CN1773307A - Aperture Expansion and Spatial Signal Processing Method of Small Antenna Array - Google Patents

Abstract

The present invention relates to a small type antenna array aperture expanding and space signal processing method. Said method includes the following steps: making antenna array into any array form: dividing scanning area of whole antenna into several subregions, finely-dividing any one subregion of said several subregions; in the finely-divided subregion constructing actual array flow form of antenna array and array flow form of expanded array so as to further obtain the array expansion conversion matrix; utilizing the obtained array expansion conversion matrix to reconstitute the signal of expanded antenna array; then making said signal undergo the process of digital beam formation or adaptive beam formation or utilizing space high resolution algorithm to obtain space arrival angle of signal.

Description

Aperture Expansion and Spatial Signal Processing Method of Small Antenna Array

technical field

The invention relates to a small antenna array aperture expansion and space signal processing method.

Background technique

According to the classical antenna theory, the angular resolution of the radar is related to the antenna beam width, and the radar antenna beam width is approximately inversely proportional to the electrical length of the antenna, that is

${\mathrm{\θ}}_{3\mathrm{dB}}\∝\frac{\mathrm{\λ}}{L}$ (radian)

Among them, θ _3dB is the half-power width of the antenna beam, λ is the working wavelength of the radar, and L is the length of the antenna array. When the spacing between antenna elements is about half of the operating wavelength of the radar, a large phased array antenna and a large number of receiving channels are required to obtain high angular resolution. However, in the case where the radar antenna position is limited, for example, for high-frequency ground wave or high-frequency sky-wave over-the-horizon radar, in order to obtain high-resolution ocean surface current and sea surface target orientation information, (Multiple Signal Classification Algorithm) MUSIC etc. are generally used High-resolution algorithm, in order to obtain high-resolution sea surface wind and wave information, DBF technology is generally used. For example, if a high-frequency radar operates at a frequency of 5MHz, to obtain an angular resolution of 1°, the radar antenna array length will reach 3000 meters. Moreover, in order to effectively excite the surface waves propagating along the sea surface, the radar antenna array should be set as close to the sea water as possible. This distance usually requires 1 to 2 radar wavelengths. Under actual coastal conditions, it is not only difficult to find Antenna sites that are matched with large antenna arrays also greatly increase the cost of radar station construction and daily maintenance costs. At the same time, because the radar antenna array is too large, its far-field range reaches several kilometers, and it is extremely difficult to calibrate the radar system. All of these will limit the popularization and application of many radars including high frequency surface wave radar.

Contents of the invention

Aiming at the limitations of the existing multi-channel large-scale phased array radar, the purpose of the present invention is to provide a small-scale antenna array aperture expansion and space signal processing method, which uses the array expansion technology to realize what only large-scale phased-array radars can achieve. Function.

In order to achieve the above object, the present invention provides a small antenna array aperture expansion and space signal processing method, the antenna array is set to any array form; the entire antenna scanning area is divided into several areas, and any sub-area is subdivided: Construct the actual array manifold A=[a(θ ₁ ), a(θ ₁ +Δθ), a(θ ₂ +2Δθ),…, a(θ _r )] and extended Array manifold A=[a(θ ₁ ), a(θ ₁ +Δθ), a(θ ₂ +2Δθ),…, a(θ _r )], and then the array expansion transformation matrix B= AA ^-1 , where θ ₁ and θ _r are the left and right boundaries of the sub-region respectively, and Δθ is the subdivision step size of the sub-region; the signal Z=BX of the extended array antenna is reconstructed by using the obtained array expansion transformation matrix, where X is The signal received by the original antenna array; the signal Z of the reconstructed extended array antenna is subjected to digital beamforming or adaptive beamforming, or the spatial angle of arrival of the signal is obtained by using a spatial high-resolution algorithm.

According to the present invention, when the actual matrix manifold A is a non-square matrix, the matrix A is loaded diagonally.

In addition, the present invention can utilize ‖ The minimum value of the norm of A-BA‖, which determines the number of elements of the extended array.

