CN115015831A - Large-scale array target orientation estimation method under combined influence of impulse noise and non-uniform noise - Google Patents

Large-scale array target orientation estimation method under combined influence of impulse noise and non-uniform noise Download PDF

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CN115015831A
CN115015831A CN202210507058.XA CN202210507058A CN115015831A CN 115015831 A CN115015831 A CN 115015831A CN 202210507058 A CN202210507058 A CN 202210507058A CN 115015831 A CN115015831 A CN 115015831A
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CN115015831B (en
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张亮
郭昆
殷敬伟
曹然
李迎松
赵欧南
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Harbin Engineering University
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    • G—PHYSICS
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    • 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
    • G01S3/80—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 ultrasonic, sonic or infrasonic waves
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Abstract

The invention provides a large-scale array target orientation estimation method under the combined influence of impulse noise and non-uniform noise, and belongs to the field of underwater acoustic array signal processing. According to the method, a variational sparse Bayesian signal model based on an underwater acoustic array is built, an S-t distribution pulse noise model is built, Bernoulli distribution is adopted to detect the existence probability of pulse noise, the designed model is closer to real application, non-uniform Gaussian noise is simulated by non-uniform Gaussian distribution of background noise, and the high-precision direction-of-arrival estimation method under the condition of mixing of the non-uniform noise and the pulse noise is provided. Compared with the existing similar wave direction estimation method, the method has higher estimation precision and stronger adaptability.

Description

Large-scale array target orientation estimation method under combined influence of impulse noise and non-uniform noise
Technical Field
The invention relates to a large-scale array target orientation estimation method under the combined influence of impulse noise and non-uniform noise, and belongs to the field of underwater acoustic array signal processing.
Background
Direction of Arrival (DOA) estimation is a popular research Direction in array signal processing, and in the field of underwater acoustic detection, an array is formed by a plurality of hydrophones to observe or receive underwater acoustic signals, so that DOA estimation of underwater targets is realized. The traditional beam forming methods such as CBF, MVDR and the like are limited in resolution and not enough to meet the high-precision requirement, and subspace type orientation estimation methods such as MUSIC, ESPRIT and the like can obtain high-resolution orientation estimation, but the number of information sources and the like are required to be used as prior information.
Because the marine environment is more complex and changeable, when the array arrangement operation is carried out underwater, the interference of seawater flow velocity, high pressure, big waves and the like can be encountered, the environment where the array is located cannot be directly assumed as a Gaussian noise environment, uneven Gaussian noise possibly exists among array elements of the array, pulse noise exists sometimes, and the complex noise environment reduces the direction-finding precision of the traditional method. Although a sparse estimation method for impulse noise exists at present, the situation that non-uniform noise and impulse noise exist simultaneously cannot be processed.
Disclosure of Invention
The invention aims to provide a large-scale array target orientation estimation method under the combined influence of impulse noise and non-uniform noise.
The purpose of the invention is realized as follows: the method comprises the following steps:
the method comprises the following steps: the method comprises the following steps of (1) building an underwater uniform long-line array, uniformly distributing the underwater uniform long-line array at a half-wavelength interval during distribution, and comprehensively building incident signal models received by each array element module:
Figure BDA0003636441310000011
wherein, Y is an observation signal,
Figure BDA0003636441310000012
is an overcomplete matrix, X is a sparse representation matrix of the incident signal, Z is an impulse noise decision matrix, V is a Gaussian noise matrix, 1 M×L The matrix is a matrix with the value of 1, and E is an impulse noise matrix;
step two: establishing prior distribution of each variable in a signal model, and carrying out layered prior distribution on array receiving signals:
