CN104820216A - Multipath signal angle-of-arrival estimation method based on array response rotation invariance - Google Patents

Abstract

The invention discloses a multipath signal angle-of-arrival estimation method based on array response rotation invariance to solve the technical problem that the existing multipath signal angle-of-arrival estimation method is low in practicability. According to the technical scheme, the method comprises the following steps: first, calculating an observation response matrix in received signals of two different sub-matrixes respectively, namely, carrying out demodulation and low-pass filtering on the output signals of antenna oscillators of each sub-matrix to obtain baseband receiving signals; performing sliding correlation on the baseband received signals and a pseudo-random sequence to obtain observation impulse responses of the two sub-matrixes, discretizing the observation impulse responses of the two sub-matrixes and extracting a multipath to obtain observation impulse response vectors of the two sub-matrixes, and integrating the observation impulse response vectors of all the antenna oscillators of the two sub-matrixes to obtain an observation response matrix of the two sub-matrixes; and finally, estimating the angle of arrival of a multipath signal based on rotation invariance of a received signal covariance matrix of the two sub-matrixes. The method of the invention is simple, high in precision of computation results, and highly practical.

Description

Based on the multipath signal direction of arrival estimation method of array response rotational invariance

Technical field

The present invention relates to a kind of multipath signal direction of arrival estimation method, particularly a kind of multipath signal direction of arrival estimation method based on array response rotational invariance.

Background technology

ESPRIT (the Estimation of Signal Parameters Via Rotational Invariance Techniques) method mentioned in document " ESPRIT algorithm estimated performance is analyzed; " infotech "; the 3rd phase in 2011,100 pages to 102 pages " is a kind of typical signal direction of arrival estimation method.Array is divided into two identical submatrixs by the method, utilizes the rotational invariance of two submatrix Received signal strength covariance matrixes to carry out DOA estimate.But there are the following problems for the method, the method requires that multipath number is less than the oscillator number of aerial array.And multipath number in actual channel is very large, consider hardware cost, be difficult to realize more massive receiving antenna array, therefore ESPRIT method is difficult to be generalized to practical application.

Summary of the invention

In order to overcome the deficiency of existing multipath signal direction of arrival estimation method poor practicability, the invention provides a kind of multipath signal direction of arrival estimation method based on array response rotational invariance.The method is difference calculating observation response matrix in the Received signal strength of two different submatrixs first, namely after solution mediation low-pass filtering being carried out to the output signal of the antenna oscillator of each submatrix, obtain baseband receiving signals, again baseband receiving signals and pseudo-random sequence are done slide relevant, obtain the observation impulse response of two submatrixs, by carrying out discretize to the observation impulse response of two submatrixs, extract multipath, obtain the observation impulse response vector of two submatrixs, respectively the observation impulse response vector of all antenna oscillators of two submatrixs is integrated the observed responses matrix obtaining two submatrixs, the rotational invariance of two submatrix Received signal strength covariance matrixes is finally utilized to estimate the Bo Dajiao of multipath signal.The inventive method is simple, computational complexity is low, operation result precision is high, is applicable to reach angle to a large amount of multipath signal ripple and estimates, breach existing ESPRIT method for antenna shake unit restriction, practical.

The technical solution adopted for the present invention to solve the technical problems is: a kind of multipath signal direction of arrival estimation method based on array response rotational invariance, is characterized in comprising the following steps:

(A) definition signal.

Select length be the pseudo-random sequence of X as base-band detection signal a (t), its expression formula is

$a\left(t\right)=\underset{n=0}{\overset{X-1}{\mathrm{\Σ}}}{b}_n{\mathrm{rect}}_{T_b}(t-n{T}_b),b_n\∈\{+1,-1\}$

Wherein, X represents the length of pseudo-random sequence, and t represents the time, expression width is T _brectangular pulse signal.K PN sequence forms explore frame u (t), and its expression formula is

$u\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}a(t-k{T}_p)---\left(2\right)$

Wherein, T _p=XT _b.Explore frame u (t) is basic detectable signal, goes out after the modulation of this explore frame through antenna transmission.

For the communication environments containing L bar multipath, its space channel impulse response model h (t) is expressed as follows

$h\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}\mathrm{\δ}(t-{\mathrm{\τ}}_l)---\left(3\right)$

Wherein, L is the multipath number in communication environments, representing the channel complex response of l article of multipath, is a complex constant, τ _lbe the time delay value of l article of multipath, δ (t) represents impulse function.

Receiving antenna array is one dimension wire antenna array, comprises M+1 antenna oscillator, wherein, and M > 1.The equidistant arrangement of this M+1 antenna oscillator, element spacing is expressed as d, and the directional diagram of antenna oscillator is all identical.From geometrically aerial array being divided into identical two submatrix: submatrix Z _αwith submatrix Z _β.Wherein, submatrix Z _αbe made up of the 1st ~ M antenna oscillator in primary antenna array, submatrix Z _βbe made up of 2nd ~ M+1 antenna oscillator in primary antenna array.Two submatrixs all include M antenna oscillator, and therefore the antenna oscillator in two submatrixs all being renumberd is 1 ~ M.Submatrix Z _αon m antenna oscillator and submatrix Z _βon the distance of m antenna oscillator be expressed as Δ.

For m antenna oscillator in submatrix, then its arrival bearing θ on complex response s (θ) be

Wherein, e represents the nature truth of a matter, and j represents imaginary number, represent the wavelength of radiofrequency signal, θ represents arrival bearing.

