CN104679976B - Contraction for signal transacting is linear and shrinks the multiple least-squares algorithm of generalized linear - Google Patents
Contraction for signal transacting is linear and shrinks the multiple least-squares algorithm of generalized linear Download PDFInfo
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Abstract
In order to solve fixed renewal step-length traditional in actual applications, and convergence rate when not considering the noncircularity of signal is slow and the problems such as mean square error is big, the present invention proposes that contraction is linear and the multiple least-squares algorithm of contraction generalized linear is applicable and Adaptive beamformer, it make use of variable step size during right value update, so that the instantaneous mean error of posteriori error when not considering noise minimizes, and the multiple least-squares algorithm for shrinking generalized linear also contemplates the noncircularity of desired signal.Both approaches improve convergence rate and greatly reduce Steady State Square Error.
Description
Technical field
The present invention relates to array signal process technique field, more particularly to one kind to shrink linearly and shrink generalized linear again most
Young waiter in a wineshop or an inn's multiplication algorithm.
Background technology
Array signal processing is an important branch in field of signal processing, has been reached its maturity by the development of decades
And suffer from being widely applied in multiple military and national economy fields such as radar, biologic medical, exploration and astronomy.Its work
Principle is that spacing wave is received and handled into sensor array, and using this array by multiple sensor groups, purpose
It is to suppress interference and noise, extracts the useful information of signal.Different from general signal processing mode, array signal processing is logical
The sensor group reception signal for being arranged in space is crossed, and information is filtered and extract using the Spatial characteristic of signal.Therefore, battle array
Column signal processing is also often known as spatial domain signal transacting.In addition, array signal processing has flexible wave beam control, very strong anti-
The advantages that interference performance is with high space hyperresolution, thus receive the concern of numerous scholars, its application is not yet
Increase disconnectedly.
In array signal processing field, most important two research directions are adaptive-filtering and Estimation of Spatial Spectrum, wherein
Auto-adaptive filtering technique prior to Estimation of Spatial Spectrum produce, and its apply it is quite varied in engineering system.However, for sky
Power estimation has obtained quick development in nearly 30 years between although, and correlative study content is quite varied, its engineer applied system
It is rare.Here, auto-adaptive filtering technique is a key concept in array signal processing field.
Adaptive-filtering may apply in modelling, equilibrium, control, Echo Canceller and Adaptive beamformer.It is multiple
The least-squares algorithm of value is a kind of ART network and Predicting Technique, it is possible to achieve performance converges to optimal wiener solution.It is adaptive
Beam-former weight vector is answered to be calculated based on different design criterias, conventional criterion has least mean-square error, minimum
Variance and constant holdup model, the criterion of the invention using least mean-square error.
The complex value wave filter for being round signal, being commonly found a LTI that classical adaptive array utilizes
W, the output of wave filterSecond order criterion is optimized under conditions of certainty constraint.But actually should
In, not rounded signal has been widely applied in many Modern Communication System.Due to the adaptive beam former pair of classics
It is optimal for circle signal, but is suboptimum for not rounded signal.Therefore, broad sense complex value least square method is believed using extension
NumberThe mean square error between lower wave beam device output and desired signal can be obtained.And
Complex value least square method is also suboptimum for the not rounded signal of second order, therefore how to ensure to obtain the optimal value of not rounded signal, with
And convergence rate and output Signal to Interference plus Noise Ratio are improved, reduce the emphasis that mean square error is problem.
The content of the invention
During in order to solve fixed renewal step-length traditional in actual applications and not consider the noncircularity of signal, convergence speed
The problems such as slow and mean square error is big is spent, the present invention proposes a kind of shrink and linearly calculated with the complex value least square for shrinking generalized linear
Method, this method is under conditions of noncircularity is considered so that do not consider that the instantaneous mean error of posteriori error during noise minimizes,
The variable step size of near-optimization is obtained, so as to improve convergence rate, reduces mean square error.
