w _ASS=w _AS, wherein, w _ASBe the beam vector that antenna selecting method obtains, ρ (w _AS) be beam vector w _ASThe corresponding power of accepting.

Only also can use other searching method such as dichotomy, unimodal function search methods (claiming Fibonacci method or 0.618 method again) etc. that can reduce complexity as a reference in the step 3 with typical evenly searching method.Because such scheme is that λ is evenly searched for, the complexity of calculating in order to reduce calculation times, can obtain to make ρ (w than higher through dichotomy (being the method for binary search) _k(the λ)) λ of optimum.Adopt the antenna subset selection scheme of dichotomy to compare with the antenna subset selection scheme that adopts even search method like this, the cycle-index of step 3 has tangible reduction.In the step 3, because ρ is (w _k(λ)) at interval 0＜λ _jThe≤1st, unimodal function utilizes the unimodal function searching method, further reducing under the situation of computation complexity, can guarantee systematic function.

Be example with the dichotomy below, corresponding step 3 is made following modification:

Step 3: loop iteration number of times J and M are set, loop initialization variable j=1, λ ₀=0, λ ₁=1,

λ _J+1=(λ _a+ λ _b)/2, λ _jBe the weights coefficient of the j time circulation time antenna T, w _{0, k}(λ _j) be the j time compression initial beam vector on the circulation time k number of sub-carrier, loop initialization variable m=1.

Beneficial effect of the present invention: method of the present invention is at first carried out day line options and compression initial beam vector; Method through exhaustive search obtains numerical solution and approaches optimal solution then; Can further reduce the power loss of radiation peak direction; Compare with level and smooth reduction scheme, the received power that obtains is bigger, and the bit error rate performance of system is more excellent.Compared with prior art, the present invention has the following advantages:

(1) beam-forming method of the present invention utilizes more antenna through selection, thereby has utilized channel matrix information more fully;

(2) beam-forming method of the present invention is applicable to the EIRP constrained system, is not only applicable to radio ultra wide band system, is equally applicable to the 60GHz system.

Beam-forming method of the present invention is compared with conventional method, can further improve systematic function.

Description of drawings

Fig. 1 is the aerial array sketch map of the first line array of N of the present invention.

Fig. 2 is a beam-forming method schematic flow sheet of the present invention.

Fig. 3 is the errored bit performance comparison figure of beam-forming method of the present invention and conventional method under the radio ultra wide band system CM3 channel circumstance.

Fig. 4 is the errored bit performance comparison figure that beam-forming method of the present invention adopts different search spacings under the radio ultra wide band system CM3 channel circumstance.

Fig. 5 is the errored bit performance sketch map of beam-forming method of the present invention under the CM1 of the 60GHz system channel circumstance.

Embodiment

To combine accompanying drawing below, provide specific embodiment of the present invention.Need to prove: the parameter among the embodiment does not influence generality of the present invention.

The channel that adopts among Fig. 3 and Fig. 4 is the CM3 channel in the 802.15.3a standard channel model; Relevant this channel is narrated in more detail; Can list of references: IEEE P802.15 Working Group for Wireless Personal Area Networks; Channel Modeling Sub-committee Report Final, IEEE P802.15-02/368r5-SG3a, Dec.2002.The channel that adopts among Fig. 5 is the CM1 channel in the 802.15.3c standard channel model; Relevant this channel is narrated in more detail; Can list of references: IEEE P802.15 Working Group for Wireless Personal Area Networks; TG3cChannel Modeling Sub-committee Report Final, IEEE 15-07-0584-01-003c, Sep.2010.

As shown in Figure 3, under radio ultra wide band system CM3 channel circumstance, the errored bit performance comparison of beam-forming method of the present invention and conventional method.Wherein, beam-forming method adopts even searching method, and the search spacing is 0.1, and searching times is 10 times, and promptly maximum cycle is J=10.When the search spacing is 0.1, at the j time circulation time, λ _jValue do

λ _jValue can be from 0.1,0.2, and 0.3 gets 1 always, at BER=10 ^-4The time, its errored bit performance is compared with level and smooth reduction scheme and has been improved about 0.7dB.