For submatrix Z _αon m antenna oscillator, according to formula (5) generate radio frequency receiving signal

$y_{{\mathrm{\α}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(5\right)$

Wherein, N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation.To submatrix Z _αon all antenna oscillators, all generate corresponding Received signal strength.

For submatrix Z _βon m antenna oscillator, according to formula (6) generate radio frequency receiving signal

$y_{{\mathrm{\β}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}{u}^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(6\right)$

Wherein, N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation.For submatrix Z _βon all antenna oscillators, all generate corresponding Received signal strength.

Submatrix Z _αon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(7\right)$

Wherein, N _mt () is the noise signal received by m antenna oscillator, θ _lthe Bo Dajiao of l article of footpath incoming signal, s _m(θ _l) be that to reach angle to ripple be θ to m antenna oscillator _lthe complex response of incoming signal.

Submatrix Z _βon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(8\right)$

Wherein, e is the nature truth of a matter, and j represents imaginary number.Due to submatrix Z _βon m antenna oscillator relative to submatrix Z _αon m antenna oscillator there is distance, delta, this distance result in the wave path-difference on arrival bearing.And then be θ for incident angle _lmultipath signal, submatrix Z _βreceived signal strength relative to submatrix Z _αreceived signal strength there is phase shift ξ (θ _l).ξ (θ _l) expression formula be

Wherein, represent the wavelength of radiofrequency signal.

(B) calculating observation impulse response matrix.

For submatrix Z _αwith submatrix Z _βon m antenna oscillator, calculating observation impulse response matrix is divided into following four steps:

(B-1), after demodulation, low-pass filtering being carried out to the output signal of antenna oscillator, baseband receiving signals is obtained respectively with

(B-2) detectable signal will be received respectively with to the local pseudo-random sequence a (t) of standard do slide relevant, can obtain observe impulse response with expression formula is as follows respectively

${\tilde{h}}_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}\mathrm{\ρ}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(10\right)$

${\tilde{h}}_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}\mathrm{\ρ}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(11\right)$

Wherein, represent spreading gain, N ' _m(t) be relevant to local pseudo-random sequence after noise signal.

(B-3) impulse response will be observed respectively with discretize.L the peak point that selective value is maximum, is expressed as with will composition observation impulse response vector will composition observation impulse response vector with mathematic(al) representation be respectively

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\α}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\α}}_m}\left({\mathrm{\τ}}_L\right)\end{array}\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(12\right)$

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\β}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\β}}_m}\left({\mathrm{\τ}}_L\right)\end{array}\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_1\right)}& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_L\right)}\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(13\right)$

Wherein, ${\tilde{N}}_m^{\′}=\left[\begin{array}{ccc}{N}_m^{\′}\left({\mathrm{\τ}}_1\right)& ...& N_m^{\′}\left({\mathrm{\τ}}_L\right)\end{array}\right]$ Represent noise vector.

(B-4) for submatrix Z _αwith submatrix Z _βupper 1st Received signal strength to M receiving antenna oscillator all performs above three steps, obtains their observation impulse response vector respectively, is expressed as with these observation impulse response vectors are organized into observation impulse response matrix H according to following form _αand H _β

$H_{\mathrm{\α}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\α}}_M}\end{array}\right]---\left(14\right)$

$H_{\mathrm{\β}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\β}}_M}\end{array}\right]---\left(15\right)$

H _αthe matrix of the capable L row of M, H _βbe the matrix of the capable L row of M, their mathematic(al) representation is

See from formula (16) and formula (17), the capable observation impulse response vector representing m oscillator of m of observation impulse response matrix, the observation impulse response of l article of footpath on whole array is shown in l list, is called array impulse response vector.

(C) estimate that ripple reaches angle.

Space channel comprises L bar multipath, and once estimate p bar wherein, 1≤p≤M, concrete step is as follows.

(C-1) from observation impulse response matrix H _αmiddle selection p arranges, and is p by this p column number ₁..., p _p, and form matrix mathematic(al) representation be

Wherein, represent noise vector, represent noise matrix, A is steering vector matrix, is defined as follows

Matrix $X=\mathrm{diag}\left[\begin{array}{ccc}\mathrm{\ρ}{h}_{p_1}^{\mathrm{CH}}& ...& \mathrm{\ρ}{h}_{p_p}^{\mathrm{CH}}\end{array}\right],$ Wherein diag [] represents structure diagonal matrix.

Again from observation impulse response matrix H _βin same position select p row, be p by this p column number equally ₁..., p _p, and form matrix expression formula as follows

Wherein, $\mathrm{\Φ}=\mathrm{diag}\left[\begin{array}{ccc}{e}^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_1}\right)}& ...& e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_p}\right)}\end{array}\right].$ Matrix and matrix all M × p matrixes.If the incoming signal angle in the selected p bar footpath corresponding to p row is different, then prove matrix and matrix order be p.

Ask for matrix and matrix order, if result is equal to p, then illustrate that the incoming signal ripple in selected p bar footpath reaches angle different, then continue perform (C-2) step.If result is not equal to p, then illustrate in selected p bar footpath that the Bo Dajiao of the incoming signal having at least two footpaths corresponding is identical.Now the value of p is deducted 1, and re-execute this sub-step.Prove when p finally reduces to 1, matrix and matrix order necessarily equal 1, and during p=1, this method is still set up.Therefore this method must be set up.

