CN102736063A - Near-field sound source positioning method - Google Patents

Abstract

The invention relates to the technical field of array signal processing and discloses a near-field sound source positioning method for estimating a direction of arrival (DOA) of a near-field sound source signal and signal source distance by adopting a speed sensor array. The method has the advantages that a high-order accumulated value is not required to be calculated, so that relatively low operation amount is achieved; automatic paring in parameter estimation can be realized; a distance between two array elements does not need to be limited in the 1/4 wavelength range; the aperture of the array can be expanded by increasing the distance between two array elements; and therefore, the parameter estimation precision of an algorithm can be improved.

Description

Near-field sound source positioning method

Technical Field

The invention relates to the technical field of array signal processing, in particular to a near-field sound source positioning method for estimating DOA (direction of arrival) and signal source distance of a near-field sound source signal by using a speed sensor array.

Background

The underwater acoustic vector sensor is composed of 2-3 speed sensors and an optional pressure sensor. The speed sensor is orthogonally oriented and can respectively measure the vibration velocity components of sound waves in a Cartesian coordinate system. In the past decades, acoustic vector sensors have been widely regarded in both theoretical and engineering applications, and vector arrays formed by acoustic vector sensors have become important tools for underwater sound source localization, and many effective algorithms have been proposed. However, these algorithms are mainly directed to far-field signals, and when the signal source is located near the array in its near field, the near-field source needs to describe the wavefront with spherical waves (rather than parallel waves) and describe the phase difference between the array elements with Fresnel (Fresnel) approximation, which makes many far-field source localization algorithms no longer applicable.

The method aims at the near-field source positioning problem, namely the joint estimation problem of the near-field source distance and the incidence angle. The incident angle is also referred to as the angle of arrival (DOA). Scholars at home and abroad also provide a plurality of algorithms, such as a maximum likelihood algorithm, a multi-dimensional MUSIC algorithm, a high-order ESPRIT algorithm and the like, and the algorithms require multi-dimensional parameter search or high-order statistic calculation and have higher operation amount. For this purpose, some near-field source parameter estimation algorithms based on second-order statistics are proposed. However, these near-field source localization algorithms require that the array element spacing must be less than a quarter wavelength, which would otherwise lead to ambiguity in the estimates. It should be noted that for large aperture arrays, the signal source is often in the near field, so it is more practical to study the near field signal source localization of the aperture extended array.

Disclosure of Invention

Aiming at the defects that the existing near-field sound source positioning method requires multi-dimensional parameter search or high-order statistic calculation and requires array element interval delta to be smaller than 1/4 wavelengths, the invention provides a method which does not need multi-dimensional parameter search or high-order statistic calculation, only relates to second-order statistic, realizes automatic pairing of parameter estimation, does not need array element interval delta to be limited in the range of 1/4 wavelengths, and can improve parameter estimation precision by increasing array element interval delta to amplify array apertures.

In order to achieve the purpose, the invention can adopt the following technical scheme:

the near-field sound source positioning method is used for estimating DOA and signal source distance of a near-field sound source signal and comprises the following steps:

step A: a uniform linear array is set up and,

the uniform linear array is composed of array elements which are linearly arranged, have an interval of delta and are 2M in number, each array element comprises a pair of speed sensors which respectively point to a y axis and a z axis, the y axis is an axis where the linear array is located, the z axis is perpendicular to the y axis, the near-field sound source is located on a y-z axis plane, and the speed sensors can be used for receiving a near-field sound source signal and outputting azimuth information of the near-field sound source signal;

and B: establishing a signal model, which comprises the following specific steps:

measuring azimuth information through a speed sensor:

c_m，k＝[sinθ_m，k，cosθ_m,k]^T （1），

wherein-pi/2<θ_m，kPi/2 or less represents DOA of the kth near-field source signal relative to the mth array element, the 0 th array element is set as a reference array element, and c is set_k＝c_0，kAnd theta_k=θ_0，k；

② demodulating to intermediate frequency and sampling, the kth near-field sound source signal is

