CN109375156A - The research method of sensing system single goal Cramér-Rao lower bound based on information theory - Google Patents
The research method of sensing system single goal Cramér-Rao lower bound based on information theory Download PDFInfo
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- CN109375156A CN109375156A CN201811158965.8A CN201811158965A CN109375156A CN 109375156 A CN109375156 A CN 109375156A CN 201811158965 A CN201811158965 A CN 201811158965A CN 109375156 A CN109375156 A CN 109375156A
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
- G01S3/02—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using radio waves
- G01S3/14—Systems for determining direction or deviation from predetermined direction
- G01S3/16—Systems for determining direction or deviation from predetermined direction using amplitude comparison of signals derived sequentially from receiving antennas or antenna systems having differently-oriented directivity characteristics or from an antenna system having periodically-varied orientation of directivity characteristic
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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- Y02D—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN INFORMATION AND COMMUNICATION TECHNOLOGIES [ICT], I.E. INFORMATION AND COMMUNICATION TECHNOLOGIES AIMING AT THE REDUCTION OF THEIR OWN ENERGY USE
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Abstract
The present invention proposes a kind of research method of sensing system single goal Cramér-Rao lower bound (CRB) based on information theory, belongs to information transmission and processing technology field.Lower bound as spacing wave arrival direction (DOA) estimation performance.By utilizing Shannon (Shannon) information theory, the theoretical model of the maximum likelihood communicationization under multiple additive white Gaussian noise (CAWGN) environment is proposed.Estimate for single information source, the posterior probability Density Distribution of angle to be measured is deduced by likelihood function, obtains the expression formula of Posterior probability distribution Gaussian approximation under the conditions of high s/n ratio (SNR), variance, that is, CRB therein.Simulation result shows that the CRB that the present invention derives is the approximate lower bound of maximum likelihood algorithm (MLE).Simulation results show the correctness of theory analysis.Conclusion of the invention has important theory directive significance to the DOA estimation based on MLE algorithm.
Description
Technical field
A kind of research method of sensing system single goal Cramér-Rao lower bound based on information theory of the present invention, belongs to
Information transmission and processing technology field.
Background technique
In decades, far field information source is positioned using sensor array and estimates interested parameter in various fields
In obtained extensive research, such as radar, sonar, wireless communication etc..In the spacing wave arrival direction of even linear array
In (Direction of Arrival, DOA) estimation, it has been proposed that algorithm of many performances based on mean square error, it is such as maximum
Likelihood (Maximum Likelihood, ML) method and its modification, i.e. MODE method, it is mainly excellent by the multidimensional of likelihood function
Change and realizes asymptotic optimality performance.
One kind is based on the DOA estimation method for receiving signal covariance matrix feature decomposition, such as Multiple Signal Classification
(Multiple Signal Classification, MUSIC) algorithm passes through rotation invariant technology (Estimation of
Signal Parameters via Rotational Invariance Techniques, ESPRIT) algorithm estimation signal ginseng
Number.Higher resolution ratio may be implemented since eigendecomposition, DOA estimation is utilized in these algorithms.On the other hand,
The very big concern of people is also resulted in corresponding performance statistics and error analysis these problems.Wherein, by for large sample
In the case of second-order properties analysis, and compared with maximum likelihood and Cramér-Rao lower bound, this is provided for the precision of unbiased esti-mator
Lower bound.Maximal possibility estimation (Maximum Likelihood Estimator, MLE) algorithm is by searching in observation section
The maximum change location of rope likelihood function estimates incoming wave angle, this is one and solves the problems, such as that non-linear multidimensional optimizes,
Calculating process very complicated.Petre Stocia et al. thinks, when array element number is seldom, even if sample is big again, and MLE
It can not level off to Cramér-Rao lower bound (Cramer-Rao Bound, CRB)；When array element number is gradually increased, MLE can gradually reach
To CRB lower bound performance.From the point of view of this result, it selects array element number and sampling number to provide in practical applications for us
Standard.In addition, Petre Stocia also proposed the CRB matrix form of DOA unbiased esti-mator, it was demonstrated that DOA estimation can approach
Theoretical error limit CRB.