constructing the layered prior distribution of the array receiving signals at the ith moment as follows:
Figure BDA0003636441310000013
wherein H, eta represents the hidden variable matrix of the control impulse noise, beta represents the variance vector of the non-uniform noise, and the lower corner is marked (-) to m,l The matrix element of the m-th array element at the first time is shown as a subscript (.) l The matrix vector representing the moment l, subscript (.) m Matrix vectors representing the m-th array element, e.g. y m,l Indicating the data received by the m-th array element at time l, y l Representing the received data of the entire array at time I, y at all times l Jointly forming a data matrix Y; the meanings of the other variables with the same subscript and case distinction are the same;
defining the noise variance β as a layered Gamma distribution:
Figure BDA0003636441310000021
wherein, g m And h m Respectively representing the noise variance beta at the m-th array element m The shape parameter and inverse scale parameter of (d);
defining the desired signal as a complex gaussian distribution:
Figure BDA0003636441310000022
where α is the variance matrix of the desired signal, subscript (·) n Representing the matrix vector at the nth scan orientation, e.g. alpha n The variance of the expected signal at the nth scanning position;
implicit variable matrix element eta for variance matrix alpha and ith time l And H l And respectively constructing layered Gamma distribution:
Figure BDA0003636441310000023
Figure BDA0003636441310000024
Figure BDA0003636441310000025
Figure BDA0003636441310000026
wherein, a n ,ξ m,l /2,c m,l And gamma m,l Respectively representing shape parameters of the corresponding distribution, b n ,ξ m,l /2,d m,l And ρ m,l Respectively representing inverse scale parameters, ξ, of the corresponding distributions l For constraining latent variable matrix element eta l α is the variance matrix of the desired signal;
noise occurrence state vector z for time I l The bernoulli distribution was constructed as:
Figure BDA0003636441310000027
wherein, pi l Represents a state vector z l The corresponding lower case letter is a specific element corresponding to the position of the lower corner mark, and the occurrence probability pi is l The layered Beta distribution is constructed as follows:
Figure BDA0003636441310000031
wherein p is m,l And q is m,l Respectively obeying Beta distribution parameters of the mth array element at the ith moment;
step three: initializing parameters: setting initial iteration iter ═1, initializing maximum, N, a 0 ,b 0 ,c 0 ,d 0 ,g 0 ,h 0 ,γ 0 ,ρ 0 ,z m,l Maximum number of iterations, N is the number of meshes divided in the whole spatial azimuth range, a 0 Shape parameter being variance of desired signal distribution, b 0 Inverse scale parameter of variance of desired signal distribution, c 0 、g 0 And gamma 0 Shape parameter of noise variance distribution, d 0 、h 0 And ρ 0 As inverse scale parameter of the noise variance distribution, p 0 And q is 0 To control the probability of occurrence pi l ；
Step four: iterative computation, namely obtaining an iterative formula of each parameter according to the distribution in the step two, and then updating;
step five: judging whether the iteration condition is met, if so, jumping out of the iteration and outputting mu x If not, continuing to perform the step four;
step six: searching a spectrum peak, and carrying out azimuth estimation according to the recovery signal:
Figure BDA0003636441310000032
step seven: outputting a bearing estimation result P VB (θ)。
The invention also includes such structural features:
1. the fourth step specifically comprises:
updating
Figure BDA0003636441310000033
Figure BDA0003636441310000034
Updating
Figure BDA0003636441310000035
Figure BDA0003636441310000036
Update a n ：
Figure BDA0003636441310000041
Update b n ：
Figure BDA0003636441310000042
Update c m,l ：
Figure BDA0003636441310000043
Update d m,l ：
Figure BDA0003636441310000044
Updating
Figure BDA0003636441310000045
Figure BDA0003636441310000046
Updating
Figure BDA0003636441310000047
Figure BDA0003636441310000048
Update g m,l ：
Figure BDA0003636441310000049
Update h m,l ：
Figure BDA00036364413100000410
Updating gamma m,l ：
Figure BDA00036364413100000411
Updating rho m,l ：
Figure BDA00036364413100000412
Updating the mean value of each variable:
Figure BDA0003636441310000051
let iter be iter + 1.
2. Updating variance
Figure BDA0003636441310000052
Updating variables of time<D Δ >Comprises the following steps:
D Δ ＝diag(z l ⊙β l +(1 M×1 -z l )⊙η l ⊙H l )
wherein diag (·) is a diagonal operation.
3. The iteration termination condition of the step five is as follows:
iter≥maxiter
Figure BDA0003636441310000053
wherein tol is a set termination threshold, and the value of tol is 0.001; α is the desired signal variance, defined as:
Figure BDA0003636441310000054
where < > represents the expectation of the variable to be solved.