Wherein s is_k(t) represents the complex amplitude of the signal, ω_k＝2πf_k/f_samp，f_sampTo sample frequency, f_k≠f_l(k ≠ l) is the carrier frequency of the signal;

third, the received signal at the m-th array element can be expressed as a 2-dimensional vector:

<math> <mrow> <msub> <mi>x</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> <mi>t</mi> </mrow> </msup> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>+</mo> <msub> <mi>n</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

wherein n is_m(t) denotes a Gaussian noise vector, τ_m，k≈γ_km+φ_km²Between element m and reference element 0 for the ith sourcePropagation delay of gamma_k＝-2πΔsinθ_k/λ_k，φ_k=πΔ²cos²θ_k/λ_kl_k，λ_kAnd l_kThe wavelength of the kth signal and the distance from the kth signal to the reference array element respectively;

converting the form of the matrix of the formula (2) into:

z(t)=As(t)+w(t) （3），

wherein,

<math> <mrow> <mi>z</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msubsup> <mi>x</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </msubsup> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mn>0</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mi>M</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo></mo> </mrow> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>&Element;</mo> <msup> <mi>C</mi> <mrow> <mn>4</mn> <mi>M</mi> <mo>&times;</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>

<math> <mrow> <mi>s</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mn>1</mn> </msub> <mi>t</mi> </mrow> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>K</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>K</mi> </msub> <mi>t</mi> </mrow> </msup> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> </mrow> </math>

$w\left(t\right)={[n_{-(M-1)}^T\left(t\right),...,n_0^T\left(t\right),n_1^T\left(t\right),...,n_M^T\left(t\right)]}^T,$ E{w(t₁)w^H(t₂)}=δ(t₁-t₂)I_4M×4M，

A＝[a₁，...，a_K]∈C^4M×K，

$a_k={[b_{-(M-1),k}^T,...,{\mathrm{<figure-callout\; id="0"\; label="b"\; filenames="120706090648.png"\; state="\{\{state\}\}">b</figure-callout>}}_{0,k}^T,{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="120706090648.png"\; state="\{\{state\}\}">b</figure-callout>}}_{1,k}^T,...,b_{M,k}^T]}^T,$ <math> <mrow> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

define l_m,kRepresents the distance between the kth near-field source signal and the mth array element and has_k＝l_0，kObtaining:

<math> <mrow> <msub> <mi>l</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>sin</mi> <mrow> <mo>(</mo> <mi>&pi;</mi> <mo>/</mo> <mn>2</mn> <mo>-</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>,</mo> </mrow> </math> m＝-(M-1)，...，-1，0，1，...，M（5）；

and C: calculating a non-fuzzy estimation of DOA and signal source distance, comprising the following steps:

firstly, acquiring N snapshot vectors z (t), t =1, and N of an array, and estimating a covariance matrix <math> <mrow> <mover> <mi>R</mi> <mo>^</mo> </mover> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>t</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>z</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>N</mi> <mo>,</mo> </mrow> </math> Wherein, <math> <mrow> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msup> <mi>z</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mi>H</mi> </msup> <mo>,</mo> </mrow> </math> obtained by eigenvalue decompositionWherein psi_k(K =1, …, K) is

K large feature values of (1), their corresponding feature vectors are U_sA column vector of (a);

secondly, according to the subspace algorithm, a nonsingular matrix T exists and satisfies B = U_sT, whereby A = U_s，1T and a Φ = U_s，2T, namely:

wherein U is_s，1=U_s(1∶4M,K)，U_s，1=U_s(1+4M∶8M,K)，

Represents the left pseudo-inverse;

(iii) frequency estimation of the signal obtained by equation (6) above

Sum wavelength estimation <math> <mrow> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mn>2</mn> <mi>&pi;c</mi> <mo>/</mo> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> <msub> <mi>f</mi> <mi>s</mi> </msub> <mo>,</mo> </mrow> </math> And

and

(ii) an estimate of (d);

obtaining the estimation of the steering matrix according to the step three:

<math> <mrow> <mover> <mi>A</mi> <mo>^</mo> </mover> <mo>=</mo> <mo>[</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mi>K</mi> </msub> <mo>]</mo> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <msup> <mover> <mi>&Phi;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

fifthly, obtaining DOA non-fuzzy estimation of near-field sound source signals according to the relation of the combination formula (4) and the combination formula (1)