But sensor array system, as a kind of Information Acquisition System, DOA estimates that can performance knowing with information theory
Know to measure, is always academia's very concern.1948, Shannon established information theory, this theory is letter
The basis of transmission, channel coding and data compression scheduling theory is ceased, therefore the communications field also achieves development at full speed.Wherein information
An important measurement is mutual information in, it refers to the correlation between two signals.1993, Bell answered mutual information
For radar waveform design method, the results showed that, optimal information extraction scheme can distribute energy between target scattering mode
Amount is to realize mutual information maximization.It ensure that receiving signal carries information as much as possible, to improve estimation, tracking and knowledge
Other accuracy.
Summary of the invention
The research method of the invention proposes a kind of sensing system single goal Cramér-Rao lower bound based on information theory, will be fragrant
Agriculture information theory is applied in DOA estimation field, obtains the carat beauty of the distribution of posterior probability and MLE algorithm under different signal-to-noise ratio
The analytical expression of sieve circle.
The present invention is to solve its technical problem to adopt the following technical scheme that
A kind of research method of the sensing system single goal Cramér-Rao lower bound based on information theory, includes the following steps:
The uniform linear array for establishing multisensor array element obtains the likelihood letter of observation using the reception signal of antenna
Number, combining information is discussed and the prior information of parameter to be measured obtains the posteriority conditional probability distribution of DOA；Under high s/n ratio, to rear
The method that the reception signal tested in distribution carries out signal decomposition and Gaussian approximation combines, the final knot for obtaining Cramér-Rao lower bound
Fruit, and with the error statistics of MLE for verifying the correctness of the result.
The reception signal using antenna obtains the likelihood function of observation, and combining information is discussed and the priori of parameter to be measured
Information obtains the posteriority conditional probability distribution of DOA, and detailed process is as follows:
First, it is assumed that observation section total length is | θ |, information source be [- | θ |/2, | θ |/2] it is equally distributed in section,
Therefore the prior distribution for obtaining angle, θ to be measured is
Wherein: p (θ) is the prior distribution form of θ to be measured；
The phase of information sourceAnd equally distributed variable, prior distribution are expressed as in [0,2 π]
Wherein:For phasePrior distribution form；
In variable θ andThe likelihood function of reception signal Y is under the conditions of known
Wherein: N0The variance of additive white Gaussian noise is answered for steady zero-mean, M is array antenna array number, and y is uniform line
The received vector of battle array, x are information source vector, and A (θ) is the guiding matrix of array, are indicated are as follows:
A (θ)=[exp (j ω0τ0(θ))exp(jω0τ1(θ))K exp(jω0τM-1(θ))]T
Wherein: ω0Indicate the angular frequency of carrier signal, τ0The time delays of upper array element, τ are put on the basis of (θ)1On the basis of (θ)
The delay of other first array element of point, τM-1(θ) is the delay of the M-1 array element；
It is obtained by the outlier for being unfolded, ignoring with θ to simplify likelihood function
WhereinExpression takes real part, and α is the decay factor of source signal, ymFor connecing in m array element
The collection of letters number, τm(θ) is the time delays that m-th of array element receives signal in array；
According to the related knowledge of prior information and probability theory, the Posterior probability distribution for obtaining θ is
It is knownThen above formula is written as
Wherein I0() indicates first kind zeroth order modified Bessel function, and the shape of posterior probability Density Distribution is by shellfish at this time
What Sai Er function determined, denominator is the normalized form of probability distribution.
The reception signal in Posterior distrbutionp carries out signal decomposition, and detailed process is as follows:
Signal will be received and be split as signal section and noise section, obtained
Wherein, wmFor the multiple additive white Gaussian noise obtained in m-th of array element；
Therefore posterior probability can also be decomposed into
Wherein:For the phase of practical information source, θ0For the practical direction DOA, τm(θ0) it is m-th of array element in the practical direction DOA
On time delays；It enablesRespectively the auto-correlation function of signal and
Cross-correlation function between signal and noise is considered as the influence that signal terms and noise item generate probability distribution, at this time
Wherein: giving signal-to-noise ratio ρ2Definition,Ε { } is to ask expectation to source signal.