(C-2) ask for according to formula (18) and formula (19) respectively with covariance matrix with

${\tilde{C}}_{\mathrm{\α}}={\hat{H}}_{\mathrm{\α}}{\hat{H}}_{\mathrm{\α}}^H---\left(21\right)$

${\tilde{C}}_{\mathrm{\β}}={\hat{H}}_{\mathrm{\β}}{\hat{H}}_{\mathrm{\β}}^H---\left(22\right)$

Wherein, with represent respectively with associate matrix.Matrix with it is all M × Metzler matrix.Prove, matrix with order are all p.

(C-3) to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (20) _α.

$U_{\mathrm{\α}}=\left[\begin{array}{ccc}{u}_{{\mathrm{\α}}_1}& ...& u_{{\mathrm{\α}}_p}\end{array}\right]---\left(23\right)$

Similarly to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (21) _β.

$U_{\mathrm{\β}}=\left[\begin{array}{ccc}{u}_{{\mathrm{\β}}_1}& ...& u_{{\mathrm{\β}}_p}\end{array}\right]---\left(24\right)$

Matrix U _αand U _βall matrixes of M × p.Prove, matrix U _αand U _βorder be p.

(C-4) matrix Ψ is asked for according to formula (22).

Ψ＝(U _α) ⁺U _β(25)

Wherein, () ⁺represent and ask group inverse matrices.Matrix Ψ is the matrix of p × p.

Eigenwert is asked for matrix Ψ, is expressed as prove namely be the diagonal element of matrix Φ.According to the signal wave can obtaining the selected p bar footpath corresponding to p row reaches angle.The method asked for is shown in formula (26).

Wherein, argsin () represents sine of negating, and argument of a complex number is asked in angle () expression.Complete the signal DOA estimate in p bar footpath.

(C-5) go to (C-1) step, then select p bar footpath to estimate, until all L bar footpaths all complete estimation.

The invention has the beneficial effects as follows: the method is difference calculating observation response matrix in the Received signal strength of two different submatrixs first, namely after solution mediation low-pass filtering being carried out to the output signal of the antenna oscillator of each submatrix, obtain baseband receiving signals, again baseband receiving signals and pseudo-random sequence are done slide relevant, obtain the observation impulse response of two submatrixs, by carrying out discretize to the observation impulse response of two submatrixs, extract multipath, obtain the observation impulse response vector of two submatrixs, respectively the observation impulse response vector of all antenna oscillators of two submatrixs is integrated the observed responses matrix obtaining two submatrixs, the rotational invariance of two submatrix Received signal strength covariance matrixes is finally utilized to estimate the Bo Dajiao of multipath signal.Due in the inventive method, an estimator process only processes the DOA estimate of a part of multipath signal, and a large amount of multipath signals can divide different estimator processes to process.Therefore estimable multipath ripple reaches angle quantity not by the restriction of receiving antenna array scale.Because the inventive method does not directly use sampled signal, but estimate according to the observation impulse response matrix that sampled signal process obtains, the data volume therefore related to is less.In addition, the inventive method does not relate to iteration or search, and therefore operand is less.Therefore the inventive method is simple, computational complexity is low, operation result precision is high, is applicable to that angle is reached to a large amount of multipath signal ripple and estimates, breach existing ESPRIT method for antenna shake unit restriction, practical.

The present invention is described in detail below in conjunction with embodiment.

Embodiment

The multipath signal direction of arrival estimation method concrete steps that the present invention is based on array response rotational invariance are as follows:

(A) definition signal; (B) calculating observation impulse response matrix; (C) estimate that ripple reaches angle.

(A) definition signal.

Select length be the pseudo-random sequence of X as base-band detection signal a (t), its expression formula is

$a\left(t\right)=\underset{n=0}{\overset{X-1}{\mathrm{\Σ}}}{b}_n{\mathrm{rect}}_{T_b}(t-n{T}_b),b_n\∈\{+1,-1\}---\left(1\right)$

Wherein, X represents the length of pseudo-random sequence, and t represents the time, expression width is T _brectangular pulse signal.K PN sequence forms explore frame u (t), and its expression formula is

$u\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}a(t-k{T}_p)---\left(2\right)$

Wherein T _p=XT _b.Explore frame u (t) is the basic detectable signal in this method, goes out after the modulation of this explore frame through antenna transmission.

For the communication environments containing L bar multipath, its space channel impulse response model h (t) is expressed as follows

$h\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}\mathrm{\δ}(t-{\mathrm{\τ}}_l)---\left(3\right)$

Wherein, L is the multipath number in communication environments, representing the channel complex response of l article of multipath, is a complex constant, τ _lbe the time delay value of l article of multipath, δ (t) represents impulse function.

Receiving antenna array is one dimension wire antenna array, comprises M+1 antenna oscillator (M > 1).The equidistant arrangement of this M+1 antenna oscillator, element spacing is expressed as d, and the directional diagram of antenna oscillator is all identical.From geometrically aerial array being divided into identical two submatrix: submatrix Z _αwith submatrix Z _β.Wherein, submatrix Z _αbe made up of the 1st ~ M antenna oscillator in primary antenna array, submatrix Z _βbe made up of 2nd ~ M+1 antenna oscillator in primary antenna array.Two submatrixs all include M antenna oscillator, and therefore the antenna oscillator in two submatrixs all being renumberd is 1 ~ M.Submatrix Z _αon m antenna oscillator and submatrix Z _βon the distance of m antenna oscillator be expressed as Δ.

