CN107102292A - A kind of target bearing tracking based on bayes method - Google Patents
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Abstract
A kind of target bearing tracking based on bayes method, belongs to array signal processing field.In the case of solving low signal-to-noise ratio, based on spatial class DOA tracking low precisions, and the known conditions that needs of particle filter DOA trackings it is excessive the problem of.Taylor expansion formula is utilized in the present invention, by the DOA tracing modelings under angle slowly change pattern into a dynamic model, and based on bayesian theory, the tracking of angle is converted into the Parameter Estimation Problem in probabilistic model, according to the observation of the estimate of the direction of arrival degree of previous moment, signal power and noise power, and current time, EM algorithms are utilized, information to previous moment is corrected, and then realizes the tracking to direction of arrival degree.Present invention is mainly used for target bearing is tracked.
Description
Technical field
The invention belongs to array signal processing field.
Background technology
In fields such as radar, communication, sonar, meteorologies, array signal processing has extensive and important application.Direction of arrival angle
(direction of arrival, DOA) estimation is a kind of lateral technology in array signal processing, and in actual conditions, mesh
Mark to be kept in motion, and DOA tracking is that moving target DOA is estimated in real time, and high-precision DOA tracking is in individual's positioning
All it is widely used in the fields such as business, battlefield mobile communication and Speech processing.However, the motion of target typically results in sky
Between compose extension, and the snap signal of moving target is difficult to long time integration, and this causes the DOA estimation method of conventional dead target
No longer it is applicable.Therefore, high-precision DOA tracking becomes research emphasis.
At present, most representational DOA trackings are：Subspace class method and Bayes's class method.
(1) subspace class method.Subspace class method is that segmentation estimates signal direction of arrival as instantaneous value, is passed through
Tracking or renewal subspace realize that DOA is tracked.In Yang B.An extension of the PASTd algorithm to
both rank and subspace tracking[J].Signal Processing Letters,IEEE,1995,2(9):
The deflation projection approximation subspace proposed in 179-182 (a kind of PASTd algorithms for realizing sum of ranks subspace tracking of extension) with
Track (projection approximation subspace tracking deflation, PASTd) algorithm is multiple due to calculating
Miscellaneous degree is relatively low and is introduced among DOA tracking problems.In Sanchez-Araujo J, Marcos S.An efficient
PASTd-algorithm implementation for multiple direction of arrival tracking[J]
.IEEE transactions on signal processing,1999,47(8):2321-2324 is (a kind of efficiently for many
The PASTd algorithms of target angle of arrival tracking) signal subspace and noise subspace are estimated in real time with PASTd algorithms in, and tie
Close Kalman filtering and realize that DOA is tracked.In the case of low signal-to-noise ratio, the tracking accuracy of this kind of method is poor.
(2) Bayes's class method.It is based on Bayes principle, to utilize unknown parameter based on Bayes tracking method
Prior information and sample information draw posterior information, then according to posterior information carry out parameter Estimation, can be in coherent
Tracked with the case of low signal-to-noise ratio.Typical method has sequential Monte-Carlo method, i.e. particle filter (particle
Filter, PF) method.In Orton M, Fitzgerald W.A Bayesian approach to tracking
multiple targets using sensor arrays and particle filters[J].Signal
Processing,IEEE Transactions on signal processing,2002,50(2):216-223 (uses sensing
Array and particle filter carry out the bayes method of multiple target tracking) in, particle filter algorithm is used for the DOA of one-dimensional array
Tracking.It is unknown in these conditions but particle filter algorithm needs to use state transition probability and observation noise probability function
In the case of, in fact it could happen that larger error.
The content of the invention
The present invention is in the case of solving low signal-to-noise ratio, be filtered based on spatial class DOA tracking low precisions, and particle
The problem of known conditions that ripple DOA trackings need is excessive, the invention provides a kind of target side based on bayes method
Position tracking.