Compared with the prior art, the invention has the beneficial effects that: the method fully considers the existence of pulse noise, the influence of the pulse noise and the influence of non-uniform noise, utilizes the distribution characteristic of Bernoulli distribution control noise to construct a variational Bayesian model, enables the model to be closer to the actual marine environment, realizes accurate azimuth estimation, has ultrahigh discrimination capability and multi-target resolution capability, resists the situation that the non-uniform noise and the pulse noise exist simultaneously, and is suitable for the azimuth estimation of the complex environment wave arrival.
Drawings
FIG. 1 is a schematic diagram of an array receive signal system according to the present invention;
FIG. 2 is a diagram of a model constructed according to the present invention;
FIG. 3 is a flow chart of the orientation estimation of the present invention;
fig. 4(a) - (b) are the comparison of the present invention (deployed method) with two other methods (conventional sparse method, impulse noise-based sparse method): FIG. 4(a) Root Mean Square Error (RMSE) results for each method; fig. 4(b) shows the detection probability results of different methods.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
The operation steps of the invention are as follows with the attached drawings:
(1) as shown in fig. 1, an M-element underwater uniform long-line array is constructed, the M-element underwater uniform long-line array is uniformly distributed at a half-wavelength interval during distribution, each array element module receives signals, and a received signal model is constructed as
Figure BDA0003636441310000061
Wherein
Figure BDA0003636441310000062
A matrix of data received for the hydrophone array,
Figure BDA0003636441310000063
in the form of an array of flow pattern matrices,
Figure BDA0003636441310000064
for the matrix of the desired signals after sampling,
Figure BDA0003636441310000065
a matrix of impulse noise occurrence states for each time instant of each channel of the array,
Figure BDA0003636441310000066
a non-uniform noise matrix of each array element of the hydrophone array, a dot product operation of matrix, 1 M×L Is a matrix of all 1 s and is,
Figure BDA0003636441310000067
the matrix is an impulse noise matrix at each array element of the hydrophone array, L is the length of the array received signal, and N is the number of the grids of the search azimuth space.
(2) The construction of the variational inference model is shown in FIG. 2.
(2.1) constructing a layered prior distribution of the array receiving signals at the ith moment as follows:
Figure BDA0003636441310000068
wherein, H, eta represents an implicit variable matrix for controlling impulse noise, beta represents a variance vector of non-uniform noise, and the lower corner is marked (-) to m,l The matrix element of the m-th array element at the first time is shown as a subscript (.) l The matrix vector representing the moment l, subscript (.) m Matrix vectors representing the m-th array element, e.g. y m,l Indicating the data received by the m-th array element at time l, y l Representing the received data of the entire array at time I, y at all times l Jointly forming a data matrix Y; similarly, the remaining variables have the same subscript and case-specificThe meanings are the same;
defining the noise variance β as a layered Gamma distribution:
Figure BDA0003636441310000069
wherein, g m And h m Respectively representing the noise variance beta at the m-th array element m The shape parameter and inverse scale parameter of (d);
defining the desired signal as a complex gaussian distribution:
Figure BDA00036364413100000610
where α is the variance matrix of the desired signal, lower corner mark (·) n Representing the matrix vector at the nth scan orientation, e.g. alpha n Is the variance of the expected signal at the nth scan orientation;
implicit variable matrix element eta for variance matrix alpha and ith time l And H l And respectively constructing layered Gamma distribution:
Figure BDA0003636441310000071
Figure BDA0003636441310000072
Figure BDA0003636441310000073
Figure BDA0003636441310000074
wherein, a n ,ξ m,l /2,c m,l And gamma m,l Respectively representing shape parameters of the corresponding distribution, b n ,ξ m,l /2,d m,l And ρ m,l Respectively representing inverse scale parameters, ξ, of the corresponding distributions l For constraining latent variable matrix element eta l α is the variance matrix of the desired signal;
noise occurrence state vector z for time I l The bernoulli distribution was constructed as:
Figure BDA0003636441310000075
wherein, pi l Represents a state vector z l The corresponding lower case letter is a specific element corresponding to the position of the lower corner mark, and the occurrence probability pi is l The layered Beta distribution is constructed as follows:
Figure BDA0003636441310000076
wherein p is m,l And q is m,l Respectively the Beta distribution parameters obeyed by the mth array element at the ith moment.