For m antenna oscillator in submatrix, then its arrival bearing θ on complex response s (θ) be

Wherein e represents the nature truth of a matter, and j represents imaginary number, represent the wavelength of radiofrequency signal, θ represents arrival bearing.

For submatrix Z _αon m antenna oscillator, according to formula (5) generate radio frequency receiving signal

$y_{{\mathrm{\α}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(5\right)$

Wherein N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation.To submatrix Z _αon all antenna oscillators, all generate corresponding Received signal strength.

For submatrix Z _βon m antenna oscillator, according to formula (6) generate radio frequency receiving signal

$y_{{\mathrm{\β}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}{u}^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(6\right)$

Wherein N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation.For submatrix Z _βon all antenna oscillators, all generate corresponding Received signal strength.

Submatrix Z _αon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(7\right)$

Wherein, N _mt () is the noise signal received by m antenna oscillator, θ _lthe Bo Dajiao of l article of footpath incoming signal, s _m(θ _l) be that to reach angle to ripple be θ to m antenna oscillator _lthe complex response of incoming signal.

Submatrix Z _βon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(8\right)$

Wherein, e is the nature truth of a matter, and j represents imaginary number.Due to submatrix Z _βon m antenna oscillator relative to submatrix Z _αon m antenna oscillator there is distance, delta, this distance result in the wave path-difference on arrival bearing.And then be θ for incident angle _lmultipath signal, submatrix Z _βreceived signal strength relative to submatrix Z _αreceived signal strength there is phase shift ξ (θ _l).ξ (θ _l) expression formula be

Wherein, represent the wavelength of radiofrequency signal.

(B) calculating observation impulse response matrix.

For submatrix Z _αwith submatrix Z _βon m antenna oscillator, calculating observation impulse response matrix is divided into following four steps:

(B-1), after demodulation, low-pass filtering being carried out to the output signal of antenna oscillator, baseband receiving signals is obtained respectively with

(B-2) detectable signal will be received respectively with to the local pseudo-random sequence a (t) of standard do slide relevant, can obtain observe impulse response with expression formula is as follows respectively

${\tilde{h}}_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}\mathrm{\ρ}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(10\right)$

${\tilde{h}}_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}\mathrm{\ρ}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(11\right)$

Wherein, represent spreading gain, N ' _m(t) be relevant to local pseudo-random sequence after noise signal.

(B-3) impulse response will be observed respectively with discretize.L the peak point that selective value is maximum, is expressed as with will composition observation impulse response vector will composition observation impulse response vector with mathematic(al) representation be respectively

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\α}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\α}}_m}\left({\mathrm{\τ}}_L\right)\end{array}\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(12\right)$

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\β}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\β}}_m}\left({\mathrm{\τ}}_L\right)\end{array}\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_1\right)}& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_L\right)}\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(13\right)$

Wherein, ${\tilde{N}}_m^{\′}=\left[\begin{array}{ccc}{N}_m^{\′}\left({\mathrm{\τ}}_1\right)& ...& N_m^{\′}\left({\mathrm{\τ}}_L\right)\end{array}\right]$ Represent noise vector.

(B-4) for submatrix Z _αwith submatrix Z _βupper 1st Received signal strength to M receiving antenna oscillator all performs above three steps, obtains their observation impulse response vector respectively, is expressed as with these observation impulse response vectors are organized into observation impulse response matrix H according to following form _αand H _β

$H_{\mathrm{\α}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\α}}_M}\end{array}\right]---\left(14\right)$

$H_{\mathrm{\β}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\β}}_M}\end{array}\right]---\left(15\right)$

H _αthe matrix of the capable L row of M, H _βbe the matrix of the capable L row of M, their mathematic(al) representation is

See from formula (16) and formula (17), the capable observation impulse response vector representing m oscillator of m of observation impulse response matrix, the observation impulse response of l article of footpath on whole array is shown in l list, is called array impulse response vector.

(C) estimate that ripple reaches angle.

Space channel comprises L bar multipath, and this method once estimates p bar (1≤p≤M) wherein, concrete step following (remaining multipath in second time, for the third time ... .. estimate in, as described in C-5).

(C-1) from observation impulse response matrix H _αmiddle selection p arranges (1≤p≤M), is p by this p column number ₁..., p _p, and form matrix mathematic(al) representation be

Wherein, represent noise vector, represent noise matrix, A is steering vector matrix, is defined as follows

Matrix $X=\mathrm{diag}\left[\begin{array}{ccc}\mathrm{\ρ}{h}_{p_1}^{\mathrm{CH}}& ...& \mathrm{\ρ}{h}_{p_p}^{\mathrm{CH}}\end{array}\right],$ Wherein diag [] represents structure diagonal matrix.

Again from observation impulse response matrix H _βin same position select p row, be p by this p column number equally ₁..., p _p, and form matrix expression formula as follows

Wherein $\mathrm{\Φ}=\mathrm{diag}\left[\begin{array}{ccc}{e}^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_1}\right)}& ...& e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_p}\right)}\end{array}\right].$ Matrix and matrix all M × p matrixes.If the incoming signal angle in the selected p bar footpath corresponding to p row is different, then prove matrix and matrix order be p.