The present invention is achieved through the following technical solutions：
The contraction of the present invention linearly answers least square method and shrinks the multiple least square method of generalized linear, different from former method
Be that not only make use of noncircularity but also make use of contraction algorithm to obtain variable renewal step value, then improved in terms of following two
Performance：
1. improve convergence rate using noncircularity and reduce mean square error.Specific implementation step is as follows：
First, when signal as desired signal is BPSK, QPSK and PAM, then now desired signal can be analyzed toThe signal then now received is related to its conjugation, i.e. conjugation the inside contains desired signal
Useful information, therefore, Cx=E [x (k) xT(k)]≠0M×M.Then extension signal now is Array output and phase are then drawn by mean-square error criteria again
The cost function of error can try to achieve weights now between prestige signal.Now, the array weight of extension isNot rounded information is then now make use of, convergence rate is improved, reduces mean square error.
2. recycle the method shunk.Specific implementation step is as follows：
Shrink linear least square seek the step-length of change during, then because step-length isUnder normal circumstances in actual applications, usual E [| | x (k) | |2] be it is known, E [| e
(k)|2] it is to pass throughEstimate, wherein λ be forgetting factor and 0 ＜ ＜ λ≤
1.However, ef(k) it is unknown, it is difficult to by solving E [ef(k)|2] solve above-mentioned step-length.Shrinkage de-noising method can be used, is led to
Cross prior uncertainty e when prior uncertainty e (k) recovers noiselessf(k).Make f [ef(k)]=0.5 | ef(k)-e(k)|2+α|ef
(k) | it is minimum, then it can recover ef(k).Now, then the value of variable step size is can obtain, so as to obtain the renewal process of weights.From
And convergence rate can be improved, reduce mean square error.
Brief description of the drawings
The contraction that Fig. 1 is the present invention is linear and shrinks the multiple least-squares algorithm A method of estimation flow charts of generalized linear；
Fig. 2 (a) and Fig. 2 (b) are algorithm of the invention and multiple least square method, generalized linear a most young waiter in a wineshop or an inn again when Q is different
The output letter drying of multiplication is than the change curve with iterations；
Fig. 3 (a) is algorithm of the invention and multiple least square method, broad sense line when Q is fixed, step-length is different with Fig. 3 (b)
Property multiple least square method output letter drying than the change curve with iterations；
Fig. 4 is that the algorithm of the present invention believes drying ratio with iteration with the output of VSS, CNLMS, WL-CNLMS and WL-VSS algorithm
The change curve of number；
Fig. 5 (a) and Fig. 5 (b) are algorithm of the invention and multiple least square method, generalized linear a most young waiter in a wineshop or an inn again when Q is different
The mean square error of multiplication with iterations change curve；
When Fig. 6 (a) with Fig. 6 (b) is that step-length is different, algorithm of the invention and multiple least square method, the multiple minimum of generalized linear
The mean square error of square law with iterations change curve；
Fig. 7 is the mean square error of algorithm and VSS, CNLMS, WL-CNLMS and WL-VSS algorithm of the present invention with iterations
Change curve.
Embodiment
The present invention is further described for explanation and embodiment below in conjunction with the accompanying drawings.
Consider the even linear array of a M array elements, receive a far field narrow band signal s0(k), corresponding direction of arrival is θd.This
Signal is zero-mean, and second order is not rounded.Then array output data can be expressed as：
X (k)=a (θd)s0(k)+n(k)
Wherein,For the steering vector of desired signal, Δ
The array spacings between adjacent array element are represented, λ represents wavelength, n (k)=[n1(k) ..., nM(k)]TFor additive noise vector, it by
Ambient noise and interference form, and can be expressed as：
Wherein, the P incoherent not rounded interference of statistics, their complex envelope is si(k), i=1,2 ..., P, and it is right
The steering vector answered is a (θi), i=1,2 ..., P, η (k) are and desired signal and the incoherent ambient noise of interference.
When weight vector is w=[w1..., wM]TWhen, then optimal weight vector can be by making output and the reason of Beam-former
Think signal sd(k) mean square error minimum obtains
Wherein, sd(k)=s0(k), can solve best initial weights by some computings is：
Wherein,Make the power J [w (k)] of instantaneous variance minimum
So as to obtain the renewal process of weights
W (k+1)=w (k)+μ e*(k)x(k)。 (3)
When desired signal is not rounded signal, such as BPSK, QPSK, PAM etc., then now desired signal vector is represented byGenerally, Cx=E [x (k) xT(k)]≠0M×MIn order to utilize noncircularity, extended vectorIt is represented by
Wherein,The steering vector and noise vector respectively extended.Similar with multiple least square method is wide
The cost function of the multiple least square method of justice is
Wherein,For the instantaneous error of extension,Represent the wave beam of extension
The output of shaper.
