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josh 85e2b1bb85 update several posts, add matlab code to figs

2023-06-20 19:17:57 +08:00

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1. 均匀加权

均匀权值为

$$\begin{equation} w_n = \frac{1}{N}, \quad n = 0,\cdots,N-1 \end{equation}$$

在 u 空间的波束方向图为

$$\begin{equation} \begin{aligned} B_u(u) &= \frac{1}{N} \frac{\sin\left( \displaystyle\frac{\pi Nd}{\lambda} u \right)}{\sin\left( \displaystyle\frac{\pi d}{\lambda} u \right)} \ &= \boldsymbol{w}^\mathrm{H}\boldsymbol{v}u(u) = e^{-j\left(\frac{N-1}{2} \right) \frac{2\pi d}{\lambda} u} \sum{n=0}^{N-1} w_n^\ast e^{j n \frac{2\pi d}{\lambda} u} \end{aligned} \end{equation}$$

对于标准线阵，有

$$\begin{equation} B_u(u) = \frac{1}{N} \frac{\sin\left( \displaystyle\frac{\pi Nu}{2} \right)}{\sin\left( \displaystyle\frac{\pi u}{2} \right)} \end{equation}$$

方向性

$$\begin{equation} D = N \end{equation}$$

由于下面经考虑实对称的权值，因此将阵元的位置也对称地进行表示，第 n 个阵元的位置表示为

$$\begin{equation} \tilde{n} = n - \frac{N-1}{2}, \quad n = 0,\cdots,N-1,\quad \tilde{n} = -\frac{N-1}{2}, \cdots, \frac{N-1}{2} \end{equation}$$

2. Cosine 加权

当 N 为奇数时，Cosine 权值为

$$\begin{equation} w(\tilde{n}) = \sin\left( \frac{\pi}{2N} \right)\cos\left( \pi\frac{\tilde{n}}{N} \right), \quad -\frac{N-1}{2} \leqslant \tilde{n} \leqslant \frac{N-1}{2} \end{equation}$$

其中归一化因子 \sin\left( \displaystyle\frac{\pi}{2N} \right) 是一个常数，使得 $B_u (0) = 1$。把 Cosine 写成指数形式，有

$$\begin{equation} w(\tilde{n}) = \sin\left( \frac{\pi}{2N} \right)\left[ \frac{e^{j\frac{\pi\tilde{n}}{N}} + e^{-j\frac{\pi\tilde{n}}{N}}}{2} \right], \quad -\frac{N-1}{2} \leqslant \tilde{n} \leqslant \frac{N-1}{2} \end{equation}$$

利用指数形式的阵列流形矢量，可得波束方向图

$$\begin{equation} B_u(u) = \frac{1}{2} \sin\left( \frac{\pi}{2N} \right) \left{ \sum_{\tilde{n} = -\frac{N-1}{2}}^{\frac{N-1}{2}} e^{j\frac{\pi \tilde{n}}{N}} e^{-j \tilde{n} \pi u} + \sum_{\tilde{n} = -\frac{N-1}{2}}^{\frac{N-1}{2}} e^{-j\frac{\pi \tilde{n}}{N}} e^{-j \tilde{n} \pi u} \right} \end{equation}$$

其中第一项和第二项分别对应指向 u_s = 1/N 和 u_s = -1/N 的常规波束方向图，因此

$$\begin{equation} B_u(u) = \frac{1}{2} \sin\left( \frac{\pi}{2N} \right) \left{ \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u - \frac{1}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u - \frac{1}{N} \right) \right)} + \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u + \frac{1}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u + \frac{1}{N} \right) \right)} \right} \end{equation}$$

下面将要介绍的很多方向图都是常规波束的叠加。图 2-1-1 给出了一个 11 阵元阵列使用 Cosine 加权得到的波束方向图，同时还给出了常规波束方向图作为参考。可以看到，旁瓣得到了降低，但是主波束变得更宽。两个波束方向图的参数为

加权方式	HPBW	`BW_{NN}`	第一旁瓣高度	$D_N$¹
Uniform	`0.89 \displaystyle\frac{2}{N}`	`2.0 \displaystyle\frac{2}{N}`	`-13.0 \text{dB}`	`1`
Cosine	`1.18 \displaystyle\frac{2}{N}`	`3.0 \displaystyle\frac{2}{N}`	`-23.5 \text{dB}`	`0.816`