Described pair of reception signal carries out Gaussian approximation processing, and the detailed process of the final result for obtaining Cramér-Rao lower bound is such as
Under:
DOA estimated result is counted, obtained statistical error is infinitely to tend to CRB under high s/n ratio, posteriority
Probability is approximately
Will | G (θ) | in θ=θ0Do Taylors approximation expansion in place
WhereinD is array element spacing, and λ is signal wavelength；In addition, Bessel function is also approximately at
It is by the processing result that two approximations combine to obtain Gaussian approximation method
Wherein k indicates normalized constant coefficient, σ2For the variance of Gaussian Profile.
Therefore under the conditions of high SNR, by estimating the variance of angle conditional probability to be measured, Cramér-Rao lower bound can be obtained
Beneficial effects of the present invention are as follows:
A kind of research method of sensor array system single goal Cramér-Rao lower bound based on information theory proposed by the present invention
The approximate lower bound of available MLE algorithm mean square error, the matrix expression side of this method calculated CRB difference and previous CRB
Method, it is derived by probability density function, can apply on design conditions entropy and information source entropy, it is mutual to finally obtain DOA
The analytical expression of information is tracked for MLE performance.Simulation result shows that signal-to-noise ratio is higher, and probability density distribution is more sharp,
Variance is smaller；Variance and signal-to-noise ratio be at negative linear relationship under high s/n ratio, and MLE Algorithm Error constantly levels off to CRB.
Detailed description of the invention
Fig. 1 is the system model of sensor of the invention array DOA estimation.
Fig. 2 is Posterior probability distribution figure under different signal-to-noise ratio of the invention.
Fig. 3 is the Cramér-Rao lower bound of the invention figure compared with MLE Algorithm Error.
Specific embodiment
Embodiments of the present invention are described below in detail, the example of the embodiment is shown in the accompanying drawings, wherein from beginning
Same or similar element or element with the same or similar functions are indicated to same or similar label eventually.Below by ginseng
The embodiment for examining attached drawing description is exemplary, and for explaining only the invention, and is not construed as limiting the claims.
It will be understood to those skilled in the art that unless otherwise defined, all terms used herein (including technical term
And scientific term) there is meaning identical with the general understanding of those of ordinary skill in fields of the present invention.It should also manage
Solution, those terms such as defined in the general dictionary, which should be understood that, to be had and the meaning in the context of the prior art
Consistent meaning, and unless defined as here, it will not be explained in an idealized or overly formal meaning.
Assuming that multisensor array system is uniform linear array, array antenna array number is M, and each array element radiation is all complete
Tropism, reception characteristic is only related and unrelated with size to its position, and position spacing is d.Far field information source and array antenna position
In in same plane, reflection coefficient is constant α, and the echo-signal of reflection is considered as plane wave when reaching array, while spatial source is believed
Number make narrowband hypothesis.The difference for then receiving signal is mainly reflected in phase difference caused by the wave path-difference for reaching each array element.Array
In m-th array element receive the time delays of signal and be(present invention is directed to the research of single goal, defaults information source number
L=1), wherein m=0,1, L, M-1, information source direction is θ, and v indicates the spread speed of signal.The points for ignoring Space domain sampling are (fast
Umber of beats N=1), then the reception signal in m array element is
Wherein: ω0Indicate that the angular frequency of carrier signal, x indicate the amplitude-phase signal obtained from information source,α
For decay factor,It is the equally distributed variable in [0,2 π], wmFor the multiple additive white Gaussian noise obtained in m-th of array element.
All array elements are integrated into matrix form expression formula:
Y=A (θ) X+W
Wherein: X is information source matrix, and Y is receipt signal matrix, and W is noise matrix, and the guiding matrix A (θ) of array is expressed as
A (θ)=[exp (j ω0τ0(θ)) exp(jω0τ1(θ)) K exp(jω0τM-1(θ))]T
Wherein: τ0The time delays of upper array element, τ are put on the basis of (θ)1The delay of other first array element, τ are put on the basis of (θ)M-1
(θ) is the delay of the M-1 array element.
Noise model is assumed to be steady zero-mean and answers additive white Gaussian noise, noise variance N0, and make an uproar between each array element
Sound is irrelevant, also uncorrelated to target source, therefore the second moment of noise vector meets
Ε[WWH]=N0I
Ε[WWT]=0
Wherein:Expectation is asked in expression, and W is noise vector, WHFor the conjugate transposition form of W, WTFor the transposition of W, I is single
Bit matrix.
