CN104330766B - Robust estimation method of direction of arrival (DOA) - Google Patents

Abstract

The invention discloses a robust estimation method of a direction of arrival (DOA), by which quite high estimation precision can be obtained. The method specifically comprises: sequentially dividing an equidistant linear array into three sub-arrays from the left to the right, i.e., a forward auxiliary array, a central main array and a backward auxiliary array respectively; dividing the central main array into a front half portion and a rear half portion according to a certain rule; then calculating a covariance matrix Rforward between the front half portion and the backward auxiliary array and a covariance matrix Rbackward between the rear half portion and the forward auxiliary array; performing singular value decomposition respectively on the Rforward and the Rbackforward, and extracting the orthogonal complement of the subspace of far-field signals received by the equidistant linear array from a singular value decomposition result; and establishing a DOA estimation operator by use of guiding vectors between the orthogonal complement and the front half portion and the rear half portion, performing search on the DOA estimation operator by use of a minimum parameter search method to obtain K minimum value points, and through the search, obtaining theta 1 to theta k of each minimum value point as k DOAs of the far-field signals.

Description

A kind of sane Wave arrival direction estimating method

Technical field

The invention belongs to linear array signal processing technology field.

Background technology

So-called uniform linear array refers to the array that multiple sensor is formed from arrangement according to linear isometry.When echo is believed Number arrive array time, multiple sensors can be formed multiple echo-signal.Therefore Array Signal Processing has spatial domain various dimensions Feature, and possess the advantage of space power synthesis.Uniform linear array as conventional array layout mode in Project Realization On there is simple and convenient, the advantage of low cost, be therefore frequently utilized for reversing radar of vehicle, train Anticollision Radar and source of sound location In field.Due to the commercial Application potentiality that it is huge, research is applicable to the various technology of linear array and has become as array signal One of main direction of studying for the treatment of technology.

In the field of linear Array Signal Processing, a demand common in engineering is exactly the location of target direction.By Being all arrowband far-field signal in common target echo, the signal that therefore echo is formed in each array element can be approximately considered width Spend equal, and between adjacent array element, phase contrast is equal.The phase contrast that the echo of different directions is formed between adjacent array element is unequal, profit The estimation to target arrival bearing can be completed by this information.

The schematic diagram of linear array is as shown in Figure 1, it is considered to array number is that the uniform line-array of M receives one from θ The far-field signal in direction.Array received is as follows to the data model of t:

Y (t)=Ax (t)+n (t)

Wherein y (t) represents the signal data that array received arrives, and the steering vector of array is $A={\left[\begin{array}{cccc}1& e^{j\frac{2\mathrm{\π}}{\mathrm{\λ}}d\sin \mathrm{\θ}}& \·\·\·& e^{j\frac{2\mathrm{\π}}{\mathrm{\λ}}(M-1)d\sin \mathrm{\θ}}\end{array}\right]}^T,$ X (t) represents the complex envelope of t signal, and n (t) represents Gauss White noise signal.

But applied environment residing for array is the most complex, the noise faced is frequently not white noise, but high The coloured noise that degree is relevant.Further, since the impact of the imperfection factor of array itself, such as array mutual coupling, currently used The DOA estimation algorithm of linear array, as MUSIC performance can be by severe exacerbation.

So-called System in Spatial Colored Noise refers to, in array, the noise between different array elements possesses dependency.Cause this dependency Factor is a lot, such as external environmental noise, array mutual coupling etc..When arriving array antenna due to spatial electromagnetic ripple, due to impedance not Coupling can cause reflection of electromagnetic wave, reappears and is transmitted in space so that around array antenna received is to this reflected signal, such battle array Coupling phenomenon complicated between unit is referred to as array mutual coupling, and wherein array mutual coupling schematic diagram is as in figure 2 it is shown, in fig. 2,0 represent from The electromagnetic wave that far field arrives, after electromagnetic wave arrives array antenna, can arrive backend load, such as in figure 1 along receiving passage Shown in.Due to antenna backend load impedance with receive the impedance mismatch of electromagnetic wave, the energy Hui Yuan road of some is returned Return, as shown in Figure 4, then by aerial radiation in space outerpace, as shown in Figure 2.The energy of this partial radiation has one Divide by received by the antenna of surrounding, as shown in Figure 3, be the formation of mutual coupling.Mutual coupling is modal a kind of phenomenon in array, The distance of two array elements in array is the nearest, and the intensity of mutual coupling is the biggest each other, and the intensity of mutual coupling is quick along with the increase of distance Decay.So the array mutual coupling coefficient between two array elements relatively far apart is approximately 0.Analyzing based on above, array mutual coupling is permissible Represented by the mutual coefficient matrix with symmetrical toeplitz matrix form.Assume to be separated by two that array number is L-1 unit The mutual coupling coefficient between individual unit is 0, then for pth unit, with it has the unit of the mutual coupling coefficient to be respectively p-L+ 1、p-L+2、…p-1、p+1…p+L-2、p+L-1.Mutual coefficient matrix is normalized so that the elements in a main diagonal Be 1, then mutual coefficient matrix can be to be expressed as form:

