CN104683006B - Beamforming Method based on Landweber iterative methods - Google Patents

Beamforming Method based on Landweber iterative methods Download PDF

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CN104683006B
CN104683006B CN201510059401.9A CN201510059401A CN104683006B CN 104683006 B CN104683006 B CN 104683006B CN 201510059401 A CN201510059401 A CN 201510059401A CN 104683006 B CN104683006 B CN 104683006B
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mrow
msup
msubsup
mfrac
landweber
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CN104683006A (en
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刘文龙
张博
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Dalian University of Technology
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0613—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission
    • H04B7/0684—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using simultaneous transmission using different training sequences per antenna
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/0408—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas using two or more beams, i.e. beam diversity
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
    • H04B7/00—Radio transmission systems, i.e. using radiation field
    • H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
    • H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
    • H04B7/06—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station
    • H04B7/0697—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the transmitting station using spatial multiplexing

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  • Computer Networks & Wireless Communication (AREA)
  • Signal Processing (AREA)
  • Radio Transmission System (AREA)

Abstract

The invention discloses a kind of Beamforming Method based on Landweber iterative methods, and it includes carrying out discrete sampling after band logical signal of communication solution is transferred into base band by antenna for base station；Using covariance matrix corresponding to top n discrete sampling data as training sequence and solution, calculate the relaxation factor α in Landweber iterative formulas and corresponding Wave beam forming weight w is solved by Landweber iterative formulas(m)；The weight w that will be obtained(m)It is brought into cost function G (m) and solves the preferable iteration stopping number based on Landweber iterative methods in cost function G (m), solve the regularization factors m for causing cost function G (m) to be minimum, and the best weight value of corresponding Landweber iterative methods；Finally using the weights of the Landweber iterative methods solved, Wave beam forming operation is carried out to subsequent sampling sequence.The present invention utilizes Landweber iterative Wave beam forming weights, determines number of iterations by simplified cost function, finally obtains the iterative solution that can resist systematic error.

Description

Beamforming Method based on Landweber iterative methods
Technical field
The invention belongs to signal of communication process field, a kind of wave beam of reception signal in smart antenna is more particularly related to Forming method.
Background technology
One of key technology in the intelligent antenna technology mobile communication later as the third generation and the third generation, have and improve Signal Interference and Noise Ratio, improve communication quality, increase power system capacity, improve number of users, improve frequency efficiency, reduce electromagnetism The many merits such as interference.Beamforming Method is the core of smart antenna, and its quality directly determines the performance of smart antenna.It is conventional Beamforming Method include sample matrix inversion (SMI) method, diagonal loading (DL) method and eigen-subspace projection (ESB) method.Wherein, SMI methods calculate simple, it is not necessary to determine experience factor size in advance.But this method is for being Error of uniting is especially sensitive, and very small error may result in method performance and decline to a great extent, it is difficult to apply in practice.DL methods are Improvement to SMI methods, but the method needs to determine the size of a load factor related to noise in advance, there is presently no Effective method calculates the size of the factor, typically using empirical value.When noise is bigger, ESB methods can be shown very well Robustness, but when signal to noise ratio is smaller, the method performance will acutely decline.
The content of the invention
In view of defects in the prior art, a kind of practical the invention aims to provide, to systematic error not Sensitive Beamforming Method, this method utilize Landweber iterative Wave beam forming weights, pass through simplified Godard generations Valency function determines number of iterations, finally obtains the iterative solution that can resist systematic error.