{width=700px}

{% note primary MATLAB Code %}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figure 3.1
% Cosine weighting: Array beam patterns
% Kristine Bell
% Modified by Xin Zhang 3/17/99
% Lillian 04/16/2001, K. Bell 7/20/01, K. Bell 9/30/01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all
close all

M=11;
d=0.5;                        % sensor spacing wrt wavelength
D = (-(M-1)/2:1:(M-1)/2)*d;   % sensor positions in wavelengths
% weights, normalized so that w(0)=1
W_unf = ones(1,M);
W_cos = cos(pi*D*2/M);

% Beampatterns
u = 0:0.001:1;
A = exp(-1i*2*pi*D.'*u);
G_unf = W_unf*A;
G_unf = G_unf/(max(abs(G_unf)));
G_cos = W_cos*A;
G_cos = G_cos/(max(abs(G_cos)));
figure
% array only
clf
plot(u,20*log10(abs(G_unf)),'--')
hold on
plot(u,20*log10(abs(G_cos)),'-')
hold off
axis([0 1 -80 0])
%title([int2str(M) ' element array'])
h=legend('Uniform','Cosine');
xlabel('$u$','Interpreter','latex')
ylabel('Beam pattern (dB)')

{% endnote %}

3. 升 Cosine 加权

可以把矩形均匀加权和 Cosine 加权组合起来，获得两种加权方法一些理想性质的组合。对应的阵列加权为

$$\begin{equation} w(\tilde{n}) = c(p) \left( p + (1-p) \cos \left( \pi \frac{\tilde{n}}{N} \right)\right), \quad \tilde{n} = -\frac{N-1}{2}, \cdots, \frac{N-1}{2} \end{equation}$$

其中归一化常数

$$\begin{equation} c(p) = \frac{p}{N} + \frac{1-p}{2} \sin\left( \frac{\pi}{2N} \right) \end{equation}$$

使得 $B_u(0) = 1$。

在图 2.1.2 中给出了 p = 0.31, 0.17, 0 时的升 Cosine 加权的波束方向图。当 p 减小时，第一旁瓣的高度减小，主波束宽度增加。波束方向图参数如下：

`p`	HPBW	`BW_{NN}`	第一旁瓣高度	`D_N`
0.31	`1.03 \displaystyle\frac{2}{N}`	`2.50 \displaystyle\frac{2}{N}`	`-20.0 \text{dB}`	`0.928`
0.17	`1.09 \displaystyle\frac{2}{N}`	`3.70 \displaystyle\frac{2}{N}`	`-22.0 \text{dB}`	`0.886`
0	`1.18 \displaystyle\frac{2}{N}`	`3.00 \displaystyle\frac{2}{N}`	`-23.5 \text{dB}`	`0.816`

{width=700px}

{% note primary MATLAB Code %}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figure 3.2
% Raised cosine weighting function
% Kristine Bell
% Modified by Xin Zhang 3/17/99
% Lillian Xu 04/16/2001, K. Bell 7/20/01, 9/30/01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all
close all

M=11;
d=0.5;                        % sensor spacing wrt wavelength
D = (-(M-1)/2:1:(M-1)/2)*d;   % sensor positions in wavelengths

% weights, normalized so that w(0)=1
p=[0;0.17;0.31;1];
W_rcos = p*ones(1,M)+(ones(4,1)-p)*cos(pi*D*2/M);

% Beampatterns
u = 0:0.001:1;
c=p+(ones(4,1)-p)*2/pi;
A = exp(-1i*2*pi*D.'*u);
G_rcos = W_rcos*A;
for m=1:4
 G_rcos(m,:) = G_rcos(m,:)/(max(abs(G_rcos(m,:))));
end

figure
clf
plot(u,20*log10(abs(G_rcos(3,:))),'-.')
hold on
plot(u,20*log10(abs(G_rcos(2,:))),'--')
plot(u,20*log10(abs(G_rcos(1,:))),'-')
hold off
axis([0 1 -80 0])
h=legend({'$p=0.31$','$p=0.17$','$p=0$'},'Interpreter','latex');
xlabel('$u$','Interpreter','latex')
ylabel('Beam pattern (dB)')

{% endnote %}

至此，我们已经可以使 HPBW 变窄，并使得第一旁瓣比均匀分布的情况要低得多。

4. Cosine$^m$ 加权

下面考虑一系列 Cosine 加权，形式为 $\cos^m(\pi \tilde{n} / N)$。阵列的权值为

$$\begin{equation} w_m(\tilde{n}) = \begin{cases} c_2 \cos^2 \left( \displaystyle \frac{\pi \tilde{n}}{N} \right),\quad m = 2 \ c_3 \cos^3 \left( \displaystyle \frac{\pi \tilde{n}}{N} \right),\quad m = 3 \ c_4 \cos^4 \left( \displaystyle \frac{\pi \tilde{n}}{N} \right),\quad m = 4 \ \end{cases} \end{equation}$$