Under mutual coupling, the signal phasor received by array is represented by:

Y=CAx+Cn

It can be seen that mutual coupling can cause steering vector to deform (being become CA from original A), simultaneously from expression above Also the dependency of noise between different array element can be caused so that space white noise originally becomes System in Spatial Colored Noise Cn.

Owing to mutual coupling is relatively big on the impact of Array Signal Processing, compensation and correction for mutual coupling have obtained substantial amounts of grinding Study carefully.But these compensate and methods of correction need to carry out the solving of nonlinear optimal problem of various dimensions.This is calculating complexity Propose the highest requirement on degree, and convergence cannot be ensured, be not suitable for the actual application of engineering.Recently, Z.Ye is relevant Document in demonstrate, by uniform line-array being divided into three submatrixs, under meeting certain constraints, mutual symplectic guiding The impact of vector can eliminate, and so may dispense with and compensates and the step of correction, thus is greatly simplified processing procedure.But base Wave arrival direction estimating method in this Subarray partition mode does not but account for the impact that mutual symplectic noise may bring, uncomfortable Application under coloured noise environment.

On the other hand, Qiang Wu and Kon Max Wong proposes direction of arrival based on space auxiliary variable method and estimates Calculating method.This algorithm utilizes two submatrix noises this characteristic uncorrelated relatively far apart to eliminate coloured noise.This algorithm is at color Preferably application is achieved in the environment of noise.But this algorithm has two big shortcomings: one is the battle array between two submatrixs of far apart Column unit is underutilized, and causes array utilization rate to decline；Two is this algorithm under the influence of not taking into account array mutual coupling, The situation of steering vector mismatch, therefore under the influence of array mutual coupling, this algorithm performance will severe exacerbation.

Summary of the invention

In view of this, the invention provides a kind of sane Wave arrival direction estimating method, to adding before and after the process employs The space auxiliary variable method of weight average, both can resist the steering vector mismatch caused by array mutual coupling and white noise sensual pleasure Problem, simultaneously also possesses amount of calculation little, it is easy to accomplish feature.

For reaching above-mentioned purpose, the technical scheme is that, the method comprises the steps:

Step 1, array number are that the uniform line-array of M receives a far-field signal, and this far-field signal has k incoming wave side To；This uniform line-array is the most sequentially divided into three submatrixs, and wherein the submatrix of the right and left is called forward direction auxiliary Battle array and backward companion matrix, middle submatrix is referred to as central main battle array.

Wherein the element number of array of forward direction auxiliary array is I, and the element number of array of backward auxiliary array is also I；The value of I is more than Or equal to L-1；Wherein L is the discreet value of the mutual coupling coefficient length of uniform line-array.

For central main battle array, wherein the I+1 array element to the M-I-Q array element is first half, and the I+1+Q array element arrives The M-I array element is latter half, and wherein Q is more than or equal to 2I.

Step 2, the covariance matrix calculated between first half and backward companion matrix are forward direction covariance matrix R_forward, with And the covariance matrix between latter half and forward direction companion matrix is backward covariance matrix R_backward；

Step 3, to forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition respectively, And in singular value decomposition, extract the orthocomplement, orthogonal complement of the subspace of described far-field signal.