To achieve these goals, the present invention adopts the following technical scheme that：
Beamforming Method based on Landweber iterative methods, it is characterised in that：
Comprise the following steps
S1, by antenna array receiver band logical signal of communication, and by the signal of communication solution be transferred to base band carry out discrete sampling；
S2, using the preceding n times sampled data of above-mentioned discrete sampling data as training sequence, solve and the training sequence pair The covariance matrix answered：
Wherein, N is the training sequence number of samples, and k is sampled point x (k) ∈ CM×1For the baseband communication signal after demodulation Vector, M are corresponding element number of array,xH(k)∈C1×MIt is x (k) conjugate transposition；
S3, to above-mentioned covariance matrixEigenvalues Decomposition is carried out, the relaxation factor required for solving in iterative formula α；
Described S3 is to above-mentioned covariance matrixEigenvalues Decomposition is carried out, the relaxation required for solving in iterative formula Factor-alpha, comprise the following steps：
S31, first to covariance matrixEigenvalues Decomposition is carried out to obtain
Wherein U ∈ CM×MFor unitary matrix, Λ ∈ CM×MFor diagonal matrix, UH∈CM×MFor U associate matrix；
S32, obtain in Λ maximally diagonal element λmax；
S33, make the λ of α=0.5/max, obtain α.
S4, by Landweber iterative formulas, solve the weight w of the m times iteration needed for this method(m),
WhereinFor the unit matrix of M ranks,For depending on antenna pattern and signal incident direction The array response vector of (i.e. weighting vector), i ∈ { 0,1,2 ..., m-1 }；
S5, the weight w that will be obtained in S4(m)It is brought into cost function G (m) and solves, wherein described cost function G (m) Refer to：
Described cost function G (m) is the Godard cost functions after simplifying, and its corresponding analysis process is：
Constant modulus property based on signal of communication, regularization factors m is determined according to the Godard cost functions of classics, then Its corresponding cost function G is (in w on mLandIn), c, a (in γ) and H2Function, and be non-convex, specific function For
Such as the optimal value of this function of (5-1) formula, using the optimization method of classics, it is difficult to solve that such as gradient descent method, which is, , therefore this method is simplified to Godard cost functions, it is as follows that it simplifies process：
A square expansion for wushu (5-1) first can obtain：
In (5-2) formula,According to optimum theory, Obtain partial differentials of the G on c：
In view of in the minimum point of (5-1) formula, corresponding (5-3) formula should be equal to 0, therefore c must is fulfilled for：
(5-4) is brought into (5-1), obtains simplified Godard cost functions：
Pass through simplification, it can be seen that simplified cost function has been changed to m one-variable function and there was only variable m's Function,Preferable iteration stopping number m is not interfered with (in w as just oneLand) constant, such as formula (5-5)；
In actual applications, constant instead of it is expected, is all typically omitted using average simultaneously, therefore this method is actual The simplification Godard cost functions used are：
The m for making formula (5-6) minimum is obtained (in wLandIn), as regularization factors of Landweber algorithms, formula (5-6) In wLandIt is w to correspond to this method(m)；
S6, apply Rule of judgment：Judge whether m is more than threshold value, otherwise make m=m+1, again repeat step S4 to step S5, until m>End during threshold value；
After S7, application Rule of judgment terminate, solve the ideal based on Landweber iterative methods in cost function G (m) and change In generation, stops number, that is, solves the regularization factors m for make it that cost function G (m) is minimized, and corresponding Landweber iteration The best weight value of method；
S8, the best weight value using the Landweber iterative methods solved in S7, the sampling to subsequent sampling data composition Sequence carries out Wave beam forming operation.
Compared with prior art, beneficial effects of the present invention：
The present invention is solved Wave beam forming weights using iterative algorithm and stopped using Landweber iterative algorithms come selective Only iteration, while the Godard cost functions of simplification a kind of are proposed easily to determine iteration stopping number, so that of the invention There is good robustness for the systematic error in Wave beam forming.
Brief description of the drawings
Fig. 1 is flow chart of steps corresponding to the present invention；
Fig. 2 is its general principles structural representation；
Fig. 3 is array physical model schematic diagram used in the present invention；
Fig. 4 is the implementing procedure figure of the embodiment of the present invention；
Fig. 5 is semiconvergent phenomenon figure when iterative algorithm solves the problems, such as Wave beam forming；
Fig. 6 a are the inventive method and existing Beamforming Method emulates obtained performance comparision figure；
Fig. 6 b are the inventive method and existing Beamforming Method emulates obtained performance comparision figure；
Fig. 6 c are the inventive method and existing Beamforming Method emulates obtained performance comparision figure；
Fig. 6 d are the inventive method and existing Beamforming Method emulates obtained performance comparision figure.