A kind of target bearing tracking based on bayes method, it comprises the following steps：
Step one：Initialize the direction of arrival θ of current tk(t), signal power pk (t), noise power σ2(t) and
Angle variable quantity △ θk(t) so that θk(t)=θk(t-1)、pk(t)=pk(t-1)、σ2(t)=σ2(t-1)、△θk(t)=0；
Wherein, θk(t-1)、pkAnd σ (t-1)2(t-1), it is known that k represents k-th of signal source, and k=1,2,3 ..., K,
K is spacing wave source number, θk(t-1) it is the direction of arrival at t-1 moment, pk(t-1) it is the signal power at t-1 moment, σ2
(t-1) it is the noise power at t-1 moment, angle variable quantity △ θk(t) it is angle change of the current time relative to last moment
Amount；
Wherein, t=1,2,3 ..., T ' -1, T ', T ' be total observation time；
Step 2：In θk(t-1) place, carries out first order Taylor expansion, so as to obtain array manifold matrix A；
Step 3：Using bayes method to the signal power p in step onek(t), noise power σ2And array manifold (t)
Matrix A is handled, and obtains Posterior Mean μ and posterior variance ∑；
Step 4：Signal power p is updated according to Posterior Mean μ and posterior variance ∑k(t), noise power σ2And angle (t)
Variable quantity △ θk(t)；
Step 5：Signal power p after m renewal is followed the trail of using EM algorithmsk(t), noise power σ2And angle change (t)
Measure △ θk(t) the signal power p after m renewal, is judgedk(t), noise power σ2(t) with angle variable quantity △ θk(t) numerical value
Whether restrain, be as a result yes, perform step 6, be as a result no, make m=m+1, perform step one；
Wherein, m is to repeat step one to the number of times of step 4；
Step 6：Under convergence state, the signal power p after the m times renewalk(t) it is current t signal power pk(t)
Estimate, the noise power σ after renewal2(t) it is current t noise power σ2(t) estimate, the angle change after renewal
Measure △ θk(t) it is current t angle variable quantity θk(t-1) estimate,
According to current t angle variable quantity △ θk(t) estimate,Obtain the direction of arrival θ of current tk(t)
Estimate, wherein, θk(t)=θk(t-1)+△θk(t)；
Step 7：Judge whether t is equal to T ', be as a result no, make t=t+1, perform step one；As a result it is yes, performs step
Eight；
Step 8：Complete that total observation time T ' is interior, the signal power p at each momentk(t), noise power σ2And signal (t)
Angle of arrival θk(t) estimation, so as to realize the real-time tracing to target bearing.
In described step two, in θk(t-1) place, carries out first order Taylor expansion, so as to obtain the tool of array manifold matrix A
Body process is：
Step 2 one：According to Taylor expansion, by the flow pattern vector a (θ of tk(t)), in θk(t-1) place carries out single order
Taylor expansion is obtained：
a(θk(t))=a (θk(t-1))+a'(θk(t-1))△θk(t) (formula one),
Wherein,
a(θk(t-1) it is) the flow pattern vector at t-1 moment, a ' (θk(t-1) derivative of the flow pattern vector at t-1 moment, e) are represented
For natural constant, j is imaginary unit, and d is array element spacing, and λ is signal wavelength, and M is element number of array, []TRepresenting matrix transposition；
Step 2 two：According to flow pattern vector a (θk(t) the array manifold matrix A of t), is obtained；
A=A1+ B Δs (formula two),
Wherein,
A1=[a (θ1(t-1)),a(θ2(t-1)),a(θ3(t-1)),...,a(θK-1(t-1)),a(θK(t-1))],
B=[a'(θ1(t-1)),a'(θ2(t-1)),a'(θ3(t-1)),...,a'(θK-1(t-1)),a'(θK(t-1))],
Δ=diag ([△ θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]),
A1For the array manifold matrix at t-1 moment, B is the matrix of the derivative composition of t-1 moment manifold vector, and Δ is angle
Correcting value, diag () represents diagonal matrix function.
In described step three, using bayes method to the signal power p in step onek(t), noise power σ2(t) and
Array manifold matrix A is handled, and the detailed process for obtaining Posterior Mean μ and posterior variance ∑ is：
μ=PAH(σ2(t)I+APAH)-1X (formula three),
∑=(P-1+σ-2(t)AHA)-1(formula four),
Wherein,
P=diag ([p1(t),p2(t),p3(t),...,pK-1(t),pK(t)]),
P is the covariance matrix of signal, and diag () represents diagonal matrix function, and I is unit matrix.
In described step four, signal power p is updated according to Posterior Mean μ and posterior variance ∑k(t), noise power σ2
(t) with angle variable quantity △ θk(t) detailed process is：
[△θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]T=U-1V (formula seven),
Wherein,
(∑)(n,n)The column element of line n n-th of ∑ is represented,
(μ)nμ line n is represented, n is positive integer；
X=AS+N,
Wherein, subscript H represents conjugate transposition,Expression takes real part, and ⊙ represents Hadamard product, and L represents fast umber of beats, diag
() represents diagonal matrix function, and v and U represent intermediate variable, X ∈ CM×LFor array received data, S ∈ CK×LFor incidence
Signal, N ∈ CM×LFor observation noise, C gathers for plural number, the mark of tr () representing matrix.