The steering vector of step 4, the orthocomplement, orthogonal complement using acquisition in step 3 and first half and latter half sets up the ripple side of reaching To estimate operator, and to this Mutual coupling operator use minimize parameter searching method scan for obtain k minimum Point, search obtains the θ corresponding to each minimum point_i, i=1～k, then θ₁～θ_kK the arrival bearing for described far-field signal.

Further, the signal phasor that described forward direction companion matrix receives is y_forward, described backward companion matrix receives Signal phasor be y_backward, the reception signal of described first half is y_main-1, the reception signal of aft section is designated as y_main-2, The most described forward direction covariance matrixDescribed backward covariance matrixWherein N represents the fast umber of beats that described uniform line-array is sampled, the number of N more than or Person is equal to 3 (M-4I).

Further, step 3 comprises the steps:

S301, to forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition respectively: $\begin{array}{cc}{R}_{\mathrm{forward}}=U_1\mathrm{\Γ}{U_2}^H& R_{\mathrm{backward}}=U_1^{\′}\end{array},{\mathrm{\Γ}}^{\′}{U}_2^{\′H}.$

Wherein U₁ R_forwardDecompose the left chief of a tribe's matrix obtained, U₂For R_forwardDecomposing the right chief of a tribe's matrix obtained, Γ is R_forward Decompose the diagonal matrix obtained； $\mathrm{\Γ}=\left[\begin{array}{cc}{\mathrm{\Γ}}_0& 0\\ 0& 0\end{array}\right],$ Γ₀=diag (γ₁ γ₂ … γ_K), γ₁≥γ₂≥…≥γ_K＞ 0；K ＞ k, γ₁ γ₂ … γ_KFor R_forwardDecompose the singular value obtained.

Wherein U'₁For R_backwardDecompose the left chief of a tribe's matrix obtained, U'₂For R_backwardDecompose the right chief of a tribe matrix R obtained_backward Decompose the diagonal matrix obtained； ${\mathrm{\Γ}}^{\′}=\left[\begin{array}{cc}{{\mathrm{\Γ}}^{\′}}_0& 0\\ 0& 0\end{array}\right],$ Γ'₀=diag (γ '₁ γ'₂ … γ'_K'), γ '₁≥γ'₂≥… ≥γ'_K'＞ 0；K ＇ ＞ k, γ '₁ γ'₂ … γ'_K'For R_backwardDecompose the singular value obtained.

S302, U₁With U '₁It is respectively divided into two parts: U₁=[U_s|U_v] U′₁=[U_s′|U_v′]；Wherein U_sWith U '_sPoint Do not represent U₁With U '₁Front k row, U_vWith U '_vRepresent U the most respectively₁With U '₁Remaining row, then U_vWith U '_vFor described far-field signal The orthocomplement, orthogonal complement of subspace.

Further, i-th arrival bearing θ_iParticularly as follows:

${\mathrm{\θ}}_i=\underset{{\mathrm{\θ}}_i}{\arg \min }[{\mathrm{\λ}}_1{a}_{\mathrm{main}-1}{\left({\mathrm{\θ}}_i\right)}^H{U}_v{U_v}^H{a}_{\mathrm{main}-1}\left({\mathrm{\θ}}_i\right)+{\mathrm{\λ}}_2{a}_{\mathrm{main}-2}{\left({\mathrm{\θ}}_i\right)}^H{U^{\′}}_v{{U^{\′}}_v}^H{a}_{\mathrm{main}-2}\left({\mathrm{\θ}}_i\right)]$

Wherein a_main-1(θ_i) it is the steering vector of described first half, a_main-2(θ_i) be described latter half guiding vow Amount, λ₁、λ₂It is respectively arithmetic number, λ₁=λ₂, arg min [] is the minimum solved.

Preferably, minimizing parameter searching method is steepest descent method, Newton method or exhaustive search algorithm.

Beneficial effect:

A kind of sane Wave arrival direction estimating method the most provided by the present invention, can either resist what array mutual coupling was brought Steering vector distorts, again can in System in Spatial Colored Noise effectively, and obtain estimates the situation that operator exists at above non-ideal factor Under, its estimated accuracy is better than traditional method.

2. the method need not do array the mutual coupling coefficient any compensation or estimation, simultaneously also without estimation space color Noise parameter, so can be substantially reduced algorithm complex and operand so that this algorithm really possesses engineering realizability.