As shown in Figure 4, under radio ultra wide band system CM3 channel circumstance, the errored bit performance comparison of beam-forming method of the present invention when adopting difference search spacing, the difference search spacing of employing is respectively: 0.5,0.2,0.1,0.05; Corresponding maximum cycle is respectively 2,5,10,20.Along with dwindling of search spacing, just accurate more to the location of λ, just more can find and make ρ (w _kThe bigger pairing λ value of value (λ)), its corresponding system's errored bit performance are just better.As can be seen from the figure, when the search spacing is 0.5, λ _jValue can only get 0.5 and 1, its errored bit performance even poorer than the errored bit performance of level and smooth reduction scheme.But, when the search spacing is 0.05, λ _jValue can be from 0.05,0.1, and 0.15,0.2 gets 1 always, at BER=10 ^-4The time, its errored bit performance is compared with level and smooth reduction scheme and has been improved about 0.8dB.

As shown in Figure 5; Under the CM1 of 60GHz system channel circumstance; Adopt the errored bit performance sketch map of the beam-forming method of different searching methods; Wherein, the beam-forming method of even searching method is adopted in ASS (evenly search method) expression, and the beam-forming method of dichotomy is adopted in ASS (dichotomy) expression.

Above instance is merely preferred example of the present invention, and use of the present invention is not limited to this instance, and is all within spirit of the present invention and principle, any modification of being made, is equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (2)

1. the beam-forming method in the EIRP constrained system comprises the steps:

Step 1: day line options: the n from the EIRP constrained system _TSelect two corresponding antenna S of maximum and time big channel gain and T in the root antenna, wherein, n _T＞2, the weights of antenna S and T are respectively: p, λ p, all the other all n _TThe weights of-2 antennas are 0, n _TRoot antenna probability assignments is designated as

Wherein, λ is the weights coefficient of antenna T, and magnitude range is: 0＜λ≤1, then the initial beam vector of i root antenna 1≤i≤n _T, (H _k) _iBe the channel gain on the k number of sub-carrier on the i root antenna, ∠ representes to ask phase bit arithmetic, () _iI element got in expression;

Step 2: compression initial beam vector: be chosen in the compression initial beam vector on the k number of sub-carrier

Wherein,

w _{0, k}Corresponding direction spin matrix

${\mathrm{\&Theta;}}_{K\&times;n_T}=\left[\begin{array}{cccc}1& 1& ...& 1\\ 1& e^{j2\mathrm{\&pi;}/K}& ...& e^{j2\mathrm{\&pi;}(n_T-1)/K}\\ M& M& O& M\\ 1& e^{j2\mathrm{\&pi;}(K-1)/K}& ...& e^{j2\mathrm{\&pi;}(n_T-1)(K-1)/K}\end{array}\right],$ K representes the direction in space number got, || || _∞Represent infinite norm computing; Obtain the initial received power on the k number of sub-carrier || || ₂Represent 2 norm computings;

Step 3: loop iteration number of times J and M are set, loop initialization variable j=1,

λ _jBe the weights coefficient of the j time circulation time antenna T, w _{0, k}(λ _j) be the j time compression initial beam vector on the circulation time k number of sub-carrier, loop initialization variable m=1;

Step 4: calculate $r_{k,m}={\mathrm{\&Theta;}}_{n_T\&times;n_T}{w}_{m-1,k}\left({\mathrm{\&lambda;}}_j\right),$ Wherein, r _{K, m}Be n _T* 1 column vector, r _{K, m}The peak-to-average power ratio of amplitude does

r _i=(r _{K, m}) _iExpression r _{K, m}I element, 1≤i≤n _T, PAPR ₀=0, Δ=| PAPR _m-PAPR _M-1|;

If Δ＞ε, wherein, ε is a preset threshold value, ${\left(r_{k,m+1}\right)}_i=\left(\right|{\left(r_{k,m}\right)}_i|-\frac{{\left|{\left(r_{k,m}\right)}_i\right|}^3}{3})\mathrm{Exp}(j\&angle;{\left(r_{k,m}\right)}_i),$

Obtain The peak value direction of space radiation by

Value is definite,

Calculate

With $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$ Otherwise, w _{M, k}(λ _j)=w _{M-1, k}(λ _j), $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$

Step 5:m=m+1, if m≤M, repeating step 4; Otherwise;

obtains

and

if j＜J; J=j+1 then; M=1; Repeating step 4; Otherwise, forward step 6 to;

Step 6: calculate

wherein $\underset{x}{\arg \max }f\left(x\right):=\left\{x\right|\&ForAll;y:f\left(y\right)\&le;f\left(x\right)\};$

Step 7: if

Then beam vector does

Otherwise beam vector is w _ASS=w _AS, wherein, w _ASBe the beam vector that antenna selecting method obtains, ρ (w _AS) be beam vector w _ASThe corresponding power of accepting.