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, below in conjunction with accompanying drawing, the present invention is entered Row is further described.
Major design principle of the present invention：When solving Wave beam forming weights using iterative algorithm, it may appear that a kind of semiconvergent shows As, as shown in figure 5, therefore the present invention using Landweber iterative algorithms come selective stopping iteration, and propose one kind Simplified Godard cost functions easily determine iteration stopping number, so that the present invention misses for the system in Wave beam forming Difference has good robustness.
Wherein, the present invention can solve weights corresponding to Wave beam forming by Landweber iteration, and it is corresponding
Iteration be：
Wherein w(m)∈CM×1It is the weighted vector of iteration the m times,It is that relaxation factor (meets certain model Enclose the constant of requirement),It is M rank unit matrixs,It is to depend on antenna pattern and signal incident direction The array response vector of (weighting vector), the reason for can improving Beam-former robustness using Landweber iteration, are： When using solution by iterative method Wave beam forming weighted problem, it may appear that a kind of semiconvergent phenomenon, i.e., when number of iterations gradually increases from 0 When, iterative solution can gradual similarity to ideal solution, but when iterate to certain number, iterative solution is again away from ideal solution.As shown in figure 5, This semiconvergent phenomenon is illustrated, in Figure 5, the quality of iterative solution is weighed using output signal interference-to-noise ratio (SINR), That is output SINR is bigger, and iterative solution is closer to ideal solution；Output SINR is smaller, and iterative solution is further away from ideal solution.Landweber changes Stopping can be iterated by one preferable iterations of selection for algorithm, make iterative solution closest to ideal solution, so that Algorithm more robust.Assuming that preferably iteration stopping number is m, then the weights of Landweber algorithms can be write：
Above-mentioned principle is described in detail with practical embodiments and corresponding accompanying drawing (such as Fig. 1-Fig. 6) below：
The present invention is the improved method on beam-forming technology, and beam-forming technology is by an antenna array receiver Signal, desired signal (a) then is proposed by adjusting the weights of every antenna, and eliminate interference signal and the skill of noise (b) Art, as shown in Figure 2.
In the present invention, the physical model of its base station end reception signal used is as shown in figure 3, it includes one vertically puts The uniform array antenna put, it is base station receiving array, and the array number of corresponding antenna is M, and the spacing of array is d.
Such as Fig. 4, specific implementation step of the invention is：
S1, the narrow band communication signal of distal end is received by uniform array antenna, and is demodulated to base band and is carried out discrete adopt Sample.
It includes
S (1a) receives distal end narrow band communication signal using even linear array antenna；
The signal of communication solution received is transferred to base band by S (1b)；
The signal of communication that S (1c) is transferred to base band to solution carries out discrete sampling, obtains the signal phasor x (k) of kth time sampling.
S2, before selection then n times sampled data solves the covariance matrix of these data as training sequence
It includes
N times sampled data is as training sequence before S (2a) chooses
S (2b) solves the covariance matrix of above-mentioned training sequence
Wherein N represents training sequence number of samples, and k is sampled point x (k) ∈ CM×1For the baseband communication signal arrow after demodulation Amount, M is element number of array,xH(k)∈C1×MFor x (k) conjugate transposition.
S3, to covariance matrixEigenvalues Decomposition is carried out, obtains relaxation factor α.
It includes
S (3a) is to covariance matrixCarry out Eigenvalues Decomposition；
In formula, U ∈ CM×MFor unitary matrix, Λ ∈ CM×MFor diagonal matrix, UH∈CM×MFor U conjugate transposition；
S (3b) obtains diagonal element λ maximum in ΛmaxAnd make the λ of α=0.5/max。
S4, using Landweber iterative formulas, obtain the weights of the m times iteration.