The advantage of the present invention lies in its excellent practical performance: (1) the caliber of the antenna array and the number of elements of the antenna array are greatly expanded by the above method; Flexible use of modern digital signal processing technology for digital beamforming, beam shape change, spatial super-resolution direction finding and antenna adaptive anti-interference; (3) Since the antenna aperture expansion is based on software, it can greatly reduce the number of antennas The number of units and receiving channels greatly reduces the cost of research and development and site construction; (4) By implementing extended transformations on small arbitrary antenna arrays, compared with the original array, it not only increases the number of detectable sources, but also improves the quality of antenna arrays. The overload capability of the array can be overcome, and the signal ambiguity problem that may occur in the array can be overcome, and the decoherence ability of the array is improved.

Aiming at the special occasions where the radar antenna array is limited, the present invention breaks the old formula of general phased array radar design, and proposes a method of increasing the aperture of the antenna array by performing extended transformation on the small dense phased array antenna. Array radar reduces antenna units and receiving channels, reduces cost, and provides a new way to be simple and easy to build. The invention is not only suitable for uniform arrays, but also suitable for any non-uniform arrays, and the expanded antenna aperture is equivalent to the performance achieved by more than three times the original antenna aperture. Applying it to high-frequency ground-wave over-the-horizon radar, through the extended antenna array, not only can obtain high-resolution ocean surface currents and azimuth information of sea surface targets, but also form relatively narrow beams to obtain ocean engineering requirements. Storm information. The invention provides theoretical and engineering basis for the miniaturization of radar antenna array.

The invention is applied to engineering practice, and not only the research and development and equipment maintenance costs are relatively low, but also has functions that the current traditional large-scale phased array radar does not have under the flexible cooperation of software. The present invention has an important theory for the use of phased arrays in the occasions where antenna positions and electromagnetic compatibility are limited, for national defense construction and economic construction such as aviation, aerospace, air defense, and coastal defense, as well as for miniaturization and cost reduction of phased array radars. value and engineering significance.

A large number of theoretical analysis and radar experiments show that the array expansion proposed by the present invention realizes the effective expansion of the number of antenna array elements, that is, the extended array has more array elements through the array expansion transformation, and its advantages are obvious. It not only increases the number of detectable signal sources, which improves the overload capability of the array, but also overcomes the possible signal ambiguity problem of the array, improves the decoherence capability of the array, greatly reduces the actual antenna aperture and the number of receiving channels, and reduces the radar The development cost satisfies the requirement of the present invention to realize spatial super-resolution by using the array extension technology.

Description of drawings

Figure 1, Figure 2 and Figure 3 are the Doppler spectra of the original array, extended array and traditional large array when the scanning angle is 40°;

Figure 4, Figure 5 and Figure 6 are the Doppler spectra of the original array, extended array and traditional large array when the scanning angle is 60°;

Figure 7, Figure 8 and Figure 9 are the Doppler spectra of the original array, extended array and traditional large array when the scan angle is 80°;

Figure 10, Figure 11 and Figure 12 are the Doppler spectra of the original array, extended array and traditional large array when the scanning angle is 100°;

Figure 13, Figure 14 and Figure 15 are the Doppler spectra of the original array, extended array and traditional large array when the scanning angle is 120°;

Figure 16, Figure 17 and Figure 18 are the Doppler spectra of the original array, extended array and traditional large array when the scan angle is 140°;

Figure 19, Figure 20 and Figure 21 are the Doppler spectra of the original array, extended array and traditional large array when the signal-to-noise ratio is 25dB and the scan angle is 40°;

Figure 22, Figure 23 and Figure 24 are the Doppler spectra of the original array, extended array and traditional large array when the signal-to-noise ratio is 25dB and the scan angle is 60°;

Figure 25, Figure 26 and Figure 27 are the Doppler spectra of the original array, extended array and traditional large array when the signal-to-noise ratio is 25dB and the scan angle is 80°;

Figure 28, Figure 29 and Figure 30 are the Doppler spectra of the original array, extended array and traditional large array when the signal-to-noise ratio is 25dB and the scan angle is 100°;

Figure 31, Figure 32 and Figure 33 are the Doppler spectra of the original array, extended array and traditional large array when the signal-to-noise ratio is 25dB and the scanning angle is 120°;

Figure 34, Figure 35 and Figure 36 are the Doppler spectra of the original array, the extended array and the traditional large array when the signal-to-noise ratio is 25dB and the scanning angle is 140°;

Figure 37 is the Doppler spectrum of the original array beam pointing at 60°;

Figure 38 is the echo spectrum of the beam pointing to 60° after the array is expanded;

Figure 39 is the Doppler spectrum of the original array beam pointing at 120°;

Figure 40 is the echo spectrum of the beam pointing to 120° after array expansion;

Figure 41 is the MUSIC spectrum before array expansion;

Figure 42 is the MUSIC spectrum after array expansion.

