Conventional beam response in the desired direction of continuous circular array-microphone array series (10)

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Previous Article " continuous annular array beam patterns uniform weighting " directly setting the weighting coefficients $w^*_a=1$ , i.e. target direction $\Omega=\left( 0^\circ,0^\circ \right)$ , and when we want their own desired observation direction $\ Omega_o$ setting is required so that the $\ Omega_o$ response function of the direction of the signal sent by As a beam steering function, the function for performing conventional beam weighting can be expressed as:

$w ^ * _ a (\ vartheta) = p ^ * _ {\ vartheta} \ left (\ Omega_o \ right) = e ^ {- ikrsin \ phi_ocos (\ vartheta - \ theta_o)}$ 。

The main purpose of this article is to observe the changing trend of the beam response on the $hey$ plane and the $xoz$ plane with the direction and frequency in any desired observation direction .

1. The beam response in the desired viewing direction changes with the elevation angle $\phi$ ;

2. Expect the beam response in the viewing direction to change with frequency $f$ .

The coordinate system used in the previous article is still used:

1. The beam response in the desired viewing direction changes with the pitch angle $\phi$

Consider a $hey$ continuous circular ring array lying in a plane, suppose $kr = 2 \ pi$ , and observe its regular delay summing beam response.

The observation directions are respectively $\Omega_o=\left( \theta_o,\phi_o\right)=\left( 0^\circ,30^\circ \right),\left( 0^\circ,60^\circ \right),\left( 0^\circ,90^\circ \right)$ , using the following formula

$B (kr, \ Omega) = \ frac {1} {2 \ pi} \ int_ {0} ^ {2 \ pi} e ^ {i \ rho krcos (\ vartheta - \ beta)} d \ vartheta = J_0 ( \ rho kr)$

$\rho=\sqrt{(sin\phi-sin\phi_o)^2+4sin\phi sin\phi_osin^2\left[\left( \theta-\theta_o \right)/2 \right]}$

Calculate the delay and sum beam response, shown in Figure 1, 2, and 3 respectively. It can be seen from the figure that the main lobe of the beam points to the viewing direction, and the beam response is $hey$ mirror-symmetrical with respect to the plane.

figure 1

figure 2

image 3

Figures 1, 2, 3 and the $\ phi_o = 0 ^ \ circ$ cross- $xoz$ sectional views of the four beam responses in the plane are shown in Figure 4. It can be seen from the figure that as the elevation angle of the desired beam viewing direction changes $\ phi_o$ from $0 ^ \ circ$ to $90 ^ \ circ$ , $\phi$ the main lobe width of the beam along the direction gradually becomes wider. Until the two main lobes above and below the plane of the ring are connected together.

Figure 4

Then observe the beam response on the horizontal plane when performing conventional beamforming on the horizontal plane, that is $\ phi = \ phi_o = 90 ^ \ circ$ . Assuming that the desired beam viewing direction is $\theta=\theta_o=0^\circ$ , the above formula is still used to calculate the beam response, as shown in Figure 5, which is also $hey$ a cross-sectional view of the beam response in the plane shown in Figure 3 .

Figure 5

2. Expect the beam response in the viewing direction to change with frequency

Next, we will examine the variation of the conventional beam response of the circular array with frequency $kr=\frac{\omega}{c}r=f\left( \frac{2\pi r}{c}\right)$ . If there is no special description later, we generally only examine the beam response in the plane where the ring lies, that is, hypothesis $\ phi = \ phi_o = 90 ^ \ circ$ .

Assuming the wavenumber radius product range $kr\in[0,10]$ and the horizontal angle value range $\theta\in[-180^\circ,180^\circ]$ , the horizontal plane beam response graph obtained by calculating the horizontal plane conventional beamforming using the formula described above is shown in Figure 6, where Figure 6(a) is the color after the beam response amplitude is taken as the logarithm Fig. 6(b) shows the cylindrical coordinates of the beam response amplitude.

It can be seen from Fig. 6 that $\ theta_o = 0 ^ \ circ ， kr = 0$ when the main lobe of the conventional beam pattern on the toroidal surface of the continuous annular array is pointed , the beam response is a unit circle, that is, there is no directivity. As the frequency increases, the main lobe of the beam gradually narrows.

Figure 6(a)

Figure 6(b)

For a clearer observation, the figure $7(a\sim d)$ shows the $kr = 2,4,6,8$ polar coordinate display of the corresponding beam response after taking the logarithm. It can be seen that the trend characteristic of the main lobe changes with the frequency is more significant at this time.

Figure 7 (a)

Figure 7 (b)

Figure 7 (c)

Figure 7 (d)

Reference books:

"Optimizing Array Signal Processing", Yan Shefeng

Conventional beam response in the desired direction of continuous circular array-microphone array series (10)

Read the original text, please move to my knowledge column: https://www.zhihu.com/column/c_1287066237843951616

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Read the original text, please move to my knowledge column:
https://www.zhihu.com/column/c_1287066237843951616