Ask for matrix and matrix order, if result is equal to p, then illustrate that the incoming signal ripple in selected p bar footpath reaches angle different, then continue perform (C-2) step.If result is not equal to p, then illustrate in selected p bar footpath that the Bo Dajiao of the incoming signal having at least two footpaths corresponding is identical.Now the value of p is deducted 1, and re-execute this sub-step.Prove when p finally reduces to 1, matrix and matrix order necessarily equal 1, and during p=1, this method is still set up.Therefore this method must be set up.

(C-2) ask for according to formula (18) and formula (19) respectively with covariance matrix with

${\tilde{C}}_{\mathrm{\α}}={\hat{H}}_{\mathrm{\α}}{\hat{H}}_{\mathrm{\α}}^H---\left(21\right)$

${\tilde{C}}_{\mathrm{\β}}={\hat{H}}_{\mathrm{\β}}{\hat{H}}_{\mathrm{\β}}^H---\left(22\right)$

Wherein, with represent respectively with associate matrix.Matrix with it is all M × Metzler matrix.Prove, matrix with order are all p.

(C-3) to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (20) _α.

$U_{\mathrm{\α}}=\left[\begin{array}{ccc}{u}_{{\mathrm{\α}}_1}& ...& u_{{\mathrm{\α}}_p}\end{array}\right]---\left(23\right)$

Similarly to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (21) _β.

$U_{\mathrm{\β}}=\left[\begin{array}{ccc}{u}_{{\mathrm{\β}}_1}& ...& u_{{\mathrm{\β}}_p}\end{array}\right]---\left(24\right)$

Matrix U _αand U _βall matrixes of M × p.Prove, matrix U _αand U _βorder be p.

(C-4) matrix Ψ is asked for according to formula (22).

Ψ＝(U _α) ⁺U _β(25)

Wherein, () ⁺represent and ask group inverse matrices.Matrix Ψ is the matrix of p × p.

Eigenwert is asked for matrix Ψ, is expressed as prove namely be the diagonal element of matrix Φ.According to the signal wave can obtaining the selected p bar footpath corresponding to p row reaches angle.The method asked for is shown in formula (26).

Wherein, argsin () represents sine of negating, and argument of a complex number is asked in angle () expression.This completes the signal DOA estimate in p bar footpath.

(C-5) go to (C-1) step, then select p bar footpath to estimate, until all L bar footpaths all complete estimation.(C-1) to (C-4) step is called an estimator process above.It should be noted that, the p value of different estimator process is not necessarily identical.

Below introduce a kind of emulation embodiment specifically.

(a) definition signal.

According to following steps definition signal.

(a-1) explore frame signal is generated.Use length be the m sequence of 1023 as pseudo-random sequence, the bit rate of base-band detection signal a (t) is 100 mbit, the T also namely in formula (1) _b=10ns, wherein ns represents nanosecond.Explore frame u (t) is connected to form by two pseudo-random sequences, is also K=2 in formula (2).Explore frame is modulated by BPSK, and carrier frequency is 2.5GHz.Explore frame after modulation is expressed as u ' (t).

(a-2) multi-path information is generated.20 multipaths are comprised in assumptions' environment.The time delay of these 20 multipaths, complex response and incoming signal angle are unknown to transmitting terminal and receiving end.For emulating, the time delay in these 20 footpaths and complex response are arranged according to table 1.

The time delay of the different multipath of table 1 and complex response

Sequence number	Complex response	Time delay (ns)		Sequence number	Complex response	Time delay (ns)
							1	ξ	100		11	0.80ξ	200
2	0.98ξ	110		12	0.78ξ	210
							3	0.96ξ	120		13	0.76ξ	220
4	0.94ξ	130		14	0.74ξ	230
							5	0.92ξ	140		15	0.72ξ	240
6	0.90ξ	150		16	0.70ξ	250
							7	0.88ξ	160		17	0.68ξ	260
8	0.86ξ	170		18	0.66ξ	270

9	0.84ξ	180		19	0.64ξ	280
							10	0.82ξ	190		20	0.62ξ	290

Wherein, ξ is a complex constant, can freely arrange.The unit of time delay is nanosecond.The incident angle of these 20 multipaths can stochastic generation, and angular resolution is 1 degree, and span is-90 ~ 90 degree.

(a-3) receiving antenna array is generated.Receiving antenna is wire antenna array, and comprise 6 pieces of antenna oscillators, the distance of adjacent antenna elements is expressed as d, and d equals the half of radiofrequency signal wavelength, is also 6 centimetres.Then submatrix Z _αbe made up of 1 ~ No. 5 antenna oscillator, submatrix Z _βbe made up of 2 ~ No. 6 antenna oscillators.By submatrix Z _αwith submatrix Z _βon antenna oscillator all to renumber be 1 ~ No. 5.Then in two submatrixs, the distance of identical numbering oscillator is Δ, easily knows Δ=d=6 centimetre in the present embodiment.

(a-4) submatrix Z is generated _αwith submatrix Z _βsteering vector on different direction of arrival.Because the oscillator of two submatrixs is identical with array structure, therefore steering vector is also identical.For m antenna oscillator in submatrix, then its arrival bearing θ on complex response s (θ) be

Wherein e represents the nature truth of a matter, and j represents imaginary number, representing the wavelength of radiofrequency signal, is 12 centimetres (0.12 meters) in this example.To the incoming signal in all 20 footpaths, all to calculate in submatrix 1 ~ No. 5 antenna oscillator at its incoming wave θ according to formula (4) ₁..., θ ₂₀on complex response s ₁(θ ₁) ..., s ₁(θ ₂₀), s ₂(θ ₁) ..., s ₅(θ ₂₀).

