finish post josh-oap-2-1
This commit is contained in:
@@ -300,7 +300,6 @@ plot(u,20*log10(abs(G_cos4)),'-.')
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hold off
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axis([0 1 -80 0])
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h=legend({'$m=2$','$m=3$','$m=4$'},'Interpreter','latex');
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set(h,'Fontsize',12)
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xlabel('$u$','Interpreter','latex')
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ylabel('Beam pattern (dB)')
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```
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@@ -441,28 +440,21 @@ $$\begin{equation}
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其分子和分母部分可分别写为
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$$\begin{equation}
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\begin{aligned}
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&\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 \\
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& &&= \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} \\
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& &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w} \\
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$$\begin{align}
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&\alpha_N &&\xlongequal{B(\psi) = \boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi)} \int_{-\psi_0}^{\psi_0}\boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi)\boldsymbol{w}\,\mathrm{d}\psi \notag \\
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& &&= \boldsymbol{w}^\mathrm{H} \left[ \int_{-\psi_0}^{\psi_0} \boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi) \,\mathrm{d}\psi \right]\boldsymbol{w} \notag \\
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& &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w} \\
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&\alpha_D &&\xlongequal{B(\psi) = \boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi)} \int_{-\pi}^{\pi}\boldsymbol{w}^\mathrm{H}\boldsymbol{v}_\psi(\psi) \boldsymbol{v} \\
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& &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w}
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\end{aligned}
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\end{equation}$$
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$$\begin{equation}
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\end{equation}$$
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&\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} \notag \\
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& &&= \boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w}
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\end{align}$$
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其中
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$$\begin{equation}
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\begin{aligned}
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\boldsymbol{A} &\triangleq \int_{-\psi_0}^{\psi_0} \boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi) \,\mathrm{d}\psi \\
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\boldsymbol{B} &\triangleq \int_{-\pi}^{\pi} \boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi) \,\mathrm{d}\psi = 2\pi \boldsymbol{I}
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\end{aligned}
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\end{equation}$$
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$$\begin{align}
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\boldsymbol{A} &\triangleq \int_{-\psi_0}^{\psi_0} \boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi) \,\mathrm{d}\psi \\
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\boldsymbol{B} &\triangleq \int_{-\pi}^{\pi} \boldsymbol{v}_\psi(\psi) \boldsymbol{v}_\psi^\mathrm{H}(\psi) \,\mathrm{d}\psi = 2\pi \boldsymbol{I}
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\end{align}$$
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$$\begin{equation}
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[\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)
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@@ -474,6 +466,335 @@ $$\begin{equation}
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\alpha = \frac{\boldsymbol{w}^\mathrm{H}\boldsymbol{A}\boldsymbol{w}}{2\pi \boldsymbol{w}^\mathrm{H}\boldsymbol{w}}
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\end{equation}$$
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为了使 $\alpha$ 最大，求取矩阵 $\boldsymbol{A}$ 的特征值和特征矢量
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$$\begin{equation}
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2\pi \lambda \boldsymbol{w} = \boldsymbol{A} \boldsymbol{w}
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\end{equation}$$
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并选择对应于最大特征值 $\lambda$ 的特征矢量，将 $[\boldsymbol{A}]_{mn}$ 代入得到
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$$\begin{equation}
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\label{DPSS}
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\sum_{n=1}^N \frac{\sin ((m-n)\psi_0)}{m-n} w_n = \pi \lambda \omega_m,\quad m = 1,2,\cdots,N
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\end{equation}$$
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利用式 $\eqref{DPSS}$ 计算出的权值（或序列）称为**{% label primary @离散长球序列（Discrete Prolate Spheroidal Sequences, DPSS %}**，对应的波束方向图称为**{% label primary @离散长球函数（Discrete Prolate Spheroidal Functions）%}**。离散长球序列也称为**{% label primary @Slepian 序列 %}**。
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  [图 2.1.5](#fig.2-1-5) 给出了几个代表性的离散长球序列（归一化到 $w_6$），[图 2.1.6](#fig.2-1-6) 给出了对应的波束方向图：
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- 当 $\psi_0$ 接近于零的时候, $w_0$ 趋近于均匀加权，因为此时使方向性最大；
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- 对于 $\psi_0 = 0.2\pi$，大多数的能量集中在主波束，波束宽度略增大，旁瓣在 $-20\text{dB}$ 以下；
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- 对于$\psi_0 = 0.4\pi$，旁瓣为 $-53\text{dB}$, 但波束更宽。
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DPSS 加权的波束方向图参数为:
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| $\psi_0$ | HPBW | $BW_{NN}$ | 第一旁瓣高度 | $D_N$ |
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| :--: | :--: | :--: | :--: | :--: |
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| $0.1\pi$ | $\displaystyle\frac{0.02}{N}$ | $\displaystyle\frac{1.40\pi}{N}$ | $-15.6 \text{dB}$ | $0.981$ |
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| $0.2\pi$ | $\displaystyle\frac{2.20}{N}$ | $\displaystyle\frac{1.79\pi}{N}$ | $-24.7 \text{dB}$ | $0.869$ |
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| $0.4\pi$ | $\displaystyle\frac{2.86}{N}$ | $\displaystyle\frac{2.97\pi}{N}$ | $-52.2 \text{dB}$ | $0.665$ |
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<a id="fig.2-1-5"></a>
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{width=700px}
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<a id="fig.2-1-6"></a>
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{width=700px}
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{% note primary MATLAB Code %}
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``` matlab fig3_05_06_13.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 3.5