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>tan</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein,

（m＝-(M-1)，...，-1，0，1，...，M；k＝1，...，K）；

combination ofAnd equation (5) obtaining a blur-free estimate of the signal source distance of the near-field acoustic source signal

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>+</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </math> （k=1，...，K）（9）；

Step D: calculating fuzzy estimation of DOA and signal source distance, which comprises the following steps:

firstly, firstlyCalculating q_m，kIs estimated by

<math> <mrow> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>/</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>m</mi> <mo>&NotEqual;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein

The result of the formula (8) is used to obtain the estimation of the formula (1), and the electron angle alpha is calculated_kAnd beta_k；

② calculating by using symmetric relation of array structure

And

due to estimation of

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mo>[</mo> <mn>2</mn> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>]</mo> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mrow> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> </mrow> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </msup> </math>

And is <math> <mrow> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <mn>2</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> <math> <mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <mi>c</mi> <msup> <mi>os</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> Thus can obtain

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>4</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <mi>co</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>M</mi> <mo>-</mo> <mn>4</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

Thirdly, obtaining fuzzy estimation of DOA according to the calculation result

And fuzzy estimation of signal source distance

Wherein

And

represent unambiguous estimates of DOA and distance obtained using equations (8) and (9), respectively;

and (3) obtaining a deblurring estimation of the DOA and the signal source distance by taking the unambiguous estimation of the DOA and the signal source distance as a reference estimation:

making <math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> ${\hat{l}}_k={\hat{l}}_k\left(i_0\right);$

② obtaining the deblurring estimate of DOA

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mi>&pi;&Delta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mrow> <mn>2</mn> <mi>&Delta;</mi> </mrow> </mfrac> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> <math> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi> </mi> <mi>min</mi> </mrow> <mi>n</mi> </munder> <mo>|</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

And deblurring estimation of signal source distance

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> </mrow> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msub> <mi>i</mi> <mn>0</mn> </msub> <mi>&pi;</mi> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </math> $i_0=\underset{i}{\arg \min }|{\hat{l}}_k\left(i\right)-{\hat{l}}_k^{\mathrm{ref}}|---\left(18\right),$

Preferably, the array element interval Δ is 1/4 greater than the wavelength λ of the near-field sound source.

Preferably, the method further comprises increasing the accuracy of the deblurred estimates of the DOA and the near-field source distance by increasing the array element spacing Δ.

The patent application of the invention is derived from the research result of fund projects of the education hall in Zhejiang province.

Due to the adoption of the technical scheme, the invention has the remarkable technical effects that:

the method only utilizes the second-order statistic, can avoid a large amount of calculation required when calculating the high-order statistic, and can also reduce the possibility of errors in the calculation process by reducing the calculation amount; meanwhile, the array element interval delta of the adopted array is not required to be limited within the range of 1/4 wavelengths, the configuration mode of the array elements is expanded, the array element interval delta can be randomly configured as required, and the 1/4 wavelength is not limited to the setting of the array interval delta any more; furthermore, the array aperture can be expanded by increasing the array element interval delta of the array, so that the estimation precision of the obtained parameters is improved; finally, the method of the invention can realize automatic pairing of the finally output parameter estimation.

Drawings

FIG. 1 is a schematic of the present invention employing a uniform linear array;

FIG. 2 is a schematic diagram of the process steps of the present invention;

FIG. 3 shows a near-field sound source k₁A plot of root mean square error versus signal ratio for the DOA estimate of (a);

FIG. 4 shows a near-field sound source k₂A plot of root mean square error versus signal ratio for the DOA estimate of (a);

FIG. 5 shows a near-field sound source k₁A plot of root mean square error versus signal ratio for the signal source range estimate of (1);

FIG. 6 shows a near-field sound source k₂Is plotted against the signal ratio of the root mean square error of the signal source distance estimate.

Detailed Description

The present invention will be described in further detail with reference to examples.

Example 1

The use of a uniform linear array of pairs of velocity sensors can improve the estimation performance of parameters through aperture expansion; furthermore, multi-dimensional parameter search or calculation of high order statistics is not required, and automatic pairing of parameter estimation can be achieved.