(a-5) submatrix relative phase shift is generated.So-called submatrix relative phase shift, refers to that for incident angle be θ _ll article of footpath incoming signal, submatrix Z _βon oscillator relative to submatrix Z _αphase shift ξ (the θ that upper same numbering oscillator outputs signal _l).The submatrix relative phase shift ξ (θ of the incoming signal in all footpaths is calculated according to formula (9) ₁) ..., ξ (θ ₂₀).

(a-6) Received signal strength is generated.For submatrix Z _αon m antenna oscillator, according to formula (5) generate radio frequency receiving signal

$y_{{\mathrm{\α}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(5\right)$

Wherein N ' _mt () is the white noise of Gaussian distributed, signal to noise ratio (S/N ratio) is set to 0dB. and τ _lbe complex response and the time delay in the footpath shown in table 1.To submatrix Z _αon all 5 pieces of antenna oscillators, all generate corresponding Received signal strength.

For submatrix Z _βon m antenna oscillator, according to formula (6) generate radio frequency receiving signal

$y_{{\mathrm{\β}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}{u}^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(6\right)$

For submatrix Z _βon all 5 pieces of antenna oscillators, all generate corresponding Received signal strength.

(b) calculating observation impulse response matrix.

Calculating observation impulse response matrix in accordance with the following steps.

(b-1) arranging antenna oscillator sequence number m is 1.

(b-2) difference radio frequency Received signal strength with carry out BPSK demodulation, low-pass filtering (filter bandwidht 100MHz), obtain base-band detection frame with

(b-3) by base-band detection frame with base-band detection signal a (t) do slide relevant, obtain observation impulse response try to achieve maximal value, be expressed as threshold values Thr is set _α, its value is

Again by base-band detection frame with base-band detection signal a (t) do slide relevant, obtain observation impulse response try to achieve maximal value, be expressed as threshold values Thr is set _β, its value is

(b-4) from t=0, find out satisfied 20 peak points, the value of these 20 peak points is the observation impulse response of discretize, and the value of these 20 points is formed a row vector according to the form of formula (12), is observation impulse response vector

Again from t=0, find out satisfied 20 peak points, the value of these 20 peak points is the observation impulse response of discretize, and the value of these 20 points is formed a row vector according to the form of formula (13), is observation impulse response vector

(b-5) antenna oscillator sequence number m adds 1, rotates back into (b-2) step and performs, until the observation impulse response vector of all 5 antenna oscillators has all been asked for

(b-6) after completing above sub-step, according to formula (14) Suo Shi submatrix Z _αthe observation impulse response vectorial combination of upper oscillator becomes an observation impulse response matrix, is expressed as H _α.Again according to formula (15) Suo Shi submatrix Z _βthe observation impulse response vectorial combination of upper oscillator becomes an observation impulse response matrix, is expressed as H _β.H in this example _αand H _βall matrixes of 5 × 20.

C () estimates that ripple reaches angle.

Carry out the estimation that multipath signal ripple reaches angle in accordance with the following steps.

(c-1) represent the sequence number of estimator process with η, arranging η is 1.The multipath signal ripple that setting is once estimated reaches angle quantity p and is set to 3.

(c-2) observation impulse response matrix H is taken out _α(η-1) × p+1 to η × p arrange.If η × p is greater than 20, then get observation impulse response matrix H _α(η-1) × p+1 to the 20th row, and the value upgrading p is 20-(η-1) × p.

Be 1 ~ p by this p column number.Matrix is formed according to formula (18) take out observation impulse response matrix H again _β((η-1) × p+1) to (η × p) row, by this p row be numbered 1 ~ p equally.Matrix is formed according to formula (20) ask for matrix order, be expressed as R _α, then ask for matrix order, be expressed as R _β.If R _αor R _βbe not equal to p, then the value of p subtracted 1, re-execute this sub-step.

(c-3) respectively according to formula (21) and formula (22) compute matrix with covariance matrix with in this example, matrix with all matrixes of 5 × 5.

(c-4) to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask eigenwert again characteristic of correspondence vector, is expressed as matrix U is built according to formula (23) _α.

Similarly to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask eigenwert again characteristic of correspondence vector, is expressed as matrix U is built according to formula (24) _β

(c-5) according to formula (25) compute matrix Ψ, ask for the eigenwert of matrix Ψ, be expressed as the signal wave obtaining this p bar footpath according to formula (26) respectively reaches angle, is expressed as and preserve.

(c-6) value that estimator crosses program number η adds 1, and the value arranging p is 3, and transfers back to (c-2) step and perform, until the signal wave in all footpaths reaches angle all complete estimation.

By the set of the signal DOA estimate value of all estimator the output of process composition, the incoming signal ripple being required 20 footpaths reaches the estimated value set at angle.