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% DPSS psi0=0.1pi,0.2pi, and 0.4pi
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% Figure 3.6
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% Beampattern of a standard 11-element linear array as a function
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% Figure 3.13
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% Zero plots for DPSS(0.1pi,0.2pi,0.4pi) weightings
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%
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% Lillian Xiaolan Xu
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% Updated 4/14/99
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% Lillian Xu 04/16/2001, K. Bell 7/22/01, 9/30/01
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% Functions called: DPSS
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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close all
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clear all
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N=11;
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us=0; % should change us between 0 and 0.2
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u=-1:0.001:1;
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n=conj(-(N-1)/2:(N-1)/2)';
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vs=exp(1i*n.*pi.*us);
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%------------------------ DPSS Weighting
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psi0=[0.1 0.2 0.4]*pi;
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w1=DPSS(psi0(1),N);
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w2=DPSS(psi0(2),N);
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w3=DPSS(psi0(3),N);
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z1=roots(w1);
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z2=roots(w2);
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z3=roots(w3);
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w1=w1/max(w1);
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w2=w2/max(w2);
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w3=w3/max(w3);
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m=-5:5;
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figure
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h1 = plot(m,w1);
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hold on;
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h2 = plot(m,w1,'o');
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h3 = plot(m,w2,'--');
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h4 = plot(m,w2,'o');
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h5 = plot(m,w3,'-.');
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h6 = plot(m,w3,'o');
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hold off
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h=legend([h1 h3 h5],{'$\psi_0=0.1\pi$','$\psi_0=0.2\pi$', ...
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'$\psi_0=0.4\pi$'},Interpreter="latex");
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%title('DPSS for 11 elements');
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xticks(-5:5);
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xlabel('Sensors');
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ylabel('Weighting');
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grid on
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%-----------------------------------------------------------%
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% ---------------- Zero Plot.
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figure
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%%%%%%%%%%%% plot 1
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subplot(1,2,1);
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plot(real(z1),imag(z1),'o');
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axis([-1.2 1.2 -1.2 1.2])
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axis('square')
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grid on;
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title('$\psi_0=0.1\pi$','Interpreter','latex');
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xlabel('Real')
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ylabel('Imaginary')
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theta=2*pi*(0:0.01:1);
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hold on
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plot(cos(theta),sin(theta),'--');
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hold off
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%%%%%%%%%%%% plot 2
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subplot(1,2,2);
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plot(real(z3),imag(z3),'o');
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grid on;
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axis([-1.2 1.2 -1.2 1.2])
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axis('square')
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title('$\psi_0=0.4\pi$','Interpreter','latex');
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xlabel('Real')
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ylabel('Imaginary')
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theta=2*pi*(0:0.01:1);
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hold on
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plot(cos(theta),sin(theta),'--');
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hold off
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w1=w1./sum(w1);
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w2=w2./sum(w2);
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w3=w3./sum(w3);
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b=0;
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for m=1:N
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b=b+w1(m)*exp(1i*(-(N+1)/2+m)*pi*u);
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end
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bdb1=20*log10(abs(b));
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b1=real(b);
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b=0;
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for m=1:N
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b=b+w2(m)*exp(1i*(-(N+1)/2+m)*pi*u);
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end
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bdb2=20*log10(abs(b));
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b2=real(b);
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b=0;
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for m=1:N
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b=b+w3(m)*exp(1i*(-(N+1)/2+m)*pi*u);
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end
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bdb3=20*log10(abs(b));
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b3=real(b);
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figure
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plot(u,bdb1,u,bdb2,'--',u,bdb3,'-.');
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hold on
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h=legend({'$\psi_0=0.1\pi$','$\psi_0=0.2\pi$','$\psi_0=0.4\pi$'}, ...