It includes
S (4a) utilizes Landweber iterative method formula
The weights of the m times iteration can be obtained, whereinIt is the unit matrix of M ranks,It is and letter Number weighting vector and the related array response of array configuration, it can be estimated by DOA estimate technology.
S5, the weights that step 4 is obtained bring simplified Godard cost function G (m) into, obtain G (m) now；
S (5a) brings the weights for the m times iteration obtained in step 4 in G (m) into, wherein G (m) is expressed as：
Wherein, (w(m))HIt is w(m)Conjugate transposition, i, k are sampled point,
Wherein simplified Godard cost function G (m) are obtained by following processes：
In view of the output of Beam-former can be written as：
Wherein, k is sampled point, x (k)=s (k)+i (k)+n (k), s (k)=cs (k) a ∈ CM×1It is to be used comprising expectation Family information s (k) desired signal vector, i (k) ∈ CM×1It is interference signal vector, n (k) ∈ CM×1It is noise signal vector, wLand ∈CM×1It is the weighted vector obtained using this method, corresponding to S (4a) w(m), c is channel fading coefficient, a ∈ CM×1It is the phase The ideal array response of signal is hoped,, both sides simultaneously divided by c γ, can obtain：
Because s (k) is constant modulus signals, if so Beam-former can sufficiently filter out interference and noise, wave beam shape The output grown up to be a useful person should accurately meet constant modulus property.Therefore, the permanent mould error for minimizing output signal can be alternatively The criterion of Landweber Beam-former regularization factors.Godard perseverance mould cost functions are one and are based on signal constant modulus property Classical cost function, be widely used in the fields such as blind equalization.Determined just using Godard cost functions herein Then change factor m, its cost function can be written as：
In formula,It is the constant related to the modulation system of signal.For 2-PAM Signal, i.e. s (k) ∈ {+1, -1 }, H2=1.It is obvious that if Beam-former can accurately recover s (k), i.e.,Then G=0.
From the foregoing, it will be observed that preferably regularization factors m should be able to make cost function G minimize cost letter closest to 0 Number.The minimum of a function is solved, most directly considers to be exactly using traditional optimization method, such as gradient descent method.But From above formula (6) it can be seen that not only m is (in wLandIn) it is unknown, and c and a (in γ) is also unknown.And on Modulation intelligence can not know in advance sometimes, i.e. H2It is and unknown.In this case, the cost function G be on m, C, a and H2Function, and be non-convex.Specific function, which can be write, to be done：
It is difficult to solve that the optimal solution of this function is using traditional optimization method.In order to determine Landweber A kind of regularization factors of algorithm, based on Godard cost functions G, it is proposed that cost function of simplification.Pass through the flat of wushu (7) Fang Zhankai can be obtained：
In formula,According to optimum theory, can obtain Obtain partial differentials of the G on c：
In the minimum point of (7) formula, (9) formula should be equal to 0.Therefore c must is fulfilled for：
(10) are brought into (7), just obtain simplified cost function：
It can be seen that, γ has been cancelled from formula (11),As just one do not interfere with preferable regularization because Sub- m is (in wLand) constant.(11) the cost function G in has had changed into only unknown parameter m function of a single variable.
In actual applications, constant instead of it is expected, is all typically omitted using average, can be actually used Simplify cost function：
Obtain the m, as Landweber algorithms that make formula (12) minimum regularization factors.
S6, make m=m+1；Repeat step 4 is to step 5 again, until m>End during threshold value, the preferred empirical value of threshold value 1000；
S7, obtain the m, as Landweber iterative methods that enable to G (m) to minimize in G (m) preferable iteration stopping Number, and obtain the best weight value of Landweber iterative methods.
S8, Wave beam forming is carried out to follow-up sample sequence using the best weight value obtained.