Detailed ways

The present invention will be described in more detail below in conjunction with the accompanying drawings and embodiments.

In order to achieve the above object, the present invention adopts an arbitrary antenna array aperture expansion and space super-resolution method, and the antenna array is set in an arbitrary array form; the entire antenna scanning area is divided into several areas, and then a certain sub-area is subdivided; through actual The relationship between the array manifold A and the extended array manifold A is obtained by obtaining the array extended transformation matrix B=AA ^-1 ; using the obtained array extended matrix to reconstruct the signal Z=BX of the extended array antenna (wherein, X is the original The signal received by the antenna array); beamforming the data Z received by the reconstructed extended array, or using a spatial high-resolution algorithm to obtain the spatial super-resolution angle of arrival of the signal.

1. Original array signal model

Without loss of generality, it is assumed that the space array is composed of m array elements, and there are d signal sources, and the signals are incident on the array in the form of plane waves. Then the data received by the i-th array element is:

${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; d</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; ik</span>}}}{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Then the vector formed by the output snapshot data of m array elements at time t is as follows:

X(t)=[x ₁ (t), x ₂ (t), . . . , x _m (t)] ^T (2)

A is the array manifold

A=[a(θ ₁ ), a(θ ₂ ), . . . , a(θ _d )] (3)

Where θ _k is the direction angle of the kth incident signal, a(θ _i ) is expressed as follows:

$\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&tau;</span>}}_{\mathrm{<span\; class="notranslate">\; mi</span>}}}{<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>}$

τ _ki =(x _k cos(θ _i )+y _k sin(θ _i ))/c is the delay of the i-th signal on the k-th array element, and (x _k , y _k ) is the position of this array element coordinates, c is the speed of light.

That is, the array receives the data matrix:

X＝AS (5)

2. Design of extended array

2.1 Array expansion transformation method

For irregular dense arrays or non-uniform arrays, the original array data is transformed into an output of an extended uniform linear array by means of digital transformation.

Let the signal direction vector of the expanded uniform linear array be

`` a(θ)=[1, exp(-jkd cosθ),..., exp(-jk(N-1)dcosθ)] (6) where d is the element spacing of the extended antenna array. Suppose there is a non-singular matrix T(Θ) of order N×N that satisfies the following relationship

B(Θ)A(Θ)＝[ a(θ ₁ ), a(θ ₂ ),..., a(θ _P )]＝ A(Θ) (7) Through the transformation of the above formula, the matrix B(Θ ) transforms the direction matrix of the original array into a direction matrix of a new uniform array, we call the new uniform array obtained in this way an extended array, and once the B(Θ) matrix is obtained, the extended uniform line can be obtained output vector of array

z(t)=B(Θ)x(t) (8)

2.2 Obtaining the array transformation matrix

The processing idea of array extension is to divide the entire antenna scanning area into several sub-areas, and then subdivide a certain sub-area. Assuming that the signal is located in the area Θ, the area Θ is equally divided into

Θ=[θ ₁ , θ ₁ +Δθ, θ ₁ +2Δθ,...,θ _r ] (9)

θ ₁ , θ _r is Θ is the left and right boundaries, Δθ is the step size. Then the array manifold matrix of the actual array is

A＝[a(θ ₁ ), a(θ ₁ +Δθ), a(θ ₂ +2Δθ),..., a(θ _r )] (10)

And the array manifold matrix A of the extended uniform linear array in the same area Θ is

A＝[ a(θ ₁ ), a(θ ₁ +Δθ), a(θ ₂ +2Δθ),…, a(θ _r )] (11)

a(θ _i ) is represented by the following formula:

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">j</figure-callout></span><figure-callout\; id="0"\; label="j\; w"\; filenames="A20051001963300155.png"\; state="\{\{state\}\}">\; </figure-callout>}{\mathrm{<span\; class="notranslate">\; </span>}}_{}{\mathrm{<span\; class="notranslate">w</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&Delta;</span>}}_{\mathrm{<span\; class="notranslate">\; ni</span>}}}{<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; )</span>$

Among them, n is the number of array elements in the extended array, Δk _i =(xx _k cos(θ _i )+yy _k sin(θ _i ))/c is the i-th signal on the k-th array element of the extended array Delay, (xx _k , yy _k ) is the position coordinate of the extended array element, and c is the speed of light.