Here signal-to-noise ratio (Signal to Noise Ratio, SNR) ρ is given2Definition,Ε { } is pair
Source signal asks expectation.Assuming that observation section total length is | θ |, information source be [- | θ |/2, | θ |/2] be uniformly distributed in section
, therefore the prior distribution for obtaining θ to be measured is
Wherein: p (θ) is the prior distribution form of θ to be measured.
The phase of information sourceAnd equally distributed variable, prior distribution are expressed as in [0,2 π]
Wherein:For phasePrior distribution form.
In view of W is multiple Gauss variable, in variable θ andUnder the conditions of the probability distribution of reception signal Y be
Wherein: y is received vector, and x is information source vector.
It is obtained by the outlier for being unfolded, ignoring with θ to simplify above formula
WhereinExpression takes real part, ymFor the reception signal in m-th of array element.According to prior information and
The related knowledge of probability theory, the Posterior probability distribution for obtaining θ are
It is knownAH(θ) is the transposition that array is oriented to matrix A (θ),
Then above formula can be written as
Wherein I0() indicates first kind zeroth order modified Bessel function.The shape of posterior probability Density Distribution is by shellfish at this time
What Sai Er function determined, denominator is the normalized form of probability distribution.The form for receiving signal is substituted into posterior probability expression formula
In, rewriteeing above formula is
Wherein:For the phase of practical information source, θ0For the practical direction DOA, τm(θ0) it is m-th of array element in the practical direction DOA
On time delays.
It enablesRespectively auto-correlation function (the direction of signal
Figure) and signal and noise between cross-correlation function, in fact also can be considered the shadow that signal terms and noise item generate probability distribution
It rings.At this time
DOA estimated result is counted, obtained statistical error is infinitely to tend to CRB under high s/n ratio, is considered
It is dimerous with noise by signal to Posterior probability distribution, therefore in high snr cases, noise section can be neglected
Slightly, posterior probability can be approximated to be
Utilize | G (θ) | in θ=θ0The Taylor expansion at place
Whereinλ is signal wavelength, and M is array element number.Under the conditions of high SNR, normalization is general
Peak value in rate distribution is concentrated mainly on θ0Near, therefore (θ-θ can be ignored0) high-order term.Furthermore the approximation of Bessel function
Expression has
Will | G (θ) | Taylor expansion substitute into Bezier approximate expression in, it is clear that can obtain
Wherein k indicates normalized constant coefficient, ρ2For signal-to-noise ratio, σ2For the variance of Gaussian Profile.Therefore in high SNR condition
Under, posterior probability is approximately Gaussian Profile form, at this time variance, that is, Cramér-Rao lower bound of Gaussian Profile
System model of the present invention is uniform linear array sensing system model, and detects just for single information source, empty
Domain samples number of snapshots and selects single snap.
Fig. 1 is the system model of sensor of the invention array DOA estimation；
Fig. 2 is Posterior probability distribution figure under different signal-to-noise ratio of the invention, and the practical direction DOA is set to 0 degree, for the ease of
Observation, only intercepted partial section here, since posterior probability Density Distribution is approximately Gaussian Profile, in figure section with
Outer part can be ignored, not impact analysis；In low SNR, posterior probability has not met Gaussian Profile form.
Fig. 3 is the Cramér-Rao lower bound of the invention figure compared with MLE Algorithm Error, and simulation parameter is set as, the practical side DOA
To θ0=0, reflection coefficient α=1, element number of array is 32, each array element spacing d=1, multiple additive white Gaussian noise (Complex
Additive White Gaussian Noise, CAWGN) channel.Space search range is [- 90 °, 90 °].
There are many application concrete application approach, and the foregoing is merely the preferred embodiments of the application, are not intended to limit
The embodiment and protection scope of this patent are made under the premise of this patent principle to those skilled in the art
Change with replacement and obviously obtained scheme, should all be included in the protection scope of patent.