The near-field sound source positioning method, as shown in fig. 1, adopts the following steps:

the method comprises the following steps: setting a uniform linear array

A uniform linear array of 2M array elements is provided (K < 2M). The array elements are spaced apart by Δ, and each array element is an array of a pair of velocity sensors, pointing in the y-axis and z-axis respectively.

Step two: establishing a signal model

Considering K near-field, narrow-band, incoherent acoustic source signals incident on the uniform linear array, -pi/2<θ_m,kπ/2 denotes the kth near field sourceDOA of the signal relative to the m-th array element; l_m，kThe distance between the kth near-field source signal and the mth array element is shown, and the measurable azimuth information of the two speed sensors in the mth array element is

c_m，k＝[sinθ_m,k，cosθ_m,k]^T （1），

Wherein-pi/2<θ_m,kAnd the value of ≦ pi/2 represents the DOA of the kth near-field source signal relative to the mth array element. Using the 0 th array element as the reference array element, and order c_k＝c_0，kAnd theta_k＝θ_0，k. After demodulation to intermediate frequency and sampling, the kth near-field narrow-band sound source signal is

Wherein s is_k(t) represents the complex amplitude of the signal, ω_k=2πf_k/f_samp，f_sampTo sample frequency, f_k≠f_l(k ≠ l) is the carrier frequency of the signal. Thus, the received signal at the m-th element can be represented as a 2-dimensional vector:

<math> <mrow> <msub> <mi>x</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> <mi>t</mi> </mrow> </msup> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>+</mo> <msub> <mi>n</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

in the formula n_m(t) denotes a Gaussian noise vector, τ_m，k≈γ_km+φ_km²For propagation delay between the ith source m and the reference element 0, where gamma_k=-2πΔsinθ_k/λ_k，φ_k=πΔ²cos²θ_k/λ_kl_k，λ_kAnd l_kRespectively the wavelength of the kth signal and its distance to the reference array element. Equation (2) is written in matrix form:

z(t)=As(t)+w(t) （3），

wherein

<math> <mrow> <mi>z</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msubsup> <mi>x</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </msubsup> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mn>0</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mi>M</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo></mo> </mrow> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>&Element;</mo> <msup> <mi>C</mi> <mrow> <mn>4</mn> <mi>M</mi> <mo>&times;</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>

<math> <mrow> <mi>s</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mn>1</mn> </msub> <mi>t</mi> </mrow> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>K</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>K</mi> </msub> <mi>t</mi> </mrow> </msup> <mo>]</mo> </mrow> <mi>T</mi> </msup> </mrow> </math>

E{w(t1)w^H(t₂)}=δ(t₁-t₂)I_4M×4M

A=[a₁，...，a_K]∈C^4M×K

<math> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msubsup> <mi>b</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Definition of l_m，kRepresents the distance between the kth near-field source signal and the mth array element and has_k=l_0，k. As shown in fig. 1, a plane triangle is formed between the kth target, the mth array element and the reference array element, so that by using the sine theorem, we can obtain:

<math> <mrow> <msub> <mi>l</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>sin</mi> <mrow> <mo>(</mo> <mi>&pi;</mi> <mo>/</mo> <mn>2</mn> <mo>-</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>-</mo> <mn>1,0,1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

equation (5) will find application in the proposed algorithm.

Step three: calculating a DOA and a unambiguous estimate of the distance of a signal source

Based on the assumption that the signal is narrowband, we have

Wherein

Therefore, from formula (3), z (t +1) ═ a Φ s (t) + w (t +1) can be obtained. Definition of <math> <mrow> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msup> <mi>z</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msup> <mi>z</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mi>H</mi> </msup> <mo>,</mo> </mrow> </math> Its correlation matrix can be expressed as <math> <mrow> <mi>R</mi> <mo>=</mo> <mi>E</mi> <mo>{</mo> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mover> <mi>z</mi> <mo>&OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mi>BP</mi> <msup> <mi>B</mi> <mi>H</mi> </msup> <mo>,</mo> </mrow> </math> Wherein P = E { s (t) s^H(t) } denotes a correlation matrix of signals, and B = [ A =^H，(AΦ)^H]^H∈C^8M×K。