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Interpreter="latex");
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axis([-1 1 -80 0])
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grid on
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xlabel('$u$','Interpreter','latex')
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ylabel('Beam pattern (dB)')
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```
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{% endnote %}
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# 9. Kaiser 加权
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  Kaiser 利用 Bessel 函数提出了一个相对简单的长球序列的近似方法，在谱分析、滤波器设计和其他领域中,这种加权得到了广泛的应用。Kaiser 加权函数为
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$$\begin{equation}
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w(\tilde{n}) = I_0 \left( \beta \sqrt{1 - \left[ \frac{2\tilde{n}}{N} \right] ^2 } \right), \quad -\frac{N-1}{2} \leqslant \tilde{n} \leqslant \frac{N-1}{2}
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\end{equation}$$
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其中 $I_0(x)$ 是零阶修正 Bessel 函数，参数 $\beta$ 决定了波束方向图在旁瓣峰值高度和主波束宽度之间的性能折中程度。
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  [图 2.1.7(a)](#fig.2-1-7) 给出了 $N=11$ 情况下 $\beta=3$ 和 $\beta=6$ 时的权值（归一化到 $w_6$），[图 2.1.7(b)](#fig.2-1-7) 给出了对应的波束方向图。注意到当 $\beta$ 减小时,权函数接近于均匀加权。当 $N = 11$ 时 Kaiser 加权的波束方向图参数为：
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| $\beta$ | HPBW | $BW_{NN}$ | 第一旁瓣高度 | $D_N$ |
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| :--: | :--: | :--: | :--: | :--: |
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| $3$ | $\displaystyle\frac{2.18}{N}$ | $\displaystyle\frac{1.75\pi}{N}$ | $-23.7 \text{dB}$ | $0.882$ |
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| $6$ | $\displaystyle\frac{2.80}{N}$ | $\displaystyle\frac{2.76\pi}{N}$ | $-44.4 \text{dB}$ | $0.683$ |
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<a id="fig.2-1-7"></a>
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{width=700px}
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{% note primary MATLAB Code %}
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``` matlab fig3_07.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 3.7
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% Kaiser weighting
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% (a)weighting for beta = 3 and 6
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% (b)Beam patterns for beta = 3 and 6
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% Xin Zhang updated 3/17/99
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% Lillian Xiaolan Xu updated 09/20/2000
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% Lillian Xiaolan Xu updated 04/16/2001, K. Bell 7/20/01, 9/30/01
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all;
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close all;
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N = 11;
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Zl = (N-1)/2;
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beta1 = 3;
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beta2 = 6;
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for n = -(N-1)/2:(N-1)/2
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x = 1-(2*n/N)^2;
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wn1(n+1+(N-1)/2,1) = besseli(0,beta1*sqrt(x));
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end
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for n = -(N-1)/2:(N-1)/2
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x = 1-(2*n/N)^2;
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wn2(n+1+(N-1)/2,1) = besseli(0,beta2*sqrt(x));
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end
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nn = -Zl/2:1/2:Zl/2;
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u=(-1:0.001:1);
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amf=(-(N-1)/2:(N-1)/2)';
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Vo=exp(1i*amf*pi*u);
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w1=wn1/sum(wn1);
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w2=wn2/sum(wn2);
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Beam1=20*log10(abs(w1'*Vo));
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Beam2=20*log10(abs(w2'*Vo));
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bes1 = besseli(0,beta1);
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bes2 = besseli(0,beta2);
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wn1 = wn1/bes1;
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wn2 = wn2/bes2;
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figure
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subplot(2,1,1)
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plot(nn,wn1,'-',nn,wn2,'--')
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xlabel('$z/\lambda$','Interpreter','latex')
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ylabel('Weights')
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%title('Kaiser Window for 11 elements')
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hold on;
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plot(nn,wn1,'o')
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plot(nn,wn2,'o')
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axis([-2.5 2.5 -0.02 1.02])
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hold off;