Now,The renewal process of broad sense weight vector is
MakeMinimize, obtain optimal broad sense weight vector
Wherein,
Change the μ in formula (3) into variable step size μk, then now the renewal process of weight vector is
Assuming that sequence pair { x (k), s0(k) } it is extended stationary, therefore optimal weight vector wopt(k) it is constant when being, i.e.,
wopt(k)=wopt.If the error vector v (k) of weight vector=w (k)-wopt, then the renewal process that can obtain v (k) is
Wherein,At the k moment, output and the desired signal s of Beam-former0(k)
Between error be
Wherein,Prior uncertainty when being noiseless.
In addition, posteriori error is represented by ε (k)=∈opt(k)+εf(k), wherein
For noiseless when posteriori error, both sides after formula (8) conjugate transposition are multiplied into x (k) in the right side simultaneously, then above formula is substituted into, obtains
Arrive
εf(k)=(1- μk||x(k)||2)ef(k)-μk∈opt(k)||x(k)||2. (11)
The make an uproar energy of posteriori error of instantaneous nothing can be expressed as
By formula (12) both sides simultaneously to μkIt is equal to 0 after derivation, then obtains
Formula (9) is substituted into formula (13), obtained
It is rightThe right side multiplies x (k) and both sides while asks expectation afterwards conjugation again, then
Therefore, input signal x (k) withIt is that statistics is vertical.WhenIt is very small and in stable state change very
When slow, e*(k) it is incoherent with x (k), can obtain
E[||x(k)||2|e(k)|2]=E [| | x (k) | |2]E[|e(k)|2]. (16)
It was observed that in j ＜ k, list entries for it is independent when, v (k) only with { x (j), s0(j) it is relevant, and independently of current
Input signal x (k), can obtain
Expectation is asked to the both sides of formula (14) and convolution (15) arrives the result of (17), is obtained
Wherein, E [μk||x(k)||2|e(k)|2]=E [μk]E[||x(k)||2|e(k)|2] (19)
Work as μkWhen being constant, above formula is set up certainly.In fact, in stable state, μkChange compared with x (k), e (k) slow.
It is therefore contemplated that μkIt is approximate incoherent with x (k), e (k), i.e., formula (19) is approximate sets up.
Due to E [| ef(k)|2] it is the unnecessary mean square error as caused by the error between weight vector and optimal weight vector.
In formula (3), μ is usedkInstead of μ, the right value update process of multiple least square method can be obtained.
In actual applications, usual E [| | x (k) | |2] be it is known, E [| e (k) |2] estimated by following formula
Wherein λ is forgetting factor and 0 ＜ ＜ λ≤1.However, ef(k) it is unknown, it is difficult to by solving E [ef(k)|2] come
Solve (20).Shrinkage de-noising method can be used, prior uncertainty e during noiseless is recovered by prior uncertainty e (k)f(k)。
f[ef(k)]=0.5 | ef(k)-e(k)|2+α|ef(k)| (22)
Make above formula on ef(k) minimize, can obtain
It can thus be appreciated that α selection is extremely important.Assuming that ambient noise is white Gaussian noise, covariance isInterference is inclined
The most of energy that can suppress interference sections from the main lobe of desired signal then has
It can be obtained by (9) and (24)
In uniform linear array, in order to ensure that the beam pattern in the direction of arrival of desired signal is 1 while maximized | | wopt
||2, generally setThus obtain
In summary, we then can beWherein, Q is a parameter, for compensating above-mentioned approximation.
Similar to E [| e (k) |2], it can obtain
Result in result in above formula and (21) is substituted into (20), available right value update step-length now is
By the μ in formula (28)kInstead of the μ in (3), you can obtain the renewal process of the weight vector of multiple least square method.