The eigenvalue decomposition has R = U_sdiag(ψ₁，...，ψ_k)U_s ^HWherein ψ_k(K1.. K.) is K large feature values of R, and their corresponding feature vectors are U_sThe column vector of (2). According to the subspace algorithm, a nonsingular matrix T is existed to satisfy B = U_sT, whereby A = U_s，1T and a Φ = U_s，2T, namely have

Wherein U is_s，1=U_s(1∶4M,K)，U_s，1=U_s(1+4M∶8M,K)，

The left pseudo-inverse is indicated. Thus, we can derive the frequency ω of the signal by equation (6)_k＝arg(ψ_k) And corresponding wavelength lambda_k=2πc/ω_kf_s。

In practice, the correlation matrix R is obtained by finite snapshot estimation (covariance matrix)

Thus is paired with

After the characteristic value is decomposed, an estimated value is obtained

And

and

and

from this we can further obtain an estimate of the steering matrix

<math> <mrow> <mover> <mi>A</mi> <mo>^</mo> </mover> <mo>=</mo> <mo>[</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mi>K</mi> </msub> <mo>]</mo> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <msup> <mover> <mi>&Phi;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

According to a_kBy definition of (1), we can be defined by

Can obtain b_m，kIs estimated by(M = - (M-1), …, -1, 0, 1, …, M; K =1, …, K). Then, DOA is obtained from the relationship of the formulae (4) and (1):

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>tan</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

note that in the formula (8)

By making use of

And in combination with equation (5), we can obtain an estimate of the distance of the signal source

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>+</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

It is clear that the DOA and distance estimates obtained by equations (8) and (9) are unique and do not present the ambiguity problem.

Step four: calculating a deblurred estimate of DOA and signal source distance

In distinction from equations (8) and (9), we can also derive the spatial phase factor

A DOA and range estimate of the signal is obtained. For this purpose we first calculate q_m，kEstimation of (2):

<math> <mrow> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mn>1</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>/</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>m</mi> <mo>&NotEqual;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

wherein

The estimation of equation (1) is obtained using the result of equation (8). Next, we compute using the symmetry of the array structure

Andis estimated. Due to the fact that

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mo>[</mo> <mn>2</mn> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>]</mo> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mrow> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> </mrow> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </msup> </math>

And is <math> <mrow> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <mn>2</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> <math> <mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <mi>c</mi> <msup> <mi>os</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> Thus can obtain

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>4</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <mi>co</mi> <msup> <mi>s</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>M</mi> <mo>-</mo> <mn>4</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Thus, DOA and distance estimates can also be obtained using equations (13) and (14), respectively. However, since the array element spacing Δ is greater than 1/4 for the wavelength, the calculation of DOA and distance estimates using equations (13) and (14) is non-unique, i.e., it is not unique

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mi>&pi;&Delta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mn>2</mn> <mi>&Delta;</mi> </mrow> </mfrac> <mi>n</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mn>2</mn> <mi>&pi;&Delta;</mi> </mrow> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> </mrow> <mrow> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>i&pi;</mi> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </math>

That is, the estimates of DOA and distance in equations (15) and (16) are ambiguous in thatAnd

respectively represents the lower rounding and the upper rounding of x, and n is more than or equal to 1 and i is more than or equal to 1. In the formula (16)

And

the DOA and the distance estimates obtained by equations (8) and (9) are shown, respectively.

To solve this ambiguity we further exploit

And

for reference, the following deblurred estimates are obtained:

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mi>&pi;&Delta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mrow> <mn>2</mn> <mi>&Delta;</mi> </mrow> </mfrac> <msup> <mi>n</mi> <mn>0</mn> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math> <math> <mrow> <msup> <mi>n</mi> <mn>0</mn> </msup> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi> </mi> <mi>min</mi> </mrow> <mi>n</mi> </munder> <mo>|</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> </mrow> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <msup> <mi>i</mi> <mn>0</mn> </msup> <mi>&pi;</mi> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </math> $i^0=\underset{i}{\arg \min }|{\hat{l}}_k\left(i\right)-{\hat{l}}_k^{\mathrm{ref}}|---\left(18\right),$

the estimation methods of equations (17) and (18) both utilize aperture (by Δ metric) information of the array, and thus have higher estimation accuracy and resolution than the estimation methods of equations (8) and (9).

