Continuous circular array uniformly weighted beam pattern—microphone array series (9)

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The content includes the following:

1. Derivation of continuous ring array beamformer;

2. Observe the influence of continuous circular array beam response, wave number radius product $kr$ and vertical angle $\phi$ on the beam;

3. A kind of Bessel function characteristics.

1. Derivation of continuous ring array beamformer

Regarding the continuous ring array, consider a continuous ring array with a radius of φ $r$ , and place it on the $hey$ plane, with the center of the ring as the coordinate origin, as shown in Figure 1.

figure 1

$P _ {vartheta$ The array manifold function of each receiving point on the continuous circular array can be expressed as:

$p_ \ vartheta (\ bold k) = e ^ {- i \ bold k ^ TP_ \ vartheta} = e ^ {ikrsin \ phi cos \ left (\ vartheta- \ theta \ right)}$

$P _ {vartheta$ The point polar coordinate form is $\left( r, \vartheta \right)$ , the rectangular coordinate form is $\left[ rcos\vartheta,rsin\vartheta,0 \right]^T$ ,

$\bold k=-k\left[ sin\phi cos\theta,sin\phi sin\theta,cos\phi \right]^T,k=\omega/c$

Assuming that $P _ {vartheta$ the weighting function of the point is taken $w^*_a\left( \vartheta \right)$ , the beam response is: $\ begin {align} \\ & B (kr, me Omega) = \ frac {1} {2 \ pi} \ int_ {2 \ pi} ^ {0} w ^ * _ a \ left (\ vartheta \ right) e {ikrsin \ phi cos (\ vartheta - \ theta)} d \ vartheta \\ & \ quad \ quad \ quad \ \ \ = J_0 (krsin \ phi) \ end {align$

The form is a kind of 0-order Bessel function. The third section " Characteristics of a kind of Bessel function " describes this function in detail and studies its characteristics.

2. Observe the influence of continuous circular array beam response, wave number radius product $kr$ and vertical angle $\phi$ on the beam

Consider a continuous circular array and calculate the beam response obtained when uniform weighting is used.

Assuming that the wave number radius product is $kr = 2 \ pi$ , let $\theta\in[0^\circ,360^\circ],\phi\in[0^\circ,180^\circ]$ , using the above beam response calculation formula, its amplitude is shown in Fig. 2 in three-dimensional coordinates.

Figure 2 Three-dimensional beam response

It can be seen from Fig. 2 that the beam response obtained by uniform weighting $with$ is rotationally symmetric with respect to the axis, that is, the beam response is only related to the vertical angle $\phi$ , and $\theta$ has nothing to do with the horizontal angle . Therefore, we only need to draw the relationship between the uniformly weighted beam pattern and the vertical angle below.

Assuming the wavenumber radius product range $kr\in[0,10]$ and the vertical angle value range $\phi\in[0^\circ,180^\circ]$ , the beam response calculated by the beam response calculation formula relative to the wavenumber radius product $kr$ and the vertical angle $\phi$ is shown in Figure 3, where the beam response amplitude in Figure 3 is the color after the logarithm Figure 4 shows the beam response amplitude cylindrical coordinate display.

image 3

Figure 4

In order to better observe the beam response, Figure 6 complements the beam response in the $with$ entire $360 ^ \ circ$ range on the plane perpendicular to the ring array (the plane where the axis lies) . Figure 6 shows the $kr = 2,4,6,8$ corresponding beam response polar coordinate display diagrams in the four diagrams.

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

It can be seen from Figures 3, 4, 5 and 6, that the uniformly weighted circular ring array obtains the main lobe in the direction perpendicular to the circular ring, that is, the beam main lobe direction $\ phi = 0 ^ \ circ$ and $\ phi = 180 ^ \ circ$ direction. $kr = 0$ When 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.

3. Features of a kind of Bessel function

The first- $n$ order Bessel function is defined as:

$J_n(z)=\frac{1}{2\pi i^n}\int_{0}^{2\pi}e^{i(zcos\psi + n\psi)}d\psi,\quad n=0,\pm1,\pm2,...$

$J _ {- n} (z) = (- 1) ^ nJ_n (z)$

Figure 7 shows a graph of the $0\sim 4$ order Bessel function. It can be seen from the figure that as the order $n$ increases, the maximum amplitude of the Bessel function $max\left| J_n(z) \right|$ gradually decreases.

Figure 7

It has been deduced above that the beam response of a uniformly weighted continuous circular array is a 0-order Bessel function, which corresponds to the beam response of a continuous linear array. As mentioned $sinc$ in the previous article, $x$ the The beam response of a continuous linear array of length on the plane is:

$B(kr,\Omega)=sinc\left( k\frac{2}{L}sin\phi \right)$

This shows that when the length of the linear array is equal to the $L$ diameter of the circular ring array $2r$ , the beam response $sinc$ arguments of the two continuous arrays are equal, the former is a function, and the latter is a 0-order Bessel function.

Figure 8 shows $J_0(z)$ the $sinc(z)$ two variables in function with respect to the $with$ comparative value of FIG. It can be seen from the figure that the $z = 0$ two functions are both 1, which indicates that the main lobe response of the beam is 1, ie $0dB$ . Except for the $z = 0$ point, the peak and valley amplitudes of the 0-order Bessel function are larger than the $sinc$ peak and valley amplitudes of the function, which indicates that the beam sidelobes of the continuous circular array are higher than the continuous linear array. However, the width of the main lobe of the continuous circular array is narrower than that of the continuous linear array.

Figure 8

Reference books:

"Optimizing Array Signal Processing", Yan Shefeng