Similarly, w is individually subtracted in the both ends of two formulas in (6)1、w2Best initial weights w1, opt、w2, opt, can obtain
The renewal process of weight vector error is as follows：
Now use variable step size μkTo substitute the μ in (6).According to (5) and (28), the vector matrix of power error vector is obtained
Form
Wherein,Be extension weight vector waveform output with
Error between desired signal.(30) both sides are carried out into conjugate transposition simultaneously, and the right side multiplies again
It is respectively without the posteriori error and prior uncertainty made an uproar due to extending
With contraction linearly least square method is similar again, what the nothing instantaneously extended made an uproar posteriori error square is
Using above formula as cost function, it is asked on μkDerivative, and by its be equal to 0, can obtain
It is in the instantaneous error at k moment
(36) are substituted into (35) and obtained
Pass throughConjugate transposition then is asked again simultaneously to its both sides
The right side multipliesBoth sides ask expectation then to have simultaneously again
Assuming that also withIt is uncorrelated, from above formulaWith the input signal of extensionIt is vertical andThen can prior uncertaintyWith x (k) and incoherent, i.e.,
Know v by (29)1(k), v2(k) respectively with x (k), x*(k) it is incoherent.If
Known by the result of (33) and (38)
Expectation is asked simultaneously to (37) both sides, recycles the result of (38) and (39) to obtain
Upper formula is based on the assumption that
E [μk] it is used as μkEstimation substitute into (30), just obtain the multiple least square method of generalized linear.
In formula (41)Estimation can be obtained by following formula
The nothing of extension is made an uproar prior uncertaintySquare average
To replace in (41)Extension is without prior uncertainty of making an uproarIt can pass throughRecover
Then, it is necessary to how select thresholding α, due to interference and ambient noise be all with desired signal it is incoherent, then
It is represented by
Wherein,It is the power of desired signal.Then optimal weight vector now is
This result is similar to the undistorted response of minimum variance of Generalized optimal
Wherein, WithOnly constant component is different.When the direction of arrival of interference
When being differed greatly with the direction of arrival of desired signal, then have
Wherein,I=1,2 ..., P.WhereinIt is the initial of i-th of interference
Phase.Approximation be
Wherein,Due toTo then it be obtained in this substitution (50)
Thus, can obtainBy in these substitutions (3-41), then have
Result in (52) is substituted into (6) to the right value update process of the multiple least square method of the broad sense that can be then shunk.
As shown in Figure 1, the linear least-squares algorithm of contraction of the invention comprises the following steps：
1. calculate posteriori error ε during noiselessf(k) energy, then to by (3-12) both sides while to μkIt is equal to after derivation
(3-9) is substituted into (3-13) and then obtained with reference to (3-15) (3-16) by 0 again
2. due to usual E [| | x (k) | |2] it is known, E [| e (k) | 2] is obtained by (3-21) estimation, and can be with
With shrinkage de-noising method, even if (3-22) minimum obtains
3. by (3-27) can obtain E [| ef(k)|2] estimator；
4. the result of estimator is substituted into μ in (1)kIt can obtain in formula
5. obtain right value update process w (k+1)=w (k)+μke*(k)x(k)。
The contraction generalized linear least-squares algorithm of the present invention comprises the following steps：
1. calculate posteriori error during instantaneous extension noiselessSquare, then to by (3-34) both sides simultaneously to μkAsk
It is equal to 0 after leading, then (3-36) is substituted into (3-35) and then obtained with reference to (3-38) (3-39)
2. due to usual E [| | x (k) | |2] be it is known,It is to be obtained by (3-43) estimation, and can uses and receive
Contracting denoising method, even if (3-45) minimum obtains
It is 3. available by (3-24)Estimator；
4. the result of estimator is substituted into μ in (1)kIt can obtain in formula
5. obtain right value update process：
Consider an even linear array, array number M=4, array pitch isThe bpsk signal of four constant powers, they
Not rounded coefficient be 1, initial phase is 0 °, and direction of arrival when desired signal incides array is θd=-45 °, signal to noise ratio is
10dB, the direction of arrival of other three interference signals is respectively θ1=8 °, θ2=-13 °, θ3=30 °, dry make an uproar is fixed as than (INR)
10dB, all simulation results obtain by 500 Monte Carlo Experiments.It is in the initial value of the linear least square of contractionAnd w (0)=0M×1.Moreover, the initial value of the multiple least square method of the generalized linear shunk isw1(0)=0M×1And w2(0)=0M×1.Forgetting factor λ is fixed, λ=0.95.