Claims (1)

1., based on a multipath signal direction of arrival estimation method for array response rotational invariance, it is characterized in that comprising the following steps:

(A) definition signal;

Select length be the pseudo-random sequence of X as base-band detection signal a (t), its expression formula is

$a\left(t\right)=\underset{n=0}{\overset{X-1}{\mathrm{\Σ}}}{b}_n{\mathrm{rect}}_{T_b}(t-n{T}_b),b_n\∈\{+1,-1\}---\left(1\right)$

Wherein, X represents the length of pseudo-random sequence, and t represents the time, expression width is T _brectangular pulse signal; K PN sequence forms explore frame u (t), and its expression formula is

$u\left(t\right)=\underset{k=0}{\overset{K-1}{\mathrm{\Σ}}}a(t-k{T}_p)---\left(2\right)$

Wherein, T _p=XT _b; Explore frame u (t) is basic detectable signal, goes out after the modulation of this explore frame through antenna transmission;

For the communication environments containing L bar multipath, its space channel impulse response model h (t) is expressed as follows

$h\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}\mathrm{\δ}(t-{\mathrm{\τ}}_l)---\left(3\right)$

Wherein, L is the multipath number in communication environments, representing the channel complex response of l article of multipath, is a complex constant, τ _lbe the time delay value of l article of multipath, δ (t) represents impulse function;

Receiving antenna array is one dimension wire antenna array, comprises M+1 antenna oscillator, wherein, and M>1; The equidistant arrangement of this M+1 antenna oscillator, element spacing is expressed as d, and the directional diagram of antenna oscillator is all identical; From geometrically aerial array being divided into identical two submatrix: submatrix Z _αwith submatrix Z _β; Wherein, submatrix Z _αbe made up of the 1st ~ M antenna oscillator in primary antenna array, submatrix Z _βbe made up of 2nd ~ M+1 antenna oscillator in primary antenna array; Two submatrixs all include M antenna oscillator, and therefore the antenna oscillator in two submatrixs all being renumberd is 1 ~ M; Submatrix Z _αon m antenna oscillator and submatrix Z _βon the distance of m antenna oscillator be expressed as △;

For m antenna oscillator in submatrix, then its arrival bearing θ on complex response s (θ) be

Wherein, e represents the nature truth of a matter, and j represents imaginary number, represent the wavelength of radiofrequency signal, θ represents arrival bearing;

For submatrix Z _αon m antenna oscillator, according to formula (5) generate radio frequency receiving signal

$y_{{\mathrm{\α}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(5\right)$

Wherein, N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation; To submatrix Z _αon all antenna oscillators, all generate corresponding Received signal strength;

For submatrix Z _βon m antenna oscillator, according to formula (6) generate radio frequency receiving signal

$y_{{\mathrm{\β}}_m}^{\′}\left(t\right)=\underset{l=1}{\overset{20}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}{u}^{\′}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(6\right)$

Wherein, N ' _mt () is the white noise of Gaussian distributed, θ _lrepresent the incident angle in l article of footpath, and τ _lfor complex response and the time delay of multipath, u ' (t) is the explore frame after modulation; For submatrix Z _βon all antenna oscillators, all generate corresponding Received signal strength;

Submatrix Z _αon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(7\right)$

Wherein, N _mt () is the noise signal received by m antenna oscillator, θ _lthe Bo Dajiao of l article of footpath incoming signal, s _m(θ _l) be that to reach angle to ripple be θ to m antenna oscillator _lthe complex response of incoming signal;

Submatrix Z _βon the Received signal strength that exports of m antenna oscillator be expressed as after demodulation

$y_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{h}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}u(t-{\mathrm{\τ}}_l)+N_m\left(t\right)---\left(8\right)$

Wherein, e is the nature truth of a matter, and j represents imaginary number; Due to submatrix Z _βon m antenna oscillator relative to submatrix Z _αon m antenna oscillator there is distance △, this distance result in the wave path-difference on arrival bearing; And then be θ for incident angle _lmultipath signal, submatrix Z _βreceived signal strength relative to submatrix Z _αreceived signal strength there is phase shift ξ (θ _l); ξ (θ _l) expression formula be

Wherein, represent the wavelength of radiofrequency signal;

(B) calculating observation impulse response matrix;

For submatrix Z _αwith submatrix Z _βon m antenna oscillator, calculating observation impulse response matrix is divided into following four steps:

(B-1), after demodulation, low-pass filtering being carried out to the output signal of antenna oscillator, baseband receiving signals is obtained respectively with

(B-2) detectable signal will be received respectively with to the local pseudo-random sequence a (t) of standard do slide relevant, can obtain observe impulse response with expression formula is as follows respectively

${\tilde{h}}_{{\mathrm{\α}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{\mathrm{\ρh}}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(10\right)$

${\tilde{h}}_{{\mathrm{\β}}_m}\left(t\right)=\underset{l=1}{\overset{L}{\mathrm{\Σ}}}{\mathrm{\ρh}}_l^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_l\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_l\right)}\mathrm{\δ}(t-{\mathrm{\τ}}_l)+N_m^{\′}\left(t\right)---\left(11\right)$

Wherein, represent spreading gain, N ' _m(t) be relevant to local pseudo-random sequence after noise signal;

(B-3) impulse response will be observed respectively with discretize; L the peak point that selective value is maximum, is expressed as with will composition observation impulse response vector will composition observation impulse response vector with mathematic(al) representation be respectively

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\α}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\α}}_m}\end{array},\left({\mathrm{\τ}}_L\right)\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(12\right)$

$\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_m}=\left[\begin{array}{ccc}{\tilde{h}}_{{\mathrm{\β}}_m}\left({\mathrm{\τ}}_1\right)& ...& {\tilde{h}}_{{\mathrm{\β}}_m}\end{array},\left({\mathrm{\τ}}_L\right)\right]\\ =\mathrm{\ρ}\left[\begin{array}{ccc}{h}_1^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_1\right)e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_1\right)}& ...& h_L^{\mathrm{CH}}{s}_m\left({\mathrm{\θ}}_L\right)e^{\mathrm{j\ξ}{\mathrm{\θ}}_L)}\end{array}\right]+{\tilde{N}}_m^{\′}\end{array}---\left(13\right)$