(2.2) defining a variable set omega ═ X, alpha, H, eta, xi, beta, Z, pi } by using a variational inference theory, and sequentially substituting the distribution matrix constructed in (2.1) into the following formula to solve the posterior probability of each variable:
Figure BDA0003636441310000077
Figure BDA0003636441310000078
Figure BDA0003636441310000079
Figure BDA0003636441310000081
Figure BDA0003636441310000082
Figure BDA0003636441310000083
Figure BDA0003636441310000084
Figure BDA0003636441310000085
wherein q (-) represents the posterior probability of the variable (-) and ln (-) represents the logarithm,<·>expressing the expectation, p (· | ·) expresses the probability of the element therein, which can be derived from (2.1), q (Ω i not ≠ is) represents the computation of the portion of the set that does not contain the variable · const, which represents the constant term.
(3) And (3) calculating the mean value and the variance of each variable according to the probability distribution of each variable obtained in the step (2).
(3.1) parameter initialization: setting initial iteration iter to 1, initializing maximum, N, a 0 ,b 0 ,c 0 ,d 0 ,g 0 ,h 0 ,γ 0 ,ρ 0 ,z m,l Maximum number of iterations, N is the number of meshes divided in the whole spatial azimuth range, a 0 Shape parameter being variance of desired signal distribution, b 0 Inverse scale parameter of variance of desired signal distribution, c 0 、g 0 And gamma 0 Shape parameter of noise variance distribution, d 0 、h 0 And ρ 0 As inverse scale parameter of the noise variance distribution, p 0 And q is 0 To control the probability of occurrence pi l (ii) a Then, according to the probability distribution of each variable obtained in the step (2), each parameter is sequentially subjected to iterative updating;
(3.2) updating the variance of the desired signal in sequence
Figure BDA0003636441310000086
Sum mean value
Figure BDA0003636441310000087
Updating
Figure BDA0003636441310000088
Figure BDA0003636441310000089
Wherein D is Δ ＝diag(z l ⊙β l +(1 M×1 -z l )⊙η l ⊙H l )，D α ＝diag(α l ) Diag (·) denotes a diagonal operation;
updating
Figure BDA00036364413100000810
Figure BDA00036364413100000811
(3.3) updating the distribution parameters according to the result of (3.2), including:
update a n ：
Figure BDA00036364413100000812
Update b n ：
Figure BDA0003636441310000091
Update c m,l ：
Figure BDA0003636441310000092
Update d m,l ：
Figure BDA0003636441310000093
Updating
Figure BDA0003636441310000094
Figure BDA0003636441310000095
Updating
Figure BDA0003636441310000096
Figure BDA0003636441310000097
Update g m,l ：
Figure BDA0003636441310000098
Update h m,l ：
Figure BDA0003636441310000099
Updating gamma m,l ：
Figure BDA00036364413100000910
Updating rho m,l ：
Figure BDA00036364413100000911
(3.4) updating the mean value of each variable according to the result of (3.3), including:
Figure BDA0003636441310000101
the mean result of each variable is the result of the iteratively estimated variable, e.g. α ═<α n >The same applies otherwise.
(3.5) update iter:
iter＝iter+1
(4) judging whether an iteration termination condition is met or not, wherein the iteration termination condition is as follows
iter≥maxiter
Figure BDA0003636441310000102
Wherein tol is a set termination threshold, and the value of tol is 0.001; alpha is the desired signal variance, and when one of the above iteration termination conditions is met, the iteration is skipped and mu is output x Otherwise, continuing the iteration of (3.2) - (3.5);
(5): searching a spectrum peak, and carrying out azimuth estimation according to the recovery signal:
Figure BDA0003636441310000103
wherein | · | charging 1 Representing a matrix-norm operation, | · | count non-woven phosphor ∞ Representing an infinite norm operation of the matrix.
(6): outputting a bearing estimation result P VB (θ)。
Fig. 3 is an overall flow of the position estimation. The invention fully considers the influence of the presence or absence of impulse noise, the influence of the impulse noise and the non-uniform noise, and the constructed model is more suitable for the actual marine environment, thereby obtaining a better estimation result.