2. the beam-forming method in the EIRP constrained system comprises the steps:

Step 1: day line options: the n from the EIRP constrained system _TSelect two corresponding antenna S of maximum and time big channel gain and T in the root antenna, wherein, n _T＞2, the weights of antenna S and T are respectively: p, λ p, all the other all n _TThe weights of-2 antennas are 0, n _TRoot antenna probability assignments is designated as

Wherein, λ is the weights coefficient of antenna T, and magnitude range is: 0＜λ≤1, then the initial beam vector of i root antenna 1≤i≤n _T, (H _k) _iBe the channel gain on the k number of sub-carrier on the i root antenna, ∠ representes to ask phase bit arithmetic, () _iI element got in expression;

Step 2: compression initial beam vector: be chosen in the compression initial beam vector on the k number of sub-carrier

Wherein, w _{0, k}Corresponding direction spin matrix

${\mathrm{\&Theta;}}_{K\&times;n_T}=\left[\begin{array}{cccc}1& 1& ...& 1\\ 1& e^{j2\mathrm{\&pi;}/K}& ...& e^{j2\mathrm{\&pi;}(n_T-1)/K}\\ M& M& O& M\\ 1& e^{j2\mathrm{\&pi;}(K-1)/K}& ...& e^{j2\mathrm{\&pi;}(n_T-1)(K-1)/K}\end{array}\right],$ K representes the direction in space number got, || || _∞Represent infinite norm computing; Obtain the initial received power on the k number of sub-carrier

|| || ₂Represent 2 norm computings;

Step 3: loop iteration number of times J and M are set, loop initialization variable j=1, λ ₀=0, λ ₁=1,

λ _J+1=(λ _a+ λ _b)/2, λ _jBe the weights coefficient of the j time circulation time antenna T, w _{0, k}(λ _j) be the j time compression initial beam vector on the circulation time k number of sub-carrier, loop initialization variable m=1;

Step 4: calculate $r_{k,m}={\mathrm{\&Theta;}}_{n_T\&times;n_T}{w}_{m-1,k}\left({\mathrm{\&lambda;}}_j\right),$ Wherein, r _{K, m}Be n _T* 1 column vector, r _{K, m}The peak-to-average power ratio of amplitude does

r _i=(r _{K, m}) _iExpression r _{K, m}I element, 1≤i≤n _T, PAPR ₀=0, Δ=| PAPR _m-PAPR _M-1|;

If Δ＞ε, wherein, ε is a preset threshold value, ${\left(r_{k,m+1}\right)}_i=\left(\right|{\left(r_{k,m}\right)}_i|-\frac{{\left|{\left(r_{k,m}\right)}_i\right|}^3}{3})\mathrm{Exp}(j\&angle;{\left(r_{k,m}\right)}_i),$

Obtain

The peak value direction of space radiation by

Value is definite,

Calculate

With $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$ Otherwise, w _{M, k}(λ _j)=w _{M-1, k}(λ _j), $\mathrm{\&rho;}\left(w_{m,k}\left({\mathrm{\&lambda;}}_j\right)\right)=\frac{{\left|\right|H_k{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_2^2}{{\left|\right|{\mathrm{\&Theta;}}_{K\&times;n_T}{w}_{m,k}\left({\mathrm{\&lambda;}}_j\right)\left|\right|}_{\&infin;}^2};$

Step 5:m=m+1, if m≤M, repeating step 4; Otherwise;

obtains

and

if j＜J; J=j+1 then; M=1; Repeating step 4; Otherwise, forward step 6 to;

Step 6: calculate wherein $\underset{x}{\arg \max }f\left(x\right):=\left\{x\right|\&ForAll;y:f\left(y\right)\&le;f\left(x\right)\};$

Step 7: if

Then beam vector does

Otherwise beam vector is w _ASS=w _AS, wherein, w _ASBe the beam vector that antenna selecting method obtains, ρ (w _AS) be beam vector w _ASThe corresponding power of accepting.

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