Present invention contemplates that the DOA tracking of angle slowly under change pattern, i.e. adjacent moment direction of arrival angle value and signal
Power is varied less.The present invention is arrived in the case of direction of arrival and slowly varying power according to the signal of previous moment
Up to the observation of the estimate of angle, signal power and noise power, and current time, using EM algorithms, to previous moment
Information is corrected, and then realizes the tracking to direction of arrival degree.
The beneficial effect that the present invention is brought is,
In the present invention, using Taylor expansion formula, by the DOA tracing modelings under angle slowly change pattern into a dynamic analog
Type, and based on bayesian theory, the tracking of angle is converted into the Parameter Estimation Problem in probabilistic model, asked using EM algorithms
Solution.The advantage of tracking proposed by the present invention be can simultaneously the angle of arrival of real-time tracking signal, signal power and
Noise power, and the prior informations such as state transition probability need not be known, and compared with management loading method, amount of calculation
It is small.
A kind of target bearing tracking based on bayes method of the present invention and subspace class DOA tracking phases
Than tracking precision of the present invention improves more than 30%, and particularly evident in the case of low signal-to-noise ratio.
A kind of target bearing tracking based on bayes method of the present invention and particle filter DOA tracking phases
Than the method for the invention need not know state transition probability and observation noise probability function.
The present invention is while DOA tracking is carried out, and the present invention can also realize the multi-parameter of signal power and noise power
Joint tracking.
Brief description of the drawings
Fig. 1 is a kind of flow chart of the target bearing tracking based on bayes method of the present invention；
Fig. 2 is to have the direction of arrival obtained by the method for the present invention (DOA) under two incoming signal situations to follow the trail of figure；
Fig. 3 is to utilize a kind of target bearing tracking based on bayes method of the present invention, obtained signal
Powerinjected method figure；
Fig. 4 is to utilize a kind of target bearing tracking based on bayes method of the present invention, obtained noise
Powerinjected method figure；
Fig. 5 is the inventive method and PASTd-Kalman (tightening projection approximation subspace tracking-Kalman filtering) algorithm
Carry out the root-mean-square error of DOA tracking and the comparison diagram of Between Signal To Noise Ratio curve.
Embodiment
Embodiment one：Illustrate present embodiment referring to Fig. 1, one kind described in present embodiment is based on Bayes side
The target bearing tracking of method, it comprises the following steps：
Step one：Initialize the direction of arrival θ of current tk(t), signal power pk(t), noise power σ2(t) and
Angle variable quantity △ θk(t) so that θk(t)=θ k (t-1), pk(t)=pk(t-1)、σ2(t)=σ2(t-1)、△θk(t)=0；
Wherein, θk(t-1)、pkAnd σ (t-1)2(t-1), it is known that k represents k-th of signal source, and k=1,2,3 ..., K,
K is spacing wave source number, θk(t-1) it is the direction of arrival at t-1 moment, pk(t-1) it is the signal power at t-1 moment, σ2
(t-1) it is the noise power at t-1 moment, angle variable quantity △ θk(t) it is angle change of the current time relative to last moment
Amount；
Wherein, t=1,2,3 ..., T ' -1, T ', T ' be total observation time；
Step 2：In θk(t-1) place, carries out first order Taylor expansion, so as to obtain array manifold matrix A；
Step 3：Using bayes method to the signal power p in step onek(t), noise power σ2And array manifold (t)
Matrix A is handled, and obtains Posterior Mean μ and posterior variance ∑；
Step 4：Signal power p is updated according to Posterior Mean μ and posterior variance ∑k(t), noise power σ2And angle (t)
Variable quantity △ θk(t)；
Step 5：Signal power p after m renewal is followed the trail of using EM algorithmsk(t), noise power σ2And angle change (t)
Measure △ θk(t) the signal power p after m renewal, is judgedk(t), noise power σ2(t) with angle variable quantity △ θk(t) numerical value
Whether restrain, be as a result yes, perform step 6, be as a result no, make m=m+1, perform step one；
Wherein, m is to repeat step one to the number of times of step 4；
Step 6：Under convergence state, the signal power p after the m times renewalk(t) it is current t signal power pk(t)
Estimate, the noise power σ after renewal2(t) it is current t noise power σ2(t) estimate, the angle change after renewal
Measure △ θk(t) it is current t angle variable quantity θk(t-1) estimate,
According to current t angle variable quantity △ θk(t) estimate, obtains the direction of arrival θ of current tk(t)
Estimate, wherein, θk(t)=θk(t-1)+△θk(t)；
Step 7：Judge whether t is equal to T ', be as a result no, make t=t+1, perform step one；As a result it is yes, performs step
Eight；
Step 8：Complete that total observation time T ' is interior, the signal power p at each momentk(t), noise power σ2And signal (t)
Angle of arrival θk(t) estimation, so as to realize the real-time tracing to target bearing.