Simulation study of the invention:
simulation conditions are as follows:
two single-frequency pulse signals are used as incident signals, the incident directions are-30 degrees and-15 degrees respectively, the number of snapshots is 100, non-uniform noise is added to half of the snapshots, the noise variance of each array element randomly changes between 0.1 and 5, S alpha S pulse noise is added to the other half of the snapshots, the generalized signal-to-noise ratio (GSNR) is changed in the range of-10 and 20, and a traditional sparse method, a sparse method based on the pulse noise and the method provided by the invention are compared and analyzed.
Fig. 4(a) is a Root Mean Square Error (RMSE) variation curve of each method when it varies with GSNR under impulse noise environment. Comparing the curves of the methods, the failure of the traditional sparse method is more serious; although the sparse method based on the impulse noise has a downward trend, when the non-uniform noise and the impulse noise alternately appear, a jump phenomenon exists, so that an estimation result is unstable; the method provided by the invention has the lowest RMSE and the most stable RMSE, can adapt to the noise change condition under different SNR, obtains a better estimation result and has the minimum deviation.
Fig. 4(b) is a detection success probability curve of each method when the method changes with GSNR under the impulse noise environment, and the detection success is defined within 1 ° of the target deviation. Comparing the curves of the methods, the detection success probability of the traditional sparse method is slowly increased along with the increase of SNR, that is to say, the method can not accurately find the target direction in time under the background of impulse noise and non-uniform noise; the probability of the pulse noise-based sparse method relative to the successful detection is increased, but when non-uniform noise and pulse noise alternately appear, a jump phenomenon exists, so that the estimation result is unstable; the method provided by the invention has the highest detection success probability, is most stable, and has the strongest estimation success capability on the target.
In conclusion, the invention provides a large-scale array target orientation estimation method under the combined influence of impulse noise and non-uniform noise, and belongs to the field of underwater acoustic array signal processing. The method has the advantages that variational sparse Bayesian signal modeling based on the underwater acoustic array is established, an S-t distribution pulse noise model is established, Bernoulli distribution is adopted to detect whether pulse noise exists, the model is closer to reality, a non-uniform Gaussian noise model with non-uniform Gaussian distribution is established, and high-precision DOA estimation under the mixed condition of the non-uniform noise and the pulse noise is realized. The method is closer to real application, the background noise simulates non-uniform Gaussian noise by using non-uniform Gaussian distribution, and the high-precision direction-of-arrival estimation method under the condition of mixing the non-uniform noise and the impulse noise is provided. Compared with the existing similar wave direction estimation method, the method has higher estimation precision and stronger adaptability. The simulation research result verifies the effectiveness and feasibility of the invention.

Claims (4)

1. A large-scale array target orientation estimation method under the combined influence of impulse noise and non-uniform noise is characterized by comprising the following steps:
the method comprises the following steps: the method comprises the following steps of (1) building an underwater uniform long-line array, uniformly distributing the underwater uniform long-line array at a half-wavelength interval during distribution, and comprehensively building incident signal models received by each array element module:
Figure FDA0003636441300000011
wherein, Y is an observation signal,
Figure FDA0003636441300000012
is an overcomplete matrix, X is a sparse representation matrix of the incident signal, Z is an impulse noise decision matrix, V is a Gaussian noise matrix, 1 M×L The matrix is a matrix with the value of 1, and E is an impulse noise matrix;
step two: establishing prior distribution of each variable in a signal model, and carrying out layered prior distribution on array receiving signals:
constructing the layered prior distribution of the array receiving signals at the ith moment as follows:
Figure FDA0003636441300000013
wherein, H, eta represents an implicit variable matrix for controlling impulse noise, beta represents a variance vector of non-uniform noise, and the lower corner is marked (-) to m,l The matrix element of the m-th array element at the l-th time is shown, the lower cornerSymbol (·) l The matrix vector representing the moment l, subscript (.) m Matrix vectors representing the m-th array element, e.g. y m,l Indicating the data received by the m-th array element at time l, y l Representing the received data of the entire array at time I, y at all times l Jointly forming a data matrix Y; the meanings of the other variables with the same subscript and case distinction are the same;