Further, effect of the invention can be further illustrated by following l-G simulation test
Corresponding simulated conditions：
In emulation, reception antenna is letter using the uniform linear array of M=11 array element, array element spacing d=λ/2, wherein λ Number wavelength.Using 1 desired signal and two interference sources, their incident angle is respectively 10 °, 40 ° and 70 °.Assuming that it is expected The interference signal that signal and direction of arrival are 40 ° is bpsk signal, and direction of arrival is 70 ° of interference signal, and it is equal to meet It is worth for 0, variance is the random signal of 1 Gaussian Profile.Channel fading coefficient c=1, ideal array response vector a meet aHA= M (=11).The reception noise of each array element is independent identically distributed additivity unit white Gaussian noise.Believed using the output of signal Number interference-to-noise ratio (SINR) carrys out the performance of comparative approach, and its calculation formula is as follows：
Corresponding emulation content：
In emulation, by the inventive method and existing SMI, the SINR that DL and ESB methods obtain is compared, with explanation Advantage of the invention relative to prior art.The present invention and the SINR of other method such as scheme with training sequence length N change Shown in 6a, the present invention and the SINR of other method with the error in pointing of array response change as shown in Figure 6 b, the present invention and its The SINR of his method with the uncertainty of array response change as fig. 6 c, the present invention is with the SINR of other method with input SNR change is as shown in fig 6d.It is can be seen that from Fig. 6 a-6d under similarity condition, the present invention obtains performance than existing technology It is good a lot, and all shown very well to training sequence length change, array response mismatch change and the change of signal to noise ratio Robustness.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto, Any one skilled in the art the invention discloses technical scope in, technique according to the invention scheme and its Inventive concept is subject to equivalent substitution or change, should all be included within the scope of the present invention.

Claims (3)

  1. A kind of 1. Beamforming Method based on Landweber iterative methods, it is characterised in that：
    Comprise the following steps
    S1, by antenna array receiver band logical signal of communication, and by the signal of communication solution be transferred to base band carry out discrete sampling；
    S2, using the preceding n times sampled data of above-mentioned discrete sampling data as training sequence, solve corresponding with the training sequence Covariance matrix：
    <mrow> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <msup> <mi>x</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, N is the training sequence number of samples, and k is sampled point x (k) ∈ CM×1For the baseband communication signal arrow after demodulation Amount, M are corresponding element number of array,xH(k)∈C1×MIt is x (k) conjugate transposition；
    S3, to above-mentioned covariance matrixEigenvalues Decomposition is carried out, the relaxation factor α required for solving in iterative formula；
    S4, by Landweber iterative formulas, solve the weight w of the m times iteration needed for this method(m),
    <mrow> <msup> <mi>w</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mi>&amp;alpha;</mi> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mi>I</mi> <mo>-</mo> <mi>&amp;alpha;</mi> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mi>i</mi> </msup> <mover> <mi>a</mi> <mo>~</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    WhereinFor the unit matrix of M ranks,For the array depending on antenna pattern and signal incident direction Response vector, i ∈ { 0,1,2 ..., m-1 }；
    S5, the weight w that will be obtained in S4(m)It is brought into cost function G (m) and solves, wherein described cost function G (m) is Refer to：
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mrow> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>w</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>w</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msubsup> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mrow> <mo>|</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>w</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>4</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>;</mo> </mrow>
    S6, apply Rule of judgment：Judge whether m is more than threshold value, otherwise make m=m+1, again repeat step S4 to step S5；
    After S7, application Rule of judgment terminate, solve the preferable iteration based on Landweber iterative methods in cost function G (m) and stop Only count, that is, solve the regularization factors m for make it that cost function G (m) is minimized, and corresponding Landweber iterative methods Best weight value；
    S8, the best weight value using the Landweber iterative methods solved in S7, to the sample sequence of subsequent sampling data composition Carry out Wave beam forming operation.