Then there is a fixed transformation relationship B(T) between the extended array and the real array, so that BA= A_B= AA ^-1 (13)

In the above formula, it involves the inversion of A. In general, the matrix A is not a square matrix and cannot be directly inverted, so we use the diagonal loading technique, and a is the loading coefficient.

C＝A′*A

A ^-1 ≈(C+a*eye(size(C,1))) ^-1 *A′ (14)

Suppose a certain matrix Z satisfies Z=BX,

Z=BX=AA ^-1 X=AA ^-1 AS= AS (15)

Then Z can be regarded as the data received by the expansion array.

2.3 Determination of initial angle and array expansion unit

以使 ${\left|\right|\stackrel{\&OverBar;}{A}\left(\widehat{\mathrm{\&Theta;}}\right)-B\left(\widehat{\mathrm{\&Theta;}}\right)A\left(\widehat{\mathrm{\&Theta;}}\right)\left|\right|}_F$ 范数达到最小。From the above analysis, it can be seen that the primary problem faced by the transformation method of the extended array is the determination of the initial angle and the number of antenna elements, and the azimuth angle of the signal source is a parameter to be estimated. Therefore, the azimuth angle needs to be estimated before constructing the transformation matrix , and the quality of the prediction directly affects the quality of the transformation matrix and ultimately affects the estimation results. In practical applications, a simple method is to divide the entire antenna scanning area into several radar elements according to the beam width of the extended array when processing the spatial signal of each range element, assuming that the signal is located in the within the digital beam area, and then calculate ${\left|\right|\stackrel{\&OverBar;}{A}\left(\widehat{\mathrm{\&Theta;}}\right)-B\left(\widehat{\mathrm{\&Theta;}}\right)A\left(\widehat{\mathrm{\&Theta;}}\right)\left|\right|}_f$ If the value is not small enough, you can adjust the number of elements of the extended antenna, select the appropriate lobe width and signal initial angle, and then recalculate the extended transformation matrix

so that ${\left|\right|\stackrel{\&OverBar;}{A}\left(\widehat{\mathrm{\&Theta;}}\right)-B\left(\widehat{\mathrm{\&Theta;}}\right)A\left(\widehat{\mathrm{\&Theta;}}\right)\left|\right|}_f$ Norm reaches the minimum.

2.4 Determination of antenna element spacing

总有误差，因此，设计扩展均匀线阵时，应该尽量选择对初始角度估计的误差最不敏感、形成的变换误差也最小的的阵列为扩展阵列。实际应用中，我们关心的是矩阵

与

的列空间是否一致。在列空间的意义上，矢量 与 a(θ)之间的距离可用两者之间的夹角

表示：To perform array expansion, the initial direction of the signal must first be known, and the initial angle estimation

There are always errors. Therefore, when designing an extended uniform linear array, one should try to select the array that is least sensitive to the error of the initial angle estimation and has the smallest transformation error as the extended array. In practice, we are concerned with the matrix

and

Is the column space consistent. In the column space sense, the vector The distance between a(θ) and the angle between the two can be used

express:

$\mathrm{\&gamma;}(\mathrm{\&theta;};\widehat{\mathrm{\&theta;}})=\arccos \frac{\left|{\stackrel{\&OverBar;}{a}}^h\left(\mathrm{\&theta;}\right)B\left(\widehat{\mathrm{\&theta;}}\right)a\left(\mathrm{\&theta;}\right)\right|}{{\left|\right|\stackrel{\&OverBar;}{a}\left(\mathrm{\&theta;}\right)\left|\right|}_2\&Center\; Dot;{\left|\right|B\left(\widehat{\mathrm{\&theta;}}\right)a\left(\mathrm{\&theta;}\right)\left|\right|}_2}---\left(16\right)$ ‖·‖ ₂ and |·| represent the spectral norm and absolute value, respectively. The optimal inter-element moment d should be

最小，即

minimum, ie

${\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; opt</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; arg</span>}\underset{\mathrm{<span\; class="notranslate">\; d</span>}}{\mathrm{<span\; class="notranslate">\; min</span>}}\mathrm{<span\; class="notranslate">\; \&gamma;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; ;</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