Example 2 (Experimental example)

Setting a uniform linear array to contain 16 velocity sensors (i.e., M = 4) with a signal sampling frequency f_samp=15MHz, and takes into account two near-field sound sources (k)₁，k₂) The parameters are respectively as follows: omega₁=0.4 π rad/s and ω₂=0.5 π rad/s (i.e., wavelengths λ respectively)₁=100m and λ₂＝80m)、θ₁＝25°、θ₂＝40°、l₁=2.5λ₁、l₂=3.0λ₂. And (4) carrying out experimental result simulation of near-field source positioning by adopting 400 snapshot numbers (N). The simulation result is a statistical result of 500 monte carlo experiments, such as fig. 3, fig. 4, fig. 5, and fig. 6.

Wherein fig. 3 and 4 show the relation between the root mean square estimation error and the signal ratio of two signal sources DOA, wherein fig. 3 shows a signal k₁FIG. 4 shows a signal k₂. We consider three different cases of array element spacing Δ: Δ ═ λ_min/4、Δ＝λ_minAnd Δ ═ 2 λ_minWherein λ is_min=min{λ₁，λ₂}. In the figure, the position of the upper end of the main shaft,

represents the unambiguous estimate calculated from equation (8), and theta_kThe deblur estimation value calculated by equation (15) is shown. It can be seen that the deblurred estimates of DOA have a lower estimation error as the array element spacing Δ increases. Thus, the advantage of the algorithm herein is that the accuracy of the DOA estimation can be improved by using a larger array element spacing Δ.

FIGS. 5 and 6 show the RMS estimation error versus signal ratio for two signal source distances, where FIG. 5 is signal k₁FIG. 6 shows a signal k₂. Similarly, in the figures

Represents a non-ambiguous estimate of the distance of the signal source calculated by equation (9), and r_kThe deblur estimation value of the signal source distance calculated by equation (18) is shown. The figure also shows that the algorithm herein can improve the accuracy of the estimation of the near-field source distance by using a larger array element spacing. In addition, comparing FIGS. 3 and 5, FIGS. 4 and 6, we also found that increasing the array element spacing also improves the unambiguous distanceEstimating

(see FIGS. 5, 6) but no blurred DOA

Without significant variation in the estimated error (see fig. 3, 4). This is mainly due to the push-to-get by equation (8)Independent of the array element spacing a and therefore insensitive to variations in the array element spacing.

In summary, the above-mentioned embodiments are only preferred embodiments of the present invention, and all equivalent changes and modifications made in the claims of the present invention should be covered by the claims of the present invention.

Claims (3)

1. A near-field sound source localization method for estimating a DOA and a signal source distance of a near-field sound source signal, comprising:

step A: a uniform linear array is set up and,

the uniform linear array is composed of array elements which are linearly arranged, have an interval of delta and are 2M in number, each array element comprises a pair of speed sensors which respectively point to a y axis and a z axis, the y axis is an axis where the linear array is located, the z axis is perpendicular to the y axis, the near-field sound source is located on a y-z axis plane, and the speed sensors can be used for receiving a near-field sound source signal and outputting azimuth information of the near-field sound source signal;

and B: establishing a signal model, which comprises the following specific steps:

measuring azimuth information through a speed sensor:

c_m，k＝[sinθ_m，k，cosθ_m，k]^T (1)，

wherein-pi/2<θ_m，kPi/2 or less represents DOA of the kth near-field source signal relative to the mth array element, the 0 th array element is set as a reference array element, and c is set_k＝c_0，kAnd theta_k=θ_0，k；

② demodulating to intermediate frequency and sampling, the kth near-field sound source signal is

Wherein s is_k(t) represents the complex amplitude of the signal, ω_k＝2πf_k/f_samp，f_sampTo sample frequency, f_k≠f_l(k ≠ l) is the carrier frequency of the signal;

third, the received signal at the m-th array element can be expressed as a 2-dimensional vector:

