Uniform rectangular array beam pattern—microphone array series (eight)

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This article includes: 1. Beam pattern product theorem; 2. Beam pattern of uniform rectangular base array.

If you want to see a more detailed explanation, you can directly read the book of Teacher Yan Shefeng.

Beam pattern product theorem

The previous calculation of the popularity vector of the basic array is based on the fact that each array element is isotropic, that is, the array element has the same response to signals arriving in all directions. At this time, we assume that the array element is not isotropic, but has a certain directionality. This is the case with the continuous matrix as shown in Figure 1.

Figure 1 Array composed of non-isotropic array elements

Suppose $m$ the direction of the array element number (array element in this case is a separate line array) as the response $p_m\left( \Omega \right)$ , if $\bold p\left( \Omega \right)$ represented by the $M$ direction of the response of the array elements of the vector composition,

Namely: $\bold p\left( \Omega \right)=\left[ p_1\left( \Omega \right),...,p_m\left( \Omega \right),...,p_M\left( \Omega \right)\right]^T$ ,

Then the real array manifold of the base array can be expressed as: $\widetilde {\bold p}\left( \Omega \right)=\bar {\bold p}\left( \Omega \right)\circ \bold p\left( \Omega \right)$

Among them, $\circus$ represents the $\widetilde {\bold p}\left( \Omega \right)=\bar {\bold p}\left( \Omega \right)\circ \bold p\left( \Omega \right)$ product, that is, the dot product; $\bar {\bold p}\left( \Omega \right)$ is the array manifold vector of the basic matrix composed of isotropic array elements calculated by the following formula:

$\bold p(\bold k)=\left[ e^{-i\bold k^TP_1}, e^{-i\bold k^TP_2}, ...e^{-i\bold k^TP_M}, \right]^T$

among them:

$\bold k\left( \Omega \right)=-k\left[ sin\phi cos\theta,sin\phi sin\theta,cos\phi \right]^T,k=\omega/c$ The beam vector corresponding to the direction;

$P_m=\left[ P_{xm},P_{ym},P_{zm} \right]^T,m=1,...,M$ Is the $m$ position coordinate vector of the number array element.

If all the array elements have the same directional response, that is $p_m\left( \Omega \right)=p\left( \Omega \right),m=1,...,M$ , the beam response of the base array can be expressed as:

$\widetilde {B}\left( \Omega \right)=\bold w^H\widetilde {\bold p}\left( \Omega \right)=\bold w^H \bar {\bold p}\left( \Omega \right)p\left( \Omega \right)=\bar {B}\left(\Omega \right)p\left( \Omega \right)$

It can be seen from this formula that when the elements that make up the base array have directivity and all the elements have the same directivity, the beam response of the base array is equal to the beam response of the corresponding isotropic array element and the element The product of the directional response is the beam pattern product theorem .

Uniform rectangular array beam pattern

For the row shown in the figure 5 2 6 row 30-membered rectangular base array, array element spacing is assumed $d_x=d_y=\lambda/2$ . Observe the uniform rectangular array beam pattern with uniform weighting.

Figure 2 Rectangular array

Assuming that the rectangular base isotropic array element array, a total $\breve {M}$ row $M$ column $M\times\breve {M}$ array element, which is disposed in the $hey$ plane of array center at the origin. Then $\left( \breve m,m \right)$ the receiving response at the array element is:

$p_{m \breve m}=e^{-i\bold k^TP_{m \breve m}}=e^{-i\left( k_xx_m+k_yy_{\breve m} \right)}$

Among them, $P_{m \breve m}$ is the coordinate position. Let $\left( \breve m,m \right)$ the weighted value of the array element $w^*_{m \breve m}$ be expressed as $w^*_{m \breve m}=w^*_mw^*_{\breve m}$ . So for this base array, the following formula is used to calculate the beam response in all directions:

$B\left(\Omega \right)=B_x\left( \Omega \right)B_y\left( \Omega \right)$

Among them $B_x\left( \Omega \right)=\sum_{m=1}^{M}{w^*_me^{-ik_xx_m}}=\frac{sin\left( MK_xd_x/2 \right)}{Msin\left( K_xd_x/2 \right)}$ , $B_y\left( \Omega \right)=\sum_{m=1}^{M}{w^*_me^{-ik_yy_m}}=\frac{sin\left( \breve MK_yd_y/2 \right)}{\breve Msin\left( K_yd_y/2 \right)}$

Set the vertical azimuth angle $0 ^ \ circ$ , the amplitude response is shown in visible main lobe direction perpendicular to the rectangular plane shown in FIG.

Figure 3 Uniformly weighted rectangular array beam diagram

If the main lobe direction of the beam needs to point to other directions, it can be achieved by using a method similar to the linear array beam steering described earlier. For example, at this time, the vertical azimuth angle is set to $20 ^ \ circ$ , and the beam pattern is shown in Figure 4.

Figure 4 Uniformly weighted rectangular array beam diagram

Reference books:

"Optimizing Array Signal Processing", Yan Shefeng

Uniform rectangular array beam pattern—microphone array series (eight)

Beam pattern product theorem

Uniform rectangular array beam pattern

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