In general d _opt for θ and is not very sensitive, so for simplicity, θ can be chosen to be the maximum output direction of the beamformer. Nevertheless, the calculation of d _opt is still inconvenient. According to the previous discussion, in order to reduce the transformation error, the amplitude and phase characteristics of the main beam of the original array and the extended uniform linear array should be as close to the same as possible. Therefore, if the 3dB main beam width of the original array is W, a simple choice of uniform line array element spacing is ^[7]

$\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0.886</span>}\frac{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; W</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

where λ is the wavelength. In this way, the 3dB main beam widths of the original array and the extended uniform linear array are equal. Computer simulations show that d is generally close to d _opt .

2.5 Extended transformation error analysis

In order to examine the error of the unitary transformation matrix, here $\widehat{\mathrm{\&Theta;}}=\{{\widehat{\mathrm{\&theta;}}}_1,{\widehat{\mathrm{\&theta;}}}_2\}$ A simple case of . When the array element is omnidirectional, it is not difficult to prove that

${<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}_{\mathrm{<span\; class="notranslate">\; f</span>}}$

in,

和

在一个波束宽度内时，要使变换矩阵误差最小，原阵列和扩展阵列的主波束之幅度特性和相位特性应尽量接近；而当

和

的间隔超过一个波束宽度时，两阵列旁瓣越低，则变换矩阵误差越小。当选择酉变换矩阵(15)式时，该结论不仅指出了阵列数据变换法对原阵列的要求，而且对扩展均匀线阵的设计也指明了方向。Study (18) shows that when _ _a → _ _b or when _ _a → 0 and _ _b → 0, the left side of (19) tends to zero. That is to say, when

and

Within one beam width, to minimize the transformation matrix error, the amplitude characteristics and phase characteristics of the main beam of the original array and the extended array should be as close as possible; and when

and

When the interval of is more than one beamwidth, the lower the sidelobes of the two arrays are, the smaller the error of the transformation matrix is. When the unitary transformation matrix (15) is selected, this conclusion not only points out the requirements of the array data transformation method for the original array, but also points out the direction for the design of the extended uniform linear array.

2.6 Effect of extended transformation on noise

Assuming that the data covariance matrix of the actual array is R and the noise covariance matrix is R _N , then the data covariance matrix of the extended array is

R _T ＝NRB ^H ＝B(AR _s A ^H +R _N )T ^H

=BAR _s A ^H T ^H +BR _N T ^H (21)

When the ambient noise is white noise, then

BR _N B ^H = σ ² BB ^H (22)

It can be seen from the above formula that after the array expansion transformation, the white noise received by the original array becomes colored noise.

3. Super-resolution algorithm of extended array

From the above analysis, it can be seen that after the array is transformed by the extended array, the white noise received by the original array becomes colored noise, so when the noises received by each array element are independent of each other, the covariance matrix can be estimated by the following method:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; :</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; h</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; :</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; three</span>}<span\; class="notranslate">\; )</span>$

In the formula, M is the number of time-domain smoothing, L is the number of snapshots, where x(i:j) represents the i-th snapshot to the j-th snapshot data. Obviously, when the noise of each array element is independent of each other, the noise item of the estimated covariance matrix should be 0. By decomposing the eigenvalues of _RT and using the MUSIC spatial spectrum estimator, the super-resolution estimation of signal DOA can be obtained.

4. Algorithm performance simulation

Simulation Test 1: Uniform Linear Array Aperture Expansion

It is assumed that N=8 omnidirectional array elements are evenly distributed on a straight line with a length of λ. There are currently 6 signals moving away from the radar direction, and their radial speed and direction are {4, 8, 12, 16, 20, 24} m/s and {40°, 60°, 80°, 100°, 120 °, 140°}. Data sampling points 256. Figures 1, 4, 7, 10, 13, and 16 show the echo Doppler spectra of the array expansion front (original array) beam pointing at 40°, 60°, 80°, 100°, 120°, and 140°, respectively. Now expand the original array into an array whose antenna spacing is d=0.5λ. Figures 2, 5, 8, 11, 14, and 17 show the echo Doppler spectra of array beams pointing at 40°, 60°, 80°, 100°, 120°, and 140°, respectively, after array expansion. Figures 3, 6, 9, 12, 15 and 18 show an 8-element uniform linear array with an array element spacing of half a wavelength. From the results, it can be clearly seen that before the array expansion, the original array could not distinguish several signal directions in a beam, but after expanding the array with the scheme proposed in this paper, the direction of each signal and Doppler can be clearly distinguished. Moreover, the resolution effect of the expanded array is almost the same as that of the actual array with the same caliber. Further analysis of the reasons shows that before the array expansion, the beamwidth of the original array is about 60°, but after the array expansion, the beamwidth of the array is reduced to about 15°, that is, the resolution of the dense array is equivalent to three times the aperture of the original array.