Present embodiment, in the present invention, using Taylor expansion formula, by the DOA tracing modelings under angle slowly change pattern
For a dynamic model, and based on bayesian theory, the tracking of angle is converted into the Parameter Estimation Problem in probabilistic model, adopted
Solved with EM algorithms.The advantage of tracking proposed by the present invention be can simultaneously real-time tracking signal angle of arrival,
Signal power and noise power, and need not know the prior informations such as state transition probability, and with management loading side
Method is compared, and amount of calculation is small.
The English full name of EM algorithms is：Expectation Maximization.
Principle analysis：First according to angle become slowly it is assumed that according to first order Taylor, obtaining and last moment angle
The array manifold matrix relevant with angle renewal amount, effectively to utilize the information of last moment.Then EM iteration is carried out, every time
The posterior probability of signal is calculated in iteration according to the estimate of the last round of signal power tried to achieve, noise power and angle correct amount
Density, according to the posterior probability density, the joint probability density to signal and noise is averaged and is allowed to maximize to update
The estimate of parameter.Obtain after angle correct amount, you can calculate real-time angle of arrival.
Embodiment two：Illustrate present embodiment referring to Fig. 1, present embodiment with described in embodiment one
A kind of difference of the target bearing tracking based on bayes method is, in described step two, in θk(t-1) place, enters
Row first order Taylor deploys, so that the detailed process for obtaining array manifold matrix A is：
Step 2 one：According to Taylor expansion, by the flow pattern vector a (θ of tk(t)), in θk(t-1) place carries out single order
Taylor expansion is obtained：
a(θk(t))=a (θk(t-1))+a'(θk(t-1))△θk(t) (formula one),
Wherein,
a(θk(t-1) it is) the flow pattern vector at t-1 moment, a ' (θk(t-1) derivative of the flow pattern vector at t-1 moment, e) are represented
For natural constant, j is imaginary unit, and d is array element spacing, and λ is signal wavelength, and M is element number of array, []TRepresenting matrix transposition；
Step 2 two：According to flow pattern vector a (θk(t) the array manifold matrix A of t), is obtained；
A=A1+ B Δs (formula two),
Wherein,
A1=[a (θ1(t-1)),a(θ2(t-1)),a(θ3(t-1)),...,a(θK-1(t-1)),a(θK(t-1))],
B=[a'(θ1(t-1)),a'(θ2(t-1)),a'(θ3(t-1)),...,a'(θK-1(t-1)),a'(θK(t-1))],
Δ=diag ([△ θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]),
A1For the array manifold matrix at t-1 moment, B is the matrix of the derivative composition of t-1 moment manifold vector, and Δ is angle
Correcting value, diag () represents diagonal matrix function.
Embodiment three：Illustrate present embodiment referring to Fig. 1, present embodiment with described in embodiment one
A kind of difference of the target bearing tracking based on bayes method is, in described step three, using bayes method
To the signal power p in step onek(t), noise power σ2(t) and array manifold matrix A handled, obtain Posterior Mean μ and
The detailed process of posterior variance ∑ is：
μ=PAH(σ2(t)I+APAH)-1X (formula three),
∑=(P-1+σ-2(t)AHA)-1(formula four),
Wherein,
P=diag ([p1(t),p2(t),p3(t),...,pK-1(t),pK(t)]),
P is the covariance matrix of signal, and diag () represents diagonal matrix function, and I is unit matrix.