Accompanying drawing explanation

Fig. 1 is linear array schematic diagram；

Fig. 2 is array mutual coupling schematic diagram；

The Subarray partition schematic diagram of the equidistant even linear array of Fig. 3；

The process chart of Fig. 4 this method；

Fig. 5 new method and the traditional method effect contrast figure under the conditions of different signal to noise ratios；

Fig. 6 new method and the traditional method effect contrast figure under different snap said conditions.

Detailed description of the invention

Develop simultaneously embodiment below in conjunction with the accompanying drawings, describes the present invention.

The invention provides a kind of sane Wave arrival direction estimating method, the method is at the base of the Subarray partition mode of Z.Ye On plinth, it is contemplated that the application under coloured noise environment, to average weighted space auxiliary variable method before and after have employed, this ripple side of reaching Both can resist the steering vector mismatch caused by array mutual coupling and the problem of white noise sensual pleasure to method of estimation, also possess simultaneously Amount of calculation is little, it is easy to accomplish feature.

The method comprises the steps:

Step one, Subarray partition；The present embodiment neutron array division methods is as shown in Figure 3.

In the present embodiment, array number is that the uniform line-array of M receives a far-field signal, and the number of its arrival bearing is k； This uniform line-array is divided into three submatrixs: forward direction auxiliary array, central authorities' submatrix and backward auxiliary array.Wherein the right and left Submatrix is called forward direction companion matrix and backward companion matrix, and middle submatrix is referred to as central main battle array；Wherein forward direction auxiliary array with The element number of array of backward auxiliary array is identical, and is I, and the value of I is more than or equal to L-1.

L is the mutual coupling coefficient length of linear array, in the present embodiment, it should pre-estimate for L.

Being first half and latter half by central authorities' Subarray partition again, wherein first half is by the I+1 array element to M- I-Q array element composition, receives signal and is designated as y_main-1, aft section is made up of the I+1+Q array element to the M-I array element, receives Signal is designated as y_main-2.More than or equal to 2I, the value of Q ensures that latter half array and first half do not have array mutual coupling Impact.

Step 2, covariance matrix calculate；

Based on above-mentioned Subarray partition mode, this step calculates respectively between central main battle array first half and backward companion matrix Covariance R_forwardAnd the covariance R between central main battle array latter half and forward direction companion matrix_backward。

$R_{\mathrm{forward}}=\frac{1}{N}\underset{t=1}{\overset{N}{\mathrm{\Σ}}}{y}_{\mathrm{main}-1}\left(t\right)y_{\mathrm{backward}}{\left(t\right)}^H$

$R_{\mathrm{backward}}=\frac{1}{N}\underset{t=1}{\overset{N}{\mathrm{\Σ}}}{y}_{\mathrm{main}-2}\left(t\right)y_{\mathrm{forward}}{\left(t\right)}^H$

Here N represents the fast umber of beats that described uniform line-array is sampled, and fast umber of beats is the most, and the estimation to covariance matrix is the most accurate Really, the angle estimation value obtained is the most accurate.In the present embodiment, the number suggestion of N is not less than 3 (M-4I).

Step 3, covariance singular value decomposition.

To forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition respectively, and very Different value decomposition result is extracted the orthocomplement, orthogonal complement of the subspace of described far-field signal.

The present embodiment uses following concrete steps to realize this step:

S301, to forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition respectively: $\begin{array}{cc}{R}_{\mathrm{forward}}=U_1\mathrm{\Γ}{U_2}^H& R_{\mathrm{backward}}=U_1^{\′}\end{array},{\mathrm{\Γ}}^{\′}{U}_2^{\′H}.$

Wherein U₁For R_forwardDecompose the left chief of a tribe's matrix obtained, U₂For R_forwardDecompose the right chief of a tribe matrix R obtained_forwardDecompose The diagonal matrix obtained； $\mathrm{\Γ}=\left[\begin{array}{cc}{\mathrm{\Γ}}_0& 0\\ 0& 0\end{array}\right],$ Γ₀=diag (γ₁ γ₂ … γ_K), γ₁≥γ₂≥…≥γ_K＞ 0；K ＞ k, γ₁ γ₂ … γ_KFor R_forwardDecompose the singular value obtained.