The output letter drying of experiment 1 is than the change with iterations.
Will algorithm more proposed by the present invention and multiple least square method, the multiple least square method of generalized linear in this emulation
With the situation of change of iterations, when the step-length of multiple least square method and the multiple least square method of generalized linear is respectively 0.001,
0.0005, it is observed that in both cases, consider convergence rate during broad sense (noncircularity) to be faster than do not consider it is not rounded
Property situation, and output Signal to Interference plus Noise Ratio when considering noncircularity is also higher, from accompanying drawing 2 as can be seen that Q is answered linear contraction
The Steady-state Properties of least square method influence smaller.Moreover, the multiple least square method of the generalized linear shunk in Q=2 than Q=1 when
Performance increase.Accompanying drawing 3 is multiple least square method and the multiple least square method of generalized linear in accompanying drawing 3 (a) when Q is fixed
Step-length be respectively 0.008,0.004, and in accompanying drawing 3 (b) multiple least square method and the multiple least square method of generalized linear step-length
Obtained when respectively 0.0005,0.00025.From accompanying drawing 3 (a) as can be seen that when the step-length that step-length is made a farfetched comparison in Fig. 2 (b) is big by 8
Times when, multiple least square method is greater than accompanying drawing 2 (b) with the multiple least square method convergence rate of generalized linear.Due to larger step-length
Convergence rate can be improved.However, in stable state, output of the multiple least square method of generalized linear with multiple least square method now is believed
Drying is smaller.If we are fixed as step-length the μ of accompanying drawing 2 (b) half, from accompanying drawing 3 (b), the multiple minimum of generalized linear
Square law reaches the output Signal to Interference plus Noise Ratio same with algorithm proposed by the invention in stable state with multiple least square method.However,
Both approaches are respectively necessary for 200,400 iterationses when reaching stable state, and are then only respectively necessary for 100 in accompanying drawing 2 (b),
200 times are that can reach stable state.This is due to that step-length is smaller, reduces convergence rate.It can be seen that the algorithm of the present invention from accompanying drawing 3
About it is respectively necessary for 60,100 iteration and reaches stable state.And the Signal to Interference plus Noise Ratio of the algorithm output of the present invention reaches approximately most respectively
Excellent generalized linear Signal to Interference plus Noise Ratio, optimal Signal to Interference plus Noise Ratio.Therefore the algorithm of the present invention relative to the multiple least square method of generalized linear and
For multiple least square method, there are faster convergence rate and higher output Signal to Interference plus Noise Ratio.
Accompanying drawing 4 is that least square method and the multiple least square method of standard are linearly answered in contraction more proposed by the invention
(CNLMS) the multiple least square method of generalized linear and broad sense line are shunk in, step length changing method (VSS), and comparison proposed by the invention
Property standard multiple least square method (WL-CNLMS), the output Signal to Interference plus Noise Ratio of generalized linear step length changing method (WL-VSS).The multiple mark of setting
The step-length of quasi- least square method and the multiple standard least-squares of generalized linear is respectively 0.2,0.1.The parameter of variable step is μmin=
e-6、μmax=3e-3、WithBroad sense variable step parameter is μmin=5e-7、μmax=1.5e-3、WithFrom accompanying drawing 4 as can be seen that compared with multiple standard least-squares and the multiple least square method of broad sense, calculation of the invention
Method convergence rate is faster.Moreover, compared with step length changing method and broad sense step length changing method, there is algorithm of the invention higher output to believe
Dry ratio of making an uproar.It can also be seen that this several method during consideration noncircularity is better than algorithm when not considering noncircularity.
The relation of the output mean square error of experiment 2 and iterations.