<math> <mrow> <msub> <mi>x</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </msubsup> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>k</mi> </msub> <mi>t</mi> </mrow> </msup> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>+</mo> <msub> <mi>n</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

wherein n is_m(t) denotes a Gaussian noise vector, τ_m，k≈γ_km+φ_km²For propagation delay between the ith source m and the reference element 0, where gamma_k＝-2πΔsinθ_k/λ_k，φ_k＝πΔ²cos²θ_k/λ_kl_k，λ_kAnd l_kThe wavelength of the kth signal and the distance from the kth signal to the reference array element respectively;

converting the form of the matrix of the formula (2) into:

z(t)＝As(t)+w(t) (3)，

wherein,

<math> <mrow> <mi>z</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msubsup> <mi>x</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mn>0</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>x</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mi>M</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>&Element;</mo> <msup> <mi>C</mi> <mrow> <mn>4</mn> <mi>M</mi> <mo>&times;</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </math>

<math> <mrow> <mi>s</mi> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mn>1</mn> </msub> <mi>t</mi> </mrow> </msup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>K</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&omega;</mi> <mi>K</mi> </msub> <mi>t</mi> </mrow> </msup> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> </mrow> </mrow> </math>

<math> <mrow> <mi>w</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>[</mo> <msubsup> <mi>n</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>n</mi> <mn>0</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msubsup> <mi>n</mi> <mn>1</mn> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>n</mi> <mi>M</mi> <mi>T</mi> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <msup> <mo>]</mo> <mi>T</mi> </msup> <mo>,</mo> <mi>E</mi> <mo>{</mo> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <msup> <mi>w</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mi>&delta;</mi> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mrow> <mn>4</mn> <mi>M</mi> <mo>&times;</mo> <mn>4</mn> <mi>M</mi> </mrow> </msub> <mo>,</mo> </mrow> </math>

A＝[a₁，...，a_K]∈C^4M×K，

<math> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>[</mo> <msubsup> <mi>b</mi> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>b</mi> <mrow> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>,</mo> <msub> <mi>b</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow> <mi>j</mi> <msub> <mi>&tau;</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

define l_m，kRepresents the distance between the kth near-field source signal and the mth array element and has_k＝l_0，kObtaining:

<math> <mrow> <msub> <mi>l</mi> <mi>k</mi> </msub> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>sin</mi> <mrow> <mo>(</mo> <mi>&pi;</mi> <mo>/</mo> <mn>2</mn> <mo>-</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>=</mo> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&theta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>,</mo> </mrow> </math> m＝-(M-1)，...，-1，0，1，...，M(5)；

and C: calculating a non-fuzzy estimation of DOA and signal source distance, comprising the following steps:

firstly, acquiring N snapshot vectors z (t) of an array, wherein t is 1

Wherein,

obtained by eigenvalue decompositionWherein psi_k(K is 1, …, K) isK large feature values of (1), their corresponding feature vectors are U_sA column vector of (a);

② according to subspace algorithm, there is a nonsingular matrix T satisfying B ═ U_sT, whereby A ═ U_s，1T and A phi ═ U_s，2T, namely:

wherein U is_s，1＝U_s(1：4M，K)，U_s，1＝U_s(1+4M：8M，K)，

Represents the left pseudo-inverse;

(iii) frequency estimation of the signal obtained by equation (6) above

Sum wavelength estimation

And

and

(ii) an estimate of (d);

obtaining the estimation of the steering matrix according to the step three:

<math> <mrow> <mover> <mi>A</mi> <mo>^</mo> </mover> <mo>=</mo> <mo>[</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mover> <mi>a</mi> <mo>^</mo> </mover> <mi>K</mi> </msub> <mo>]</mo> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>U</mi> <mo>^</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mover> <mi>T</mi> <mo>^</mo> </mover> <msup> <mover> <mi>&Phi;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

fifthly, obtaining DOA non-fuzzy estimation of near-field sound source signals according to the relation of the combination formula (4) and the combination formula (1)

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>tan</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein,