Wherein U'₁For R_backwardDecompose the left chief of a tribe's matrix obtained, U'₂For R_backwardDecompose the right chief of a tribe matrix R obtained_backward Decompose the diagonal matrix obtained； ${\mathrm{\Γ}}^{\′}=\left[\begin{array}{cc}{{\mathrm{\Γ}}^{\′}}_0& 0\\ 0& 0\end{array}\right],$ Γ'₀=diag (γ '₁ γ'₂ … γ'_K'), γ '₁≥γ'₂≥… ≥γ'_K'＞ 0；K ＇ ＞ k, γ '₁ γ'₂ … γ'_K'For R_backwardDecompose the singular value obtained；

S302, the U that singular value decomposition is obtained₁With U '₁It is respectively divided into two parts: U₁=[U_s|U_v]U₁'=[U_s′| U_v′]；Wherein U_sWith U '_sRepresent U respectively₁With U '₁Front K row, U_vWith U '_vRepresent U the most respectively₁With U '₁Remaining row, then U_vWith U′_vOrthocomplement, orthogonal complement for the subspace of described far-field signal.

Step 4, cost function minimize parameter search.

The steering vector using orthocomplement, orthogonal complement and first half and latter half sets up Mutual coupling operator, and to this ripple Reach direction estimation operator use minimize parameter searching method scan for obtain k minimum point, search for obtain k minimum θ value corresponding to Dian is arrival bearing's angle of target.

I-th arrival bearing θ_iIt is specially:

${\mathrm{\θ}}_i=\underset{{\mathrm{\θ}}_i}{\arg \min }[{\mathrm{\λ}}_1{a}_{\mathrm{main}-1}{\left({\mathrm{\θ}}_i\right)}^H{U}_v{U_v}^H{a}_{\mathrm{main}-1}\left({\mathrm{\θ}}_i\right)+{\mathrm{\λ}}_2{a}_{\mathrm{main}-2}{\left({\mathrm{\θ}}_i\right)}^H{U^{\′}}_v{{U^{\′}}_v}^H{a}_{\mathrm{main}-2}\left({\mathrm{\θ}}_i\right)]$

Wherein a_main-1(θ_i) it is the steering vector of described first half, a_main-2(θ_i) be described latter half guiding vow Amount, λ₁、λ₂It is respectively arithmetic number, the two random value, λ₁=λ₂；Arg min [] is the minimum solved.

Owing to the element number of array of front and back's array is equal, therefore first half is identical with the steering vector of latter half, and all Do not affected by array mutual coupling.The search θ value obtained corresponding to k minimum point is designated as arrival bearing's angle of target.

Minimize parameter searching method and can use steepest descent method, Newton method or exhaustive search algorithm.

Emulation have employed the uniform line-array of 16 array elements, be provided with two far field narrowband target signals, wherein No. 1 targets Power compared with No. 2 big 5dB of target, No. 1 target power is-5dB, and No. 2 target powers are-10dB, I=L-1=1, N=500, figure 5 is the design sketch that the mean square error that No. 1 angle on target is estimated changes with self signal to noise ratio, and Fig. 6 is the equal of No. 1 target state estimator angle The design sketch that side's error changes with fast umber of beats, it can be seen that its mean square error of the angle on target of estimation is much smaller than traditional method.

In sum, these are only presently preferred embodiments of the present invention, be not intended to limit protection scope of the present invention. All within the spirit and principles in the present invention, any modification, equivalent substitution and improvement etc. made, should be included in the present invention's Within protection domain.

Claims (5)

1. a sane Wave arrival direction estimating method, it is characterised in that the method comprises the steps:

Step 1, array number are that the uniform line-array of M receives a far-field signal, and this far-field signal has k arrival bearing；Will This uniform line-array is the most sequentially divided into three submatrixs, wherein the submatrix of the right and left be called forward direction companion matrix and after To companion matrix, middle submatrix is referred to as central main battle array；

The element number of array of wherein said forward direction auxiliary array is I, and the element number of array of backward auxiliary array is also I；The value of I is more than Or equal to L-1；Wherein L is the discreet value of the mutual coupling coefficient length of described uniform line-array；