Simulation Experiment 2: Effect of SNR on Aperture Expansion of Uniform Linear Array

Assuming that N=8 omnidirectional array elements are evenly distributed on a straight line with a length of λ, the incident wave signal is the same as the simulation test 1, the difference is that the signal-to-noise ratio of each signal is 25dB. Figures 19, 22, 25, 28, 31 and 34 show the echo Doppler spectra of the array expansion front (original array) beam pointing at 40°, 60°, 80°, 100°, 120° and 140° respectively. Now expand the original array into an array whose antenna spacing is d=0.5λ. Figures 20, 23, 26, 29, 32 and 35 show the echo Doppler spectra of the array beams pointing at 40°, 60°, 80°, 100°, 120° and 140° respectively after the array expansion. Figures 21, 24, 27, 30, 33 and 36 show a large uniform linear array with an element spacing of half a wavelength. Similar to the conclusion obtained in simulation experiment 1, before the array expansion, the original array cannot distinguish several signal directions in a beam, but after expanding the array with the scheme proposed in this paper, the direction of each signal and the direction of Doppler can be clearly distinguished. , further analysis shows that with the increase of the signal-to-noise ratio or the increase of the antenna spacing of the original array elements, the resolution effect will be greatly improved.

Simulation Test 3: Large-Space Uniform Linear Array Aperture Expansion

It is assumed that N=8 omnidirectional array elements are evenly distributed on a straight line with a length of 10λ (obviously, the distance between the antennas is much larger than half a wavelength, and grating lobes will appear in the pattern of the original array). The signal-to-noise ratio is 25dB. The incident wave signal is the same as the simulation experiment 1. Typical simulation results are as follows: Figure 37 shows the echo Doppler spectrum with the beam pointing to 60° before array expansion. Now expand the original array into an array whose antenna spacing is d=0.5λ. Figure 38 shows the echo spectrum of the beam pointing to 60° after array expansion. From the results, it can be clearly seen that before the array expansion, multiple arrival angles appeared in one beam, which is obviously due to the direction-finding ambiguity caused by the grating lobes in the antenna pattern due to the excessive spacing of the original antenna array. However, after the array is extended by using the scheme proposed in this paper, the antenna spacing is greatly reduced, and the grating lobes appearing in the antenna pattern are eliminated, so the direction of the signal in the beam and the Doppler can be clearly distinguished.

Simulation test 4: The number of extended array elements exceeds the actual number of array elements

Assume that N=8 omnidirectional array elements are evenly distributed on a straight line with a length of λ, and the incident wave signal is the same as that of the simulation experiment 1, except that the signal-to-noise ratio of each signal is 20dB. Typical simulation results are as follows: Figure 39 shows the echo Doppler spectrum with the beam pointing at 120° before array expansion. Now expand the original array into a 12-element array with an antenna spacing of d=0.5λ. Figure 40 shows the echo spectrum after the array expands and the beam points to 120°. A large number of random mode experiments show that before the array expansion, the original array cannot distinguish several signal directions in a beam, but after expanding the array with the scheme proposed in this paper, the direction of each signal and Doppler can be clearly distinguished. Further analysis shows that the beam width of the antenna array is narrower than that of the 8 uniform linear array at this time, but the result is more affected by the signal-to-noise ratio, and the higher the signal-to-noise ratio, the better the effect.

Simulation experiment 5: non-uniform array aperture expansion

It is assumed that N=8 omnidirectional array elements are randomly distributed on a straight line with a length of λ (or uniformly distributed on a semicircle with a radius of r=0.5λ). The incident wave signal is the same as the simulation experiment 1. Now expand the original array into an array whose antenna spacing is d=0.5λ. A large number of random simulation experiments show that before the array is expanded, the original array cannot distinguish several signal directions in a beam, but after the array is expanded by using the technical solution of the present invention, the direction of each signal and Doppler can be clearly distinguished.