  2. A kind of 2. Beamforming Method based on Landweber iterative methods according to claim 1, it is characterised in that：
    Described S3 is to above-mentioned covariance matrixEigenvalues Decomposition is carried out, solves the relaxation factor α needed in iterative formula, Comprise the following steps：
    S31, first to covariance matrixEigenvalues Decomposition is carried out to obtain
    <mrow> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>=</mo> <msup> <mi>U&amp;Lambda;U</mi> <mi>H</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
    Wherein U ∈ CM×MFor unitary matrix, Λ ∈ CM×MFor diagonal matrix, UH∈CM×MFor U associate matrix；
    S32, obtain in Λ maximally diagonal element λmax；
    S33, make the λ of α=0.5/max, obtain α.
  3. A kind of 3. Beamforming Method based on Landweber iterative methods according to claim 1, it is characterised in that：
    Described cost function G (m) is the Godard cost functions after simplifying, and simplified process corresponding to it is：
    Constant modulus property based on signal of communication, regularization factors m is determined according to the Godard cost functions of classics, then its is right The cost function G answered is on m, c, a and H2Function, specific function is：
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mfrac> <mrow> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi>c</mi> <mi>&amp;gamma;</mi> </mrow> </mfrac> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Wherein, in γ, m's a existsM represents iteration stopping number, and c represents channel fading coefficient, a ∈ CM×1It is desired signal Ideal array responds,It is the weighted vector obtained using this method, H2=E | s (k) |4}/E{|s(k)|2It is the constant related to the modulation system of signal, s (k)=cs (k) a ∈ CM×1It is to be used comprising expectation Family information s (k) desired signal vector；
    This method is obtained by a square expansion for wushu (5-1)：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>&amp;xi;</mi> <mo>,</mo> <mi>&amp;zeta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfrac> <mi>&amp;xi;</mi> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>4</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <mi>&amp;zeta;</mi> </mrow> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <msubsup> <mi>H</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mfrac> <msqrt> <mi>&amp;xi;</mi> </msqrt> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mi>&amp;zeta;</mi> <msqrt> <mi>&amp;xi;</mi> </msqrt> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msubsup> <mi>H</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> <mfrac> <msup> <mi>&amp;zeta;</mi> <mn>2</mn> </msup> <mi>&amp;xi;</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    In (5-2) formula,According to optimum theory, G is obtained Partial differential on c：
    <mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>G</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>,</mo> <msub> <mi>H</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>&amp;xi;</mi> <mo>,</mo> <mi>&amp;zeta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>c</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>&amp;zeta;</mi> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> </mrow> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>4</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>4</mn> <mi>&amp;xi;</mi> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>3</mn> </msup> </mrow> <msup> <mrow> <mo>|</mo> <mi>c</mi> <mo>|</mo> </mrow> <mn>8</mn> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    In view of in the minimum point of (5-1) formula, corresponding (5-3) formula should be equal to 0, therefore c must is fulfilled for：
    <mrow> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> <mo>=</mo> <mfrac> <mi>&amp;zeta;</mi> <mi>&amp;xi;</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <msup> <mrow> <mo>|</mo> <mi>&amp;gamma;</mi> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> </mrow> <mrow> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>4</mn> </msup> <mo>}</mo> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    (5-4) is brought into (5-1), and then obtains simplified Godard cost functions：
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>H</mi> <mn>2</mn> <mn>2</mn> </msubsup> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mi>E</mi> <mo>{</mo> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>4</mn> </msup> <mo>}</mo> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    By simplification, this method has obtained only variable m function,Preferable iteration is not interfered with as just one Stop number m constant, such as formula (5-5)；
    In actual applications, constant instead of it is expected, is all typically omitted using average simultaneouslyTherefore this method actual use Simplification after Godard cost functions be：
    <mrow> <mi>G</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msubsup> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msup> <mrow> <mo>|</mo> <msubsup> <mi>w</mi> <mrow> <mi>L</mi> <mi>a</mi> <mi>n</mi> <mi>d</mi> </mrow> <mi>H</mi> </msubsup> <mi>x</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>4</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>-</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    Obtain the m, as Landweber algorithms that make formula (5-6) minimum regularization factors, the w in formula (5-6)LandCorrespond to this Method is w(m)。
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