其中 c_2, c_3, c_4 是归一化常数。

$m=2$ 时的加权有时称为是 Hann 加权²。同样，波束方向图可以利用 Cosine 函数的指数形式来计算。波束方向图在图 2.1.3 中给出。当 m 增加时，旁瓣减小，但是主波束变宽。波束方向图的参数为:

`m`	HPBW	`BW_{NN}`	第一旁瓣高度	`D_N`
2	`1.44 \displaystyle\frac{2}{N}`	`4 \displaystyle\frac{2}{N}`	`-31.4 \text{dB}`	`0.667`
3	`1.66 \displaystyle\frac{2}{N}`	`5 \displaystyle\frac{2}{N}`	`-39.4 \text{dB}`	`0.576`
4	`1.85 \displaystyle\frac{2}{N}`	`6 \displaystyle\frac{2}{N}`	`-46.7 \text{dB}`	`0.514`

{width=700px}

{% note primary MATLAB Code %}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figure 3.3
% Cosine^m weighting: m = 2,3,and 4
% Kristine Bell
% Modified by Xin Zhang 3/17/99
% Lillian Xu 04/16/2001, K. Bell 7/20/01, 9/30/01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

clear all
close all

M=11;
d=0.5;                        % sensor spacing wrt wavelength
D = (-(M-1)/2:1:(M-1)/2)*d;   % sensor positions in wavelengths

% weights, normalized so that w(0)=1
W_cos2 = cos(pi*D*2/M).^2;
W_cos3 = cos(pi*D*2/M).^3;
W_cos4 = cos(pi*D*2/M).^4;

% Beampatterns

u = 0:0.001:1;
A = exp(-1i*2*pi*D.'*u);
G_cos2 = W_cos2*A;
G_cos2 = G_cos2/(max(abs(G_cos2)));
G_cos3 = W_cos3*A;
G_cos3 = G_cos3/(max(abs(G_cos3)));
G_cos4 = W_cos4*A;
G_cos4 = G_cos4/(max(abs(G_cos4)));

figure
clf
plot(u,20*log10(abs(G_cos2)),'--')
hold on
plot(u,20*log10(abs(G_cos3)),'-')
plot(u,20*log10(abs(G_cos4)),'-.')
hold off
axis([0 1 -80 0])
h=legend({'$m=2$','$m=3$','$m=4$'},'Interpreter','latex');
set(h,'Fontsize',12)
xlabel('$u$','Interpreter','latex')
ylabel('Beam pattern (dB)')

{% endnote %}

5. 升 Cosine 平方加权

升 Cosine 平方系列的加权为

$$\begin{equation} \begin{aligned} w(\tilde{n}) &= c_2 (p) \left{ p + (1-p) \cos^2 \left( \frac{\pi \tilde{n}}{N} \right) \right} \ &= \frac{c_2(p)}{2} \left{ (1+p) + (1-p) \cos \left( \frac{2 \pi \tilde{n}}{N} \right) \right}, \quad \tilde{n} = -\frac{N-1}{2}, \cdots, \frac{N-1}{2} \end{aligned} \end{equation}$$

其中 c_2 (p) 是归一化常数。下面考虑一个特别的加权，称为 Hamming 加权。

6. Hamming 加权

Hamming 加权利用了矩形方向图和 Cosine 平方方向图的特点，在第一旁瓣的峰值上放置了一个零点。加权函数为

$$\begin{equation} w(\tilde{n}) = g_0 + g_1 \cos \left( \frac{2 \pi \tilde{n}}{N} \right), \quad \tilde{n} = -\frac{N-1}{2}, \cdots, \frac{N-1}{2} \end{equation}$$

选择系数 g_0 和 g_1 的使 u = 3/N 的位置上有一个零点，并将正侧向的响应归一化，有

$$\begin{equation} w(\tilde{n}) = 0.54 + 0.46 \cos \left( \frac{2 \pi \tilde{n}}{N} \right), \quad -\frac{N-1}{2} \leqslant \tilde{n} \leqslant \frac{N-1}{2} \end{equation}$$