In an experiment, the worth setting of each variable and last experiment are the same.It can be seen that the present invention's from accompanying drawing 5
For algorithm for the multiple least square method of generalized linear and multiple least square method, convergence rate is very fast.The shrinkage front of the present invention
Property multiple least square method and multiple least square method approximation in stable state have equal mean square error, equally, contraction of the invention is wide
It is adopted that linearly least square method and the multiple least square method approximation in stable state of broad sense have equal mean square error again.It is additionally, since and examines
Noncircularity is considered, the method based on generalized linear there are better properties.Because the array aperture of extension makes the multiple minimum of contraction broad sense
Selection of the square law to α is more sensitive, then from accompanying drawing 5 (a) as can be seen that as Q=1, the generalized linear of contraction of the invention
Multiple least square method will be slightly inferior to the multiple least square method of broad sense in stable state.From accompanying drawing 5 (b) as can be seen that as Q=2, this
The multiple least square method of the contraction generalized linear of invention will be slightly better than the multiple least square method of broad sense in stable state.And unlike based on line
Property algorithm, Q is worth size to influence very little to shrinking linearly least square method again and multiple least square method.Accompanying drawing 6 is to compare this
The algorithm of invention and the multiple least square method of multiple least square method and broad sense with different step values.It is attached compared with accompanying drawing 5 (b)
The multiple least square method of multiple least square method and generalized linear in Fig. 6 (a) has faster convergence rate, because their step-length
Value is respectively 0.008,0.004, is 8 times in accompanying drawing 5 (b).From accompanying drawing 6 (a) as can be seen that the multiple least square method of broad sense and
Multiple least square method converges to -4dB and 0dB respectively.However, in accompanying drawing 5 (b), during μ=0.001, then they restrain respectively
To -8dB and -4dB.Accompanying drawing 6 (b) is respectively μ=0.00025 for algorithm and the step-length of the display present invention, the broad sense of μ=0.0005
The comparison of linear least square method again and the mean square error of multiple least square method.Although multiple least square method and generalized linear are again most
Small square law can reach identical steady-state value with inventive algorithm, but compared with accompanying drawing 5 (b), they need bigger change
Generation number.The former is respectively necessary for 200,400 iteration, and the latter is respectively necessary for 150,250 iteration.Therefore the algorithm of the present invention
No matter in convergence rate or multiple least square method and the multiple least square method of generalized linear will be better than in terms of mean square error.
In fig. 7, it is algorithm more of the invention and the multiple standard least-squares of multiple standard least-squares, generalized linear
The mean square error of method, step length changing method and generalized linear step length changing method.The multiple standard most young waiter in a wineshop or an inn of multiple standard least-squares, generalized linear
The parameter setting of multiplication, step length changing method and generalized linear step length changing method is with being identical in experiment 1.It can be seen that from accompanying drawing 7
Least square method is linearly answered in the contraction of the present invention and the multiple least square method of contraction generalized linear is respectively necessary for 60,50 iteration and reached
To stable state.However, multiple standard least-squares and the linear standard least-squares again of justice are respectively necessary for 100,150 iteration and reached
Stable state.Therefore consider that convergence rate during noncircularity will be faster than when not considering noncircularity.Although step length changing method and generalized linear become
Step length has approximate convergence rate with this paper algorithms, and still, they have very big imbalance in stable state.It follows that this hair
Bright algorithm has higher convergence rate and less mean square error.
Above content is to combine specific preferred embodiment further description made for the present invention, it is impossible to is assert
The specific implementation of the present invention is confined to these explanations.For general technical staff of the technical field of the invention,
On the premise of not departing from present inventive concept, some simple deduction or replace can also be made, should all be considered as belonging to the present invention's
Protection domain.