Battle array main for described central authorities, wherein the I+1 array element to the M-I-Q array element is first half, and the I+1+Q array element arrives The M-I array element is latter half, and wherein Q is more than or equal to 2I；

Step 2, the covariance matrix calculated between described first half and backward companion matrix are forward direction covariance matrix R_forward, meter Calculating the covariance matrix between described latter half and forward direction companion matrix is backward covariance matrix R_backward；

Step 3, to forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition respectively, and Singular value decomposition extracts the orthocomplement, orthogonal complement of the subspace of described far-field signal；

The steering vector of step 4, the orthocomplement, orthogonal complement using acquisition in described step 3 and first half and latter half sets up the ripple side of reaching To estimate operator, and to this Mutual coupling operator use minimize parameter searching method scan for obtain k minimum Point, search obtains the θ corresponding to each minimum point_i, i=1～k, then θ₁～θ_kK the arrival bearing for described far-field signal.

A kind of sane Wave arrival direction estimating method the most as claimed in claim 1, it is characterised in that described forward direction companion matrix connects The signal phasor received is y_forward, the signal phasor that described backward companion matrix receives is y_backward, described first half Reception signal is y_main-1, the reception signal of aft section is designated as y_main-2, the most described forward direction covariance matrixDescribed backward covariance matrix Wherein N represents the fast umber of beats that described uniform line-array is sampled, and the number of N is more than or equal to 3 (M-4I).

A kind of sane Wave arrival direction estimating method the most as claimed in claim 1 or 2, it is characterised in that described step 3 includes Following steps:

S301, to forward direction covariance matrix R_forwardWith backward covariance matrix R_backwardCarry out singular value decomposition: R respectively_forward =U₁ΓU₂ ^H

Wherein U₁For R_forwardDecompose the left chief of a tribe's matrix obtained, U₂For R_forwardDecomposing the right chief of a tribe's matrix obtained, Γ is R_forwardPoint Solve the diagonal matrix obtained；Γ₀=diag (γ₁ γ₂ … γ_K), γ₁≥γ₂≥…≥γ_K＞ 0；K ＞ k,γ₁ γ₂ … γ_KFor R_forwardDecompose the singular value obtained；

Wherein U'₁For R_backwardDecompose the left chief of a tribe's matrix obtained, U'₂For R_backwardDecompose the right chief of a tribe matrix R obtained_backwardDecompose The diagonal matrix obtained；Γ'₀=diag (γ '₁ γ'₂ … γ'_K'), γ '₁≥γ'₂≥…≥γ'_K' ＞ 0；K ＇ ＞ k, γ '₁ γ'₂ … γ'_K'For R_backwardDecompose the singular value obtained；

S302, U₁With U '₁It is respectively divided into two parts: U₁=[U_s|U_v] U′₁=[U_s′|U_v′]；Wherein U_sWith U '_sRepresent respectively U₁With U '₁Front k row, U_vWith U '_vRepresent U the most respectively₁With U '₁Remaining row, then U_vWith U '_vSon for described far-field signal is empty Between orthocomplement, orthogonal complement.

A kind of sane Wave arrival direction estimating method the most as claimed in claim 1, it is characterised in that i-th arrival bearing θ_iTool Body is:

${\mathrm{\θ}}_i=\underset{{\mathrm{\θ}}_i}{\arg \min }\[{\mathrm{\λ}}_1{a}_{main-1}{\left({\mathrm{\θ}}_i\right)}^H{U}_v{U_v}^H{a}_{main-1}\left({\mathrm{\θ}}_i\right)+{\mathrm{\λ}}_2{a}_{main-2}{\left({\mathrm{\θ}}_i\right)}^H{U^{\′}}_v{U}^{\′}{{}_v}^H{a}_{main-2}\left({\mathrm{\θ}}_i\right)\]$

Wherein a_main-1(θ_i) it is the steering vector of described first half, a_main-2(θ_i) it is the steering vector of described latter half, λ₁、λ₂It is respectively arithmetic number, λ₁=λ₂, argmin [] is the minimum solved.

A kind of sane Wave arrival direction estimating method the most as claimed in claim 4, it is characterised in that described in minimize parameter and search Suo Fangfa is steepest descent method, Newton method or exhaustive search algorithm.