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legend({'$\beta = 3$','$\beta = 6$'},Interpreter="latex");
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text(-0.0625,-0.28,'(a)')
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subplot(2,1,2)
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plot(u,Beam1,'-',u,Beam2,'--')
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h=legend({'$\beta = 3$','$\beta = 6$'},Interpreter="latex");
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axis([-1 1 -60 0])
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text(-0.025,-75,'(b)')
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xlabel('$u$','Interpreter','latex')
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ylabel('Beam pattern (dB)')
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set(gcf,'Position',[0 0 600 600])
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```
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{% endnote %}
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  [图 2.1.8](#fig.2-1-8) 给出了固定 $\beta=3$，$N=11,21$ 和 $41$ 时的波束方向图，三种情况具有相同的最大旁瓣。当 $N$ 改变时，最大旁瓣对应的 $u$ 值发生改变。
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<a id="fig.2-1-8"></a>
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{width=700px}
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{% note primary MATLAB Code %}
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``` matlab fig3_08.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 3.8
|
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% Beam patterns for Kaiser weighting
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% beta = 3, N = 11, 21, and 41
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% Xin Zhang
|
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% Last updated 04/16/2001 Lillian Xu, K. Bell 7/20/01,9/30/01
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all;
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close all;
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N1 = 11;
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N2 = 21;
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N3 = 41;
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beta = 3;
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bes = besseli(0,beta);
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for n = -(N1-1)/2:(N1-1)/2
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x = 1-(2*n/N1)^2;
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wn1(n+1+(N1-1)/2) = besseli(0,beta*sqrt(x));
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end
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wn1 = wn1'/bes;
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for n = -(N2-1)/2:(N2-1)/2
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x = 1-(2*n/N2)^2;
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wn2(n+1+(N2-1)/2) = besseli(0,beta*sqrt(x));
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end
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wn2 = wn2'/bes;
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for n = -(N3-1)/2:(N3-1)/2
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x = 1-(2*n/N3)^2;
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wn3(n+1+(N3-1)/2) = besseli(0,beta*sqrt(x));
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end
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wn3 = wn3'/bes;
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u=(-1:0.001:1);
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amf1=(-(N1-1)/2:(N1-1)/2)';
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Vo1=exp(1i*amf1*pi*u);
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amf2=(-(N2-1)/2:(N2-1)/2)';
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Vo2=exp(1i*amf2*pi*u);
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amf3=(-(N3-1)/2:(N3-1)/2)';
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Vo3=exp(1i*amf3*pi*u);
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wn1=wn1/sum(wn1);
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wn2=wn2/sum(wn2);
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wn3=wn3/sum(wn3);
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Beam1=20*log10(abs(wn1'*Vo1));
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Beam2=20*log10(abs(wn2'*Vo2));
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Beam3=20*log10(abs(wn3'*Vo3));
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plot(u,Beam3,'-',u,Beam2,'--',u,Beam1,'-.')
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h=legend({'$N = 41$','$N = 21$','$N = 11$'},Interpreter="latex");
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||||
axis([-1 1 -50 0])
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||||
xlabel('$u$','Interpreter','latex')
|
||||
ylabel('Beam pattern (dB)')
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
# 参考文献
|
||||
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1. Van Trees, Harry L. _Optimum array processing: Part IV of detection, estimation, and modulation theory._ John Wiley & Sons, 2002.
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2. Van Trees, Harry L, 汤俊. _最优阵列处理技术._ 清华大学出版社. 2008.
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[^1]: 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。
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[^2]: 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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Reference in New Issue
Block a user