Claims (2)
1. least square method is linearly answered in a kind of contraction for signal transacting, methods described is applied in Wave beam forming, and it is special
Sign is：It the described method comprises the following steps：
1) consider the even linear array of a M array elements, receive a far field narrow band signal s0(k), corresponding direction of arrival is
θd, make x (k) represent its sample data matrix received, obtain x (k)=a (θd)s0(k)+n (k), whereinFor the steering vector of desired signal, Δ represents adjacent array element
Between array spacings, λ represents wavelength,Wherein, η (k) is and desired signal and interference
Incoherent ambient noise；
2) the output y=ω (k) of computing arrayHX (k), obtained by minimum mean square error criterion Make J [ω (k)] minimum, then obtain ω (k+1)=ω (k)+μ e*(k)x(k)；
3) variable step size μ is usedkTo replace above-mentioned μ values, obtain If weights error vector is v (k)=ω (k)-ωopt, ωopt(k) it is optimal weight vector,
Then obtain the renewal process of weights error vectorWherein,It is the error between the waveform output of optimal weight vector and desired signal；
4) at the k moment, the prior uncertainty between the output of wave beam device and desired signal isWherein noiseless when
Prior uncertainty be
5) similarly, posteriori error is ε (k)=εopt(k)+εf(k), wherein noiseless when posteriori error be
6) by can be calculated εfAnd e (k)f(k) relation between is：
εf(k)=(1- μk||x(k)||2)ef(k)-μkεopt(k)||x(k)||2,
To the both sides of its square simultaneously to μkDifferentiate and derivative is obtained equal to 0
7) abbreviation is carried out to step (6) according to above-mentioned formula to obtainWork as μkWhen being constant, on
Formula is set up certainly；In fact, in stable state, μkChange compared with x (k), e (k) it is slow, it is therefore contemplated that they be it is approximate not
Related, then it can obtain
8) in actual applications, usual E [| | x (k) | |2] be it is known, E [| e (k) |2] it is to pass throughEstimate, still, ef(k) it is unknown, it is difficult to by solving E [ef(k)|2]
To solve above-mentioned step size muk；
9) e (k) is made to recover e by shrinkage de-noising methodf(k)：Utilize
f[ef(k)]=0.5 | ef(k)-e(k)|2+α|ef(k) |,
Make this formula minimum, then can obtainSo as to be updated toIn obtain E [ef(k)|2] estimator；
10) by above-mentioned E [ef(k)|2]、E[|e(k)|2Estimator substitute intoThen shunk
Linear least square obtains step valueIt is linear multiple so as to replace the μ in step 2 to obtain the contraction
The right value update process of least square method.
2. a kind of multiple least square method of contraction generalized linear for signal transacting, methods described are applied in Wave beam forming,
It is characterized in that：Comprise the following steps：
1) consider the even linear array of a M array elements, receive a far field narrow band signal s0(k), corresponding direction of arrival is θd, make x (k)
Its sample data matrix received is represented, obtains x (k)=a (θd)s0(k)+n (k), whereinFor the steering vector of desired signal, Δ represents adjacent array element
Between array spacings, λ represents wavelength,Wherein, η (k) is and desired signal and interference
Incoherent ambient noise；
2) C is worked asx=E [x (k) xT(k)]≠0M×MWhen, then now need to consider the data of the noncircularity, then extension now of signal
For：
The output of computing array
3) made by minimum mean square error criterionMinimum, andThen now right value update process is
4) v is set1(k)=ω1-ωopt, v2(k)=ω2-ωopt, wherein, η (k) is and desired signal and the incoherent back of the body of interference
Scape noise, then obtain the renewal process of weight vector error
Can then be write as matrix form is：
Wherein,It is waveform output and the expectation letter of the weight vector of extension
Error between number；
5) relation that the nothing being expanded by above-mentioned formula is made an uproar between posteriori error and prior uncertainty is Whether there is again and make an uproar posteriori error and prior uncertainty is respectively：
6) by instantaneous posteriori error square, then it is asked on μkDerivative be 0, obtain
7) obtained according to above-mentioned formula and abbreviationWork as μkWhen being constant, above formula certainly into
It is vertical；In fact, in stable state, μkWith x (k),It is slow compared to changing, it is therefore contemplated that they are approximate incoherent, then
It can obtain
8) wherein,It can pass throughTry to achieve, still,Be it is unknown,
It is difficult to pass through solutionTo solve above-mentioned step size muk；By shrinkage de-noising method, obtainIt is updated in following formula i.e. availableEstimate：
9) will be above-mentionedEstimator substitute intoThen obtain shrinking extensively
It is adopted that linearly least square method obtains step value againSo as to substitute into obtain the renewal process of weights.
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