(m＝-(M-1)，...，-1，0，1，...，M；k＝1，...，K)；

combination of

And equation (5) obtaining a blur-free estimate of the signal source distance of the near-field acoustic source signal

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>+</mo> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </msubsup> <mo>|</mo> <mfrac> <mrow> <mi>m&Delta;</mi> <mi>cos</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> </mrow> <mrow> <mi>sin</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>.</mo> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>|</mo> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>K</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

Step D: calculating fuzzy estimation of DOA and signal source distance, which comprises the following steps:

firstly, calculate q_m，kIs estimated by

<math> <mrow> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>/</mo> <msubsup> <mover> <mi>c</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> <mi>T</mi> </msubsup> <msub> <mover> <mi>b</mi> <mo>^</mo> </mover> <mrow> <mn>0</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mi>m</mi> <mo>&NotEqual;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

Wherein

The result of the formula (8) is used to obtain the estimation of the formula (1), and the electron angle alpha is calculated_kAnd beta_k；

② calculating by using symmetric relation of array structure

And

due to estimation of

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mo>[</mo> <mn>2</mn> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>]</mo> </mrow> </msup> <mo>=</mo> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>m</mi> <mo>-</mo> <mi>M</mi> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msubsup> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>M</mi> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> <mo>*</mo> </msubsup> <msub> <mover> <mi>q</mi> <mo>^</mo> </mover> <mrow> <mi>M</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

And is <math> <mrow> <msub> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mo>-</mo> <mn>2</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> <math> <mrow> <msub> <mover> <mi>&phi;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>&pi;&Delta;</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>,</mo> </mrow> </math> Thus can obtain

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>4</mn> <mi>&pi;&Delta;</mi> <mi>sin</mi> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

<math> <mrow> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>/</mo> <msub> <mover> <mi>&lambda;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>M</mi> <mo>-</mo> <mn>4</mn> </mrow> </mfrac> <msubsup> <mi>&Sigma;</mi> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>M</mi> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&eta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>/</mo> <msub> <mi>&xi;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>/</mo> <mn>2</mn> <mover> <mo>=</mo> <mi>&Delta;</mi> </mover> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

Thirdly, obtaining fuzzy estimation of DOA according to the calculation result

And fuzzy estimation of signal source distance

Wherein

And

represent unambiguous estimates of DOA and distance obtained using equations (8) and (9), respectively;

and (3) obtaining a deblurring estimation of the DOA and the signal source distance by taking the unambiguous estimation of the DOA and the signal source distance as a reference estimation:

making <math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>i</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

② obtaining the deblurring estimate of DOA

<math> <mrow> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msup> <mi>sin</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <mn>4</mn> <mi>&pi;&Delta;</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mrow> <mn>2</mn> <mi>&Delta;</mi> </mrow> </mfrac> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>n</mi> </munder> <mo>|</mo> <msub> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </math>

And deblurring estimation of signal source distance

<math> <mrow> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>&pi;</mi> <msup> <mi>&Delta;</mi> <mn>2</mn> </msup> <msup> <mi>cos</mi> <mn>2</mn> </msup> <msubsup> <mover> <mi>&theta;</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> </mrow> <mrow> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> <mi>arg</mi> <mrow> <mo>(</mo> <msub> <mi>&beta;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mrow> <mn>2</mn> <mi>i</mi> </mrow> <mn>0</mn> </msub> <mrow> <mi>&pi;</mi> <msub> <mi>&lambda;</mi> <mi>k</mi> </msub> </mrow> </mrow> </mfrac> <mo>,</mo> <msub> <mi>i</mi> <mn>0</mn> </msub> <mo>=</mo> <munder> <mrow> <mi>arg</mi> <mi>min</mi> </mrow> <mi>i</mi> </munder> <mo>|</mo> <msub> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mover> <mi>l</mi> <mo>^</mo> </mover> <mi>k</mi> <mi>ref</mi> </msubsup> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </math>

2. The near field acoustic source localization method of claim 1, wherein the array element spacing Δ is 1/4 greater than the near field acoustic source wavelength λ.

3. The near-field sound source localization method according to claim 1, further comprising improving the accuracy of the deblurring estimation of the DOA and the near-field sound source distance by increasing the array element spacing Δ.

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