defining the noise variance β as a layered Gamma distribution:
Figure FDA0003636441300000014
wherein, g m And h m Respectively representing the noise variance beta at the m-th array element m The shape parameter and inverse scale parameter of (d);
defining the desired signal as a complex gaussian distribution:
Figure FDA0003636441300000015
where α is the variance matrix of the desired signal, subscript (·) n Representing the matrix vector at the nth scan orientation, e.g. alpha n The variance of the expected signal at the nth scanning position;
for variance matrix alpha and hidden variable matrix element eta at the first moment l And H l And respectively constructing layered Gamma distribution:
Figure FDA0003636441300000021
Figure FDA0003636441300000022
Figure FDA0003636441300000023
Figure FDA0003636441300000024
wherein, a n ,ξ m,l /2,c m,l And gamma m,l Respectively representing shape parameters of the corresponding distribution, b n ,ξ m,l /2,d m,l And ρ m,l Respectively representing the inverse scale parameter, ξ, of the corresponding distribution 1 For constraining latent variable matrix element eta l α is the variance matrix of the desired signal;
noise occurrence state vector z for time I l The bernoulli distribution was constructed as:
Figure FDA0003636441300000025
wherein, pi l Represents a state vector z l The corresponding lower case letter is a specific element corresponding to the position of the lower corner mark, and the occurrence probability pi is l The layered Beta distribution is constructed as follows:
Figure FDA0003636441300000026
wherein p is m,l And q is m,l Respectively obeying Beta distribution parameters of the mth array element at the ith moment;
step three: initializing parameters: setting initial iteration iter to 1, initializing maximum, N, a 0 ,b 0 ,c 0 ,d 0 ,g 0 ,h 0 ,γ 0 ,ρ 0 ,z m,l Maximum number of iterations, N is the number of meshes divided in the whole spatial azimuth range, a 0 Shape parameter being variance of desired signal distribution, b 0 Inverse scale parameter of variance of desired signal distribution, c 0 、g 0 And gamma 0 Shape parameter of noise variance distribution, d 0 、h 0 And ρ 0 As inverse scale parameter of the noise variance distribution, p 0 And q is 0 To control the probability of occurrence pi l ；
Step four: iterative computation, namely obtaining an iterative formula of each parameter according to the distribution in the step two, and then updating;
step five: judging whether the iteration condition is met, if so, jumping out of the iteration and outputting mu x If not, continuing to perform the step four;
step six: searching a spectrum peak, and carrying out azimuth estimation according to the recovery signal:
Figure FDA0003636441300000031
step seven: outputting a bearing estimation result P VB (θ)。
2. The method for estimating the orientation of the target of the large-scale array under the combined influence of the impulse noise and the non-uniform noise according to claim 1, wherein the step four specifically comprises:
updating
Figure FDA0003636441300000032
Figure FDA0003636441300000033
Updating
Figure FDA0003636441300000034
Figure FDA0003636441300000035
Update a n ：
Figure FDA0003636441300000036
Update b n ：
Figure FDA0003636441300000037
Update c m,l ：
Figure FDA0003636441300000038
Update d m,l ：
Figure FDA0003636441300000039
Updating
Figure FDA00036364413000000310
Figure FDA00036364413000000311
Updating
Figure FDA00036364413000000312
Figure FDA0003636441300000041
Update g m,l ：
Figure FDA0003636441300000042
Update h m,l ：
Figure FDA0003636441300000043
Updating gamma m,l ：
Figure FDA0003636441300000044
Updating rho m,l ：
Figure FDA0003636441300000045
Updating the mean value of each variable:
Figure FDA0003636441300000046
let iter be iter + 1.
3. The method of claim 2, wherein the variance is updated according to the estimation method of the target orientation of the large-scale array under the combined influence of impulse noise and non-uniform noise
Figure FDA0003636441300000047
Updating variables of time<D Δ >Comprises the following steps:
D Δ ＝diag(z l ⊙β l +(1 M×1 -z l )⊙η l ⊙H l )
wherein diag (·) is a diagonal operation.
4. The method for estimating the orientation of the target of the large-scale array under the combined influence of the impulse noise and the non-uniform noise according to claim 1, wherein the iteration termination condition of the step five is as follows:
iter≥maxiter
Figure FDA0003636441300000051
wherein tol is a set termination threshold, and the value of tol is 0.001; α is the desired signal variance, defined as:
Figure FDA0003636441300000052
where < > represents the expectation of the variable to be solved.
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