对应升 Cosine 平方加权中 p = 0.08 的情况，得到的波束方向图是三个常规波束方向图的和

$$\begin{equation} B_u(u) = 0.54 \frac{\sin\left(\displaystyle\frac{N\pi u}{2}\right)}{\sin\left(\displaystyle\frac{\pi u}{2}\right)} + 0.23 \left[ \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u - \frac{2}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u - \frac{2}{N} \right) \right)} + \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u + \frac{2}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u + \frac{2}{N} \right) \right)} \right] \end{equation}$$

图 2.1.4 给出了 Hamming 加权、Hann 加权和 Blackman-Harris 加权的波束方向图。第一个零点的位置在 $u = 4/N$，第一个非零旁瓣的高度为 $-39.5\text{dB}$。由于加权具有步进不连续性（step discontinuity），旁瓣的衰减渐近为 $O(1/n)$。但是，第一个旁瓣被完全对消掉了，尽管剩余的旁瓣衰减得比较慢，它们的幅度都不是很高。而且注意到 Hamming 窗的波束宽度小于 Hann 加权的波束宽度，且第一旁瓣更低。

{width=700px}

{% note primary MATLAB Code %}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figure 3.4
% Beam patterns for Hann, Hamming, and Blackman-Harris
% Kristine Bell
% Modified by Xin Zhang 3/17/99
% Lillian Xu 04/16/2001, K. Bell 7/20/01, 9/30/01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Hann, Hamming, Blackman-Harris
% M-element array, d=lambda/2

clear all
close all

M=11;
d=0.5;                        % sensor spacing wrt wavelength
D = (-(M-1)/2:1:(M-1)/2)*d;   % sensor positions in wavelengths

% weights, normalized so that w(0)=1
W_Hann = 0.5*(ones(1,M)+cos(2*pi*D*2/M));
W_Hamm = 0.54*ones(1,M)+0.46*cos(2*pi*D*2/M);
W_BH   = 0.42*ones(1,M)+0.5*cos(2*pi*D*2/M)+0.08*cos(4*pi*D*2/M);

% Beampatterns

u = 0:0.001:1;
A = exp(-1i*2*pi*D.'*u);
G_Hann = W_Hann*A;
G_Hann = G_Hann/(max(abs(G_Hann)));
G_Hamm = W_Hamm*A;
G_Hamm = G_Hamm/(max(abs(G_Hamm)));
G_BH = W_BH*A;
G_BH = G_BH/(max(abs(G_BH)));

figure
clf
plot(u,20*log10(abs(G_Hann)),'--')
hold on
plot(u,20*log10(abs(G_Hamm)),'-')
plot(u,20*log10(abs(G_BH)),'-.')
hold off
axis([0 1 -80 0])
h=legend('Hann','Hamming','Blackman-Harris');
xlabel('$u$','Interpreter','latex')
ylabel('Beam pattern (dB)')

{% endnote %}

7. Blackman-Harris 加权

Blackman-Harris 加权把加权过程扩展到了更高阶谐波的情况，并在前两个旁瓣的峰值位置上放置了零点。加权函数为

$$\begin{equation} w(\tilde{n}) = 0.42 + 0.5 \cos\left( \frac{2\pi \tilde{n}}{N} \right) + 0.08 \cos\left( \frac{4\pi \tilde{n}}{N} \right), \quad \tilde{n} = -\frac{N-1}{2}, \cdots, \frac{N-1}{2} \end{equation}$$

对应的波束方向图是多个常规波束方向图的和，已在图 2.1.4 中给出

$$\begin{equation} \begin{aligned} B_u(u) =& 0.42 \frac{\sin\left( \displaystyle\frac{N\pi u}{2} \right)}{\sin\left( \displaystyle\frac{\pi u}{2} \right)} + 0.25 \left[ \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u - \frac{2}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u - \frac{2}{N} \right) \right)} + \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u + \frac{2}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u + \frac{2}{N} \right) \right)} \right] \ &+ 0.04 \left[ \frac{\sin\left( \displaystyle\frac{N\pi}{2} \left( u - \frac{2}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{2} \left( u - \frac{2}{N} \right) \right)} + \frac{\sin\left( \displaystyle\frac{N\pi}{4} \left( u + \frac{4}{N} \right) \right)}{\sin\left( \displaystyle\frac{\pi}{4} \left( u + \frac{4}{N} \right) \right)} \right] \end{aligned} \end{equation}$$