Embodiment four：Illustrate present embodiment referring to Fig. 1, present embodiment with described in embodiment one
A kind of difference of the target bearing tracking based on bayes method is, in described step four, according to Posterior Mean μ and
Posterior variance ∑ updates signal power pk(t), noise power σ2(t) with angle variable quantity △ θk(t) detailed process is：
[△θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]T=U-1V (formula seven),
Wherein,
(∑)(n,n)The column element of line n n-th of ∑ is represented,
(μ)nμ line n is represented, n is positive integer；
X=AS+N,
Wherein, subscript H represents conjugate transposition,Expression takes real part, and ⊙ represents Hadamard product, and L represents fast umber of beats, diag
() represents diagonal matrix function, and v and U represent intermediate variable, X ∈ CM×LFor array received data, S ∈ CK×LFor incidence
Signal, N ∈ CM×LFor observation noise, C gathers for plural number, the mark of tr () representing matrix.
Embodiment five：Illustrate present embodiment referring to Fig. 1, present embodiment with described in embodiment two
A kind of difference of the target bearing tracking based on bayes method is, in described step two two, A=[a (θ1(t)),
a(θ2(t)),a(θ3(t)),...,a(θK-1(t)),a(θK(t))]。
Simulating, verifying：
L-G simulation test is carried out using a kind of target bearing tracking based on bayes method of the present invention：
One by one, simulation parameter sets as follows step：Element number of array M=12, fast umber of beats L=20, array element spacing d=λ/2, make an uproar
Acoustical power σ2(t)=1, observation time T '=100；
Step one two：Set up emulation signal movement state model：
xk(t)=Fxk(t-1)+Gvk(t)
In formula,vk(t) it is the white Gaussian of zero-mean
Noise, and variance is 0.001, whereinRepresent θk(t) speed；
Step one three：Incoming signal number K=2 is set, and initial angle is respectively 40 ° and 30 °, the change difference of signal to noise ratio
For 8dB to -3dB and 3dB to 8dB, obtained DOA, signal power and noise power aircraft pursuit course are respectively such as Fig. 2, Fig. 3 and Fig. 4
It is shown, wherein, DOA is direction of arrival；
Step one four：In order to preferably analyze the DOA tracking performances of the inventive method, it is combined into card with PASTd algorithms
The DOA trackings of Kalman Filtering are compared.100 Monte Carlo experiments are carried out, the root-mean-square error of two kinds of algorithms is obtained
It is as shown in Figure 5 with the relation curve of signal to noise ratio.
Simulation results show：
The method of the present invention can be very good to carry out direction of arrival degree, signal work(it can be seen from Fig. 2, Fig. 3 and Fig. 4
The real-time tracking of rate and noise power；As seen from Figure 5, method of the invention and PASTd algorithm combination Kalman filterings
DOA trackings, which are compared, has higher tracking accuracy, and advantage is more obvious in the case of low signal-to-noise ratio.
Claims (5)
1. a kind of target bearing tracking based on bayes method, it is characterised in that it comprises the following steps：
Step one：Initialize the direction of arrival θ of current tk(t), signal power pk(t), noise power σ2And angle (t)
Variable quantity △ θk(t) so that θk(t)=θk(t-1)、pk(t)=pk(t-1)、σ2(t)=σ2(t-1)、△θk(t)=0；
Wherein, θk(t-1)、pkAnd σ (t-1)2(t-1), it is known that k represents k-th of signal source, and k=1,2,3 ..., K, K are
Spacing wave source number, θk(t-1) it is the direction of arrival at t-1 moment, pk(t-1) it is the signal power at t-1 moment, σ2(t-1)
For the noise power at t-1 moment, angle variable quantity △ θk(t) it is angle variable quantity of the current time relative to last moment；
Wherein, t=1,2,3 ..., T ' -1, T ', T ' be total observation time；
Step 2：In θk(t-1) place, carries out first order Taylor expansion, so as to obtain array manifold matrix A；
Step 3：Using bayes method to the signal power p in step onek(t), noise power σ2(t) with array manifold matrix
A processing, obtains Posterior Mean μ and posterior variance ∑；
Step 4：Signal power p is updated according to Posterior Mean μ and posterior variance ∑k(t), noise power σ2And angle change (t)
Measure △ θk(t)；
Step 5：Signal power p after m renewal is followed the trail of using EM algorithmsk(t), noise power σ2(t) with angle variable quantity △
θk(t) the signal power p after m renewal, is judgedk(t), noise power σ2(t) with angle variable quantity △ θk(t) whether numerical value
Convergence, is as a result yes, performs step 6, be as a result no, make m=m+1, perform step one；
Wherein, m is to repeat step one to the number of times of step 4；
Step 6：Under convergence state, the signal power p after the m times renewalk(t) it is current t signal power pk(t) estimation
Value, the noise power σ after renewal2(t) it is current t noise power σ2(t) estimate, the angle variable quantity △ after renewal
θk(t) it is current t angle variable quantity θk(t-1) estimate,
According to current t angle variable quantity △ θk(t) estimate, obtains the direction of arrival θ of current tk(t) estimate
Evaluation, wherein, θk(t)=θk(t-1)+△θk(t)；
Step 7：Judge whether t is equal to T ', be as a result no, make t=t+1, perform step one；As a result it is yes, performs step 8；
Step 8：Complete that total observation time T ' is interior, the signal power p at each momentk(t), noise power σ2(t) reached with signal
Angle θk(t) estimation, so as to realize the real-time tracing to target bearing.