Claims (5)
1. a kind of research method of the sensing system single goal Cramér-Rao lower bound based on information theory, which is characterized in that including such as
Lower step:
The uniform linear array for establishing multisensor array element obtains the likelihood function of observation using the reception signal of antenna, knot
The prior information for closing information theory and parameter to be measured obtains the posteriority conditional probability distribution of DOA；Under high s/n ratio, to Posterior distrbutionp
In reception signal carry out the method that signal decomposition and Gaussian approximation combine, it is final obtain Cramér-Rao lower bound as a result, being used in combination
The error statistics of MLE are used to verify the correctness of the result.
2. the research method of the sensing system single goal Cramér-Rao lower bound according to claim 1 based on information theory,
It is characterized in that, the reception signal using antenna obtains the likelihood function of observation, and combining information is discussed and the elder generation of parameter to be measured
It tests information and obtains the posteriority conditional probability distribution of DOA, detailed process is as follows:
First, it is assumed that observation section total length is | θ |, information source be [- | θ |/2, | θ |/2] it is equally distributed in section, therefore
The prior distribution for obtaining angle, θ to be measured is
Wherein: p (θ) is the prior distribution form of θ to be measured；
The phase of information sourceAnd equally distributed variable, prior distribution are expressed as in [0,2 π]
Wherein:For phasePrior distribution form；
In variable θ andThe likelihood function of reception signal Y is under the conditions of known
Wherein: N0The variance of additive white Gaussian noise is answered for steady zero-mean, M is array antenna array number, and y is even linear array
Received vector, x are information source vector, and A (θ) is the guiding matrix of array, are indicated are as follows:
A (θ)=[exp (j ω0τ0(θ)) exp(jω0τ1(θ)) K exp(jω0τM-1(θ))]T
Wherein: ω0Indicate the angular frequency of carrier signal, τ0The time delays of upper array element, τ are put on the basis of (θ)1It is put on the basis of (θ) other
The delay of first array element, τM-1(θ) is the delay of the M-1 array element；
It is obtained by the outlier for being unfolded, ignoring with θ to simplify likelihood function
WhereinExpression takes real part, and α is the decay factor of source signal, ymFor the reception letter in m array element
Number, τm(θ) is the time delays that m-th of array element receives signal in array；
According to the related knowledge of prior information and probability theory, the Posterior probability distribution for obtaining θ is
It is knownThen above formula is written as
Wherein I0() indicates first kind zeroth order modified Bessel function, and the shape of posterior probability Density Distribution is by Bezier at this time
What function determined, denominator is the normalized form of probability distribution.
3. the research method of the sensing system single goal Cramér-Rao lower bound according to claim 2 based on information theory,
It is characterized in that, the reception signal in Posterior distrbutionp carries out signal decomposition, and detailed process is as follows:
Signal will be received and be split as signal section and noise section, obtained
Wherein, wmFor the multiple additive white Gaussian noise obtained in m-th of array element；
Therefore posterior probability is also decomposed into
Wherein:For the phase of practical information source, θ0For the practical direction DOA, τm(θ0) it is m-th of array element on the practical direction DOA
Time delays；It enablesThe respectively auto-correlation function and signal of signal
Cross-correlation function between noise is considered as the influence that signal terms and noise item generate probability distribution, at this time
Wherein: giving signal-to-noise ratio ρ2Definition,Ε { } is to ask expectation to source signal.
4. the research method of the sensing system single goal Cramér-Rao lower bound according to claim 3 based on information theory,
It is characterized in that, described pair of reception signal carries out Gaussian approximation processing, and the detailed process of the final result for obtaining Cramér-Rao lower bound is such as
Under:
DOA estimated result is counted, obtained statistical error is infinitely to tend to CRB under high s/n ratio, posterior probability
It is approximately
Will | G (θ) | in θ=θ0Do Taylors approximation expansion in place
WhereinD is array element spacing, and λ is signal wavelength；In addition, Bessel function is also approximately at
It is by the processing result that two approximations combine to obtain Gaussian approximation method
Wherein k indicates normalized constant coefficient, σ2For the variance of Gaussian Profile；
Therefore under the conditions of high SNR, Cramér-Rao lower bound is arrived by estimating the variance of angle conditional probability to be measured
5. the research method of the sensing system single goal Cramér-Rao lower bound according to claim 1 based on information theory,
It is characterized in that, the uniform linear array for establishing multisensor array element, each array element radiation is all omni-directional.
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