波束方向图的参数为

加权方式	HPBW	`BW_{NN}`	第一旁瓣高度	`D_N`
Hann	`1.44 \displaystyle\frac{2}{N}`	`4.0 \displaystyle\frac{2}{N}`	`-31.4 \text{dB}`	`0.667`
Hamming	`1.31 \displaystyle\frac{2}{N}`	`4.0 \displaystyle\frac{2}{N}`	`-39.5 \text{dB}`	`0.730`
Blackman-Harris	`1.65 \displaystyle\frac{2}{N}`	`6.0 \displaystyle\frac{2}{N}`	`-56.6 \text{dB}`	`0.577`

至此，所有的加权方法都是基于各种正弦函数进行的。现在看看其他类型的阵列加权技术，试试能否更好在波束宽度和旁瓣之间的获得折中。

8. 长球函数 (Prolate Spheroidal Functions)

在设计加权方案时，我们的目标是使得一个给定角度范围内的能量占总能量的比例最大，也即最大化如下比值

$$\begin{equation} \alpha = \frac{\displaystyle\iint_{\varOmega_1}\left| B(\theta,\varphi) \right|^2 \sin\theta,\mathrm{d}\theta,\mathrm{d}\varphi}{\displaystyle\int_0^\pi \int_0^{2\pi}\left| B(\theta,\varphi) \right|^2 \sin\theta,\mathrm{d}\varphi,\mathrm{d}\theta} \end{equation}$$

其中 \varOmega_1 是主波束周围的一个区域。对于标准线性阵列，可以写成

$$\begin{equation} \alpha = \frac{\displaystyle\int_{-\psi_0}^{\psi_0}\left| B(\psi) \right|^2,\mathrm{d}\psi}{\displaystyle\int_{-\pi}^{\pi}\left| B(\psi) \right|^2,\mathrm{d}\psi} \end{equation}$$

其分子和分母部分可分别写为

$$\begin{equation} \begin{aligned} &\alpha_N &&\xlongequal{B(\psi) = \boldsymbol{w}^\mathrm{H}\boldsymbol{v}\psi(\psi)} \int{-\psi_0}^{\psi_0}\boldsymbol{w}^\mathrm{H}\boldsymbol{v}\psi(\psi) \boldsymbol{v}\psi^\mathrm{H}(\psi)\boldsymbol{w},\mathrm{d}\psi \ & &&= \boldsymbol{w}^\mathrm{H} \left[ \int_{-\psi_0}^{\psi_0} \boldsymbol{v}\psi(\psi) \boldsymbol{v}\psi^\mathrm{H}(\psi) ,\mathrm{d}\psi \right]\boldsymbol{w} \ & &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w} \

&\alpha_D &&\xlongequal{B(\psi) = \boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi)}  \int_{-\pi}^{\pi}\boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi) \boldsymbol{v} \\
& &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w}

\end{aligned} \end{equation}$$

$$\begin{equation} \end{equation}$$

其中

$$\begin{equation} \begin{aligned} \boldsymbol{A} &\triangleq \int_{-\psi_0}^{\psi_0} \boldsymbol{v}\psi(\psi) \boldsymbol{v}\psi^\mathrm{H}(\psi) ,\mathrm{d}\psi \ \boldsymbol{B} &\triangleq \int_{-\pi}^{\pi} \boldsymbol{v}\psi(\psi) \boldsymbol{v}\psi^\mathrm{H}(\psi) ,\mathrm{d}\psi = 2\pi \boldsymbol{I} \end{aligned} \end{equation}$$

$$\begin{equation} [\boldsymbol{A}]{mn} = \int{-\psi_0}^{\psi_0} e^{jm\psi} e^{-jn\psi} ,\mathrm{d}\psi = \frac{2\sin ((m-n)\psi_0)}{m-n} = 2\psi_0 ,\mathrm{sinc}((m-n)\psi_0) \end{equation}$$

因此

$$\begin{equation} \alpha = \frac{\boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w}}{2\pi \boldsymbol{w}^\mathrm{H}\boldsymbol{w}} \end{equation}$$

参数 D_N 是阵列的归一化方向性。它是相对于一个均匀加权阵列的方向性进行归一化的。对于标准线阵, D_N 也是归一化的白噪声增益。 ↩︎
这个权值是奥地利气象学家 Von Hann 提出的，有时也称为 Hanning 加权。 ↩︎

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1. 均匀加权

2. Cosine 加权

3. 升 Cosine 加权

4. Cosine$^m$ 加权

5. 升 Cosine 平方加权

6. Hamming 加权

7. Blackman-Harris 加权

8. 长球函数 (Prolate Spheroidal Functions)

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