update several posts, add matlab code to figs
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@@ -130,6 +130,48 @@ $$\begin{equation}
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{width=1000px}
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{% note primary MATLAB Code%}
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``` matlab fig2_22.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 2.22
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% Effect of element spacing on beam pattern
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% Lillian Xu
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% Last updated by K. Bell 7/22/01, 10/4/01
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all
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close all
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N=10;
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n=(-(N-1)/2:(N-1)/2)';
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w=ones(N,1)/N;
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u=-3:0.01:3;
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d=[2/3 1/2];
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theta=[30 90];
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figure
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s = ['(a)';'(b)'];
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for m=1:2
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beam=(w.*exp(1i*2*pi*n*d(m)*sin(theta(m)/180*pi)))' ...
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*exp(1i*2*pi*n*d(m)*u);
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subplot(2,1,m);
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plot(u,20*log10(abs(beam)));
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hold on
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axis([-3 3 -25 5])
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ylabel('Beam pattern (dB)')
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xlabel('$u$','Interpreter','latex')
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text(0,-32,s(m,:),'HorizontalAlignment','center')
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% title(['N=',num2str(N),', d=',num2str(d(m),2),' lambda, theta-Bar=', ...
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% num2str(theta(m)),' degrees'])
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grid on
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end
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set(gcf,'Position',[0 0 800 600])
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```
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{% endnote %}
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  通常，为了避免栅瓣进入可视区域，需要有
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$$\begin{equation}
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@@ -161,6 +203,38 @@ $$\begin{equation}
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{width=800px}
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{% note primary MATLAB Code %}
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``` matlab fig2_23.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 2.23
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% Beam pattern for 10-element uniform array (d=lambda/2)
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% scanned to 30 degrees (60 degrees from broadside)
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% Xin Zhang 1/20/99
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% Last updated by K. Bell 6/25/01
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% Functions called: polardb
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all
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close all
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N = 10;
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n = (-(N-1)/2:(N-1)/2).';
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theta = pi*(-1:0.001:1);
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u = cos(theta);
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d = 1/2;
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vv = exp(1i*2*pi*d*n*u);
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theta_T = 30/180*pi;
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w = 1/N*ones(N,1).*exp(1i*n*pi*cos(theta_T));
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B = w'*vv;
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B = 10*log10(abs(B).^2);
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figure
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h=polardb(theta,B,-40);
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hold off
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```
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{% endnote %}
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  为了研究在 $\theta$ 空间的半功率带宽情况，我们利用式 $\eqref{SteeredBeamPatternInAngleDomain}$ 和 [$N \geqslant 10$ 时半功率带宽的估算式][]，可以得到在 $u$ 空间的左右半功率点分别为
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$$\begin{equation}
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@@ -196,6 +270,61 @@ $$\begin{equation} \label{HPBWOfUWLA}
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{width=1000px}
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{% note primary MATLAB Code %}
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``` matlab fig2_24.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 2.24
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% HPBW versus steering angle
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% Lillian Xu
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% Modified by Xin Zhang
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% Last updated by L. Xu 11/30/00, K. Bell 7/22/01, 10/4/01, 10/17/01
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all;
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close all;
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Nd_set=[1000:-5:700 700:-0.5:200 200:-0.05:20 15:-0.005:1];
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theta_deg=[2.5 5 10 20 30 45 90];
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%theta_deg=[2.5 5 10 20 30 45 60 90];
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theta_set=theta_deg*pi/180;
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figure
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for num=1:size(theta_set,2)
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theta=theta_set(num);
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bw=0;
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for num1=1:size(Nd_set,2)
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Nd=Nd_set(num1);
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r=cos(theta)+0.443/Nd;
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if abs(r)<=1