2. a kind of target bearing tracking based on bayes method according to claim 1, it is characterised in that described
The step of two in, in θk(t-1) place, carries out first order Taylor expansion, so that the detailed process for obtaining array manifold matrix A is：
Step 2 one：According to Taylor expansion, by the flow pattern vector a (θ of tk(t)), in θk(t-1) place carries out first order Taylor
Expansion is obtained：
a(θk(t))=a (θk(t-1))+a'(θk(t-1))△θk(t) (formula one),
Wherein,
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a(θk(t-1) it is) the flow pattern vector at t-1 moment, a ' (θk(t-1) derivative of the flow pattern vector at t-1 moment) is represented, e is certainly
Right constant, j is imaginary unit, and d is array element spacing, and λ is signal wavelength, and M is element number of array, []TRepresenting matrix transposition；
Step 2 two：According to flow pattern vector a (θk(t) the array manifold matrix A of t), is obtained；
A=A1+ B Δs (formula two),
Wherein,
A1=[a (θ1(t-1)),a(θ2(t-1)),a(θ3(t-1)),...,a(θK-1(t-1)),a(θK(t-1))],
B=[a'(θ1(t-1)),a'(θ2(t-1)),a'(θ3(t-1)),...,a'(θK-1(t-1)),a'(θK(t-1))],
Δ=diag ([△ θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]),
A1For the array manifold matrix at t-1 moment, B is the matrix of the derivative composition of t-1 moment manifold vector, and Δ is angle correct
Amount, diag () represents diagonal matrix function.
3. a kind of target bearing tracking based on bayes method according to claim 1, it is characterised in that described
The step of three in, using bayes method to the signal power p in step onek(t), noise power σ2(t) with array manifold matrix
A processing, the detailed process for obtaining Posterior Mean μ and posterior variance ∑ is：
μ=PAH(σ2(t)I+APAH)-1X (formula three),
∑=(P-1+σ-2(t)AHA)-1(formula four),
Wherein,
P=diag ([p1(t),p2(t),p3(t),...,pK-1(t),pK(t)]),
P is the covariance matrix of signal, and diag () represents diagonal matrix function, and I is unit matrix.
4. a kind of target bearing tracking based on bayes method according to claim 1, it is characterised in that described
The step of four in, signal power p is updated according to Posterior Mean μ and posterior variance ∑k(t), noise power σ2And angle change (t)
Measure △ θk(t) detailed process is：
[△θ1(t),△θ2(t),△θ3(t),...,△θK-1(t),△θK(t)]T=U-1V (formula seven),
Wherein,
(∑)(n,n)The column element of line n n-th of ∑ is represented,
(μ)nμ line n is represented, n is positive integer；
X=AS+N,
Wherein, subscript H represents conjugate transposition,Expression takes real part, and ⊙ represents Hadamard product, and L represents fast umber of beats, diag () table
Show expression diagonal matrix function, v and U represent intermediate variable, X ∈ CM×LFor array received data, S ∈ CK×LFor incoming signal,
N∈CM×LFor observation noise, C gathers for plural number, the mark of tr () representing matrix.
5. a kind of target bearing tracking based on bayes method according to claim 2, it is characterised in that described
The step of two or two in, A=[a (θ1(t)),a(θ2(t)),a(θ3(t)),...,a(θK-1(t)),a(θK(t))]。
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| CN114721015B (en) * | 2022-04-08 | 2022-12-09 | 中山大学 | Blind robust STAP beam forming method and device for GNSS receiver |
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