Wherein, ${\tilde{N}}_m^{\′}=\left[\begin{array}{ccc}{N}_m^{\′}\left({\mathrm{\τ}}_1\right)& ...& N_m^{\′}\left({\mathrm{\τ}}_L\right)\end{array}\right]$ Represent noise vector;

(B-4) for submatrix Z _αwith submatrix Z _βupper 1st Received signal strength to M receiving antenna oscillator all performs above three steps, obtains their observation impulse response vector respectively, is expressed as with these observation impulse response vectors are organized into observation impulse response matrix H according to following form _αand H _β

$H_{\mathrm{\α}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\α}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\α}}_M}\end{array}\right]---\left(14\right)$

$H_{\mathrm{\β}}=\left[\begin{array}{c}{\tilde{a}}_{{\mathrm{\β}}_1}\\ .\\ .\\ .\\ {\tilde{a}}_{{\mathrm{\β}}_M}\end{array}\right]---\left(15\right)$

H _αthe matrix of the capable L row of M, H _βbe the matrix of the capable L row of M, their mathematic(al) representation is

See from formula (16) and formula (17), the capable observation impulse response vector representing m oscillator of m of observation impulse response matrix, the observation impulse response of l article of footpath on whole array is shown in l list, is called array impulse response vector;

(C) estimate that ripple reaches angle;

Space channel comprises L bar multipath, and once estimate p bar wherein, 1≤p≤M, concrete step is as follows;

(C-1) from observation impulse response matrix H _αmiddle selection p arranges, and is p by this p column number ₁..., p _p, and form matrix mathematic(al) representation be

Wherein, represent noise vector, represent noise matrix, A is steering vector matrix, is defined as follows

matrix $X=\mathrm{diag}\left[\begin{array}{ccc}\mathrm{\ρ}{h}_{p_1}^{\mathrm{CH}}& ...& \mathrm{\ρ}{h}_{p_p}^{\mathrm{CH}}\end{array}\right],$ Wherein diag [] represents structure diagonal matrix;

Again from observation impulse response matrix H _βin same position select p row, be p by this p column number equally ₁..., p _p, and form matrix expression formula as follows

Wherein, $\mathrm{\Φ}=\mathrm{diag}\left[\begin{array}{ccc}{e}^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_1}\right)}& ...& e^{\mathrm{j\ξ}\left({\mathrm{\θ}}_{p_p}\right)}\end{array}\right];$ Matrix and matrix all M × p matrixes; If the incoming signal angle in the selected p bar footpath corresponding to p row is different, then prove matrix and matrix order be p;

Ask for matrix and matrix order, if result is equal to p, then illustrate that the incoming signal ripple in selected p bar footpath reaches angle different, then continue perform (C-2) step; If result is not equal to p, then illustrate in selected p bar footpath that the Bo Dajiao of the incoming signal having at least two footpaths corresponding is identical; Now the value of p is deducted 1, and re-execute this sub-step; Prove when p finally reduces to 1, matrix and matrix order necessarily equal 1, and during p=1, this method is still set up; Therefore this method must be set up;

(C-2) ask for according to formula (18) and formula (19) respectively with covariance matrix with

${\tilde{C}}_{\mathrm{\α}}={\hat{H}}_{\mathrm{\α}}{\hat{H}}_{\mathrm{\α}}^H---\left(21\right)$

${\tilde{C}}_{\mathrm{\β}}={\hat{H}}_{\mathrm{\β}}{\hat{H}}_{\mathrm{\β}}^H---\left(22\right)$

Wherein, with represent respectively with associate matrix; Matrix with it is all M × Metzler matrix; Prove, matrix with order are all p;

(C-3) to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (20) _α;

$U_{\mathrm{\α}}=\left[\begin{array}{ccc}{U}_{{\mathrm{\α}}_1}& ...& u_{{\mathrm{\α}}_p}\end{array}\right]---\left(23\right)$

Similarly to matrix carry out feature decomposition, and by eigenwert according to order arrangement from big to small, be expressed as ask for these eigenwert characteristic of correspondence vectors, be expressed as matrix U is built according to formula (21) _β;

$U_{\mathrm{\β}}=\left[\begin{array}{ccc}{U}_{{\mathrm{\β}}_1}& ...& u_{{\mathrm{\β}}_p}\end{array}\right]---\left(24\right)$

Matrix U _αand U _βall matrixes of M × p; Prove, matrix U _αand U _βorder be p;

(C-4) matrix Ψ is asked for according to formula (22);

Ψ＝(U _α) ⁺U _β(25)

Wherein, () ⁺represent and ask group inverse matrices; Matrix Ψ is the matrix of p × p;

Eigenwert is asked for matrix Ψ, is expressed as prove namely be the diagonal element of matrix Φ; According to the signal wave can obtaining the selected p bar footpath corresponding to p row reaches angle; The method asked for is shown in formula (26);

Wherein, argsin () represents sine of negating, and argument of a complex number is asked in angle () expression; Complete the signal DOA estimate in p bar footpath;

(C-5) go to (C-1) step, then select p bar footpath to estimate, until all L bar footpaths all complete estimation.