Simulation Experiment 6: High Resolution Algorithm of Extended Array

It is assumed that N=8 omnidirectional array elements are uniformly distributed on a straight line with a length of 5λ. The incident wave signal is the same as the simulation test 1, and the signal-to-noise ratio is 25dB. Now expand the original array into an array whose antenna spacing is d=0.5λ. Figure 41 shows the super-resolution spatial spectrum obtained using the MUSIC algorithm before array expansion. Figure 42 shows the super-resolution spatial spectrum obtained by using the MUSIC algorithm after array expansion. From the results, it can be seen that before the expansion of the antenna array, the direction finding ambiguity of the antenna array appears, but the method proposed in this paper is used to expand the original array, the resolution of the antenna is not only high, but also the direction finding ambiguity is eliminated.

5. Discussion

The aperture expansion method based on the small antenna array of the present invention has the following characteristics:

(a) The transformation process does not need to know the exact direction of the signal, only the initial incident angle of the signal is assumed, and the array manifold matrix of the original array and the extended array can be obtained, and then the transformation matrix of the array can be realized. In practical applications, the transformation matrix T can be stored in advance, and the expansion of the array can be realized.

(b) The general array antenna spacing is required to be less than half a wavelength. If it exceeds half a wavelength, direction finding ambiguity will appear. However, through array expansion transformation, the non-equidistant antenna array can be transformed into an antenna array whose element spacing is less than half a wavelength. In this way The signal ambiguity of the array can be resolved.

(c) The flexibility of the array is increased. Since the number of array elements (degrees of freedom) is increased by the expansion array, the number of digital array elements of the expansion array can be flexibly selected according to the actual environment when performing DOA estimation. For example, when the number of signals is relatively small, it can be Choose an extended array with a small aperture, and increase the number of array elements to increase the aperture of the antenna array for occasions that require high precision.

(d) Since the expanded array can have more array elements than the actual array, the expanded array can estimate more signal sources.

(e) The number of extended array elements cannot be infinite, and the selection of the number of array elements can only be valid within a certain range. This is because: in terms of energy, the energy and information received by the antenna are conserved, any transformation in the mathematical sense can only improve the traditional algorithm to a certain extent, and will not increase the energy and useful information; , the number of array elements increases, the condition number of the extended transformation matrix becomes larger, and the error of reconstructing the amplitude and phase of the extended array antenna elements becomes larger, which will eventually bring greater errors to the results. Moreover, as the number of extended array elements increases, in order to obtain engineering-usable results, the requirements for the signal-to-noise ratio will increase greatly. In practical applications, the number of extended array elements should be compromised among factors such as the signal-to-noise ratio, the number of actual signals, and the condition number of the extended transformation matrix.

(f) Due to the smaller aperture of the small array array, it is beneficial to reduce the calibration distance of the antenna array.

(g) Combining the invention with modern high-resolution algorithms, array space super-resolution direction finding can be realized.

Claims (3)

1. A method for aperture expansion and spatial signal processing of a small antenna array is characterized in that: setting the antenna array into any array form; dividing the whole antenna scanning area into a plurality of areas, and then subdividing any one of the sub-areas; constructing the actual array manifold a ═ a (θ) of the antenna array in subdivided sub-regions₁)，a(θ₁+Δθ)，a(θ₂+2Δθ)，...，a(θ_r)]And an array manifold a ═ a (θ) of the extended array₁)， a(θ₁+Δθ)， a(θ₂+2Δθ)，...， a(θ_r)]And further obtaining an array expansion transformation matrix B ═ AA^-1Wherein, theta₁，θ_rThe left and right boundaries of the sub-region are respectively, and delta theta is the step length of sub-region subdivision; reconstructing a signal Z ═ BX of an extended array antenna by using the obtained array extended transformation matrix, wherein X is a signal received by the original antenna array; and performing digital beam forming or adaptive beam forming on the reconstructed signal Z of the extended array antenna or solving the spatial arrival angle of the signal by using a spatial high-resolution algorithm.

2. The method of claim 1, wherein: and when the actual matrix manifold A is a non-square matrix, carrying out diagonal loading on the matrix A.

3. The method according to claim 1 or 2, characterized in that: the number of cells of the extended array is determined using a norm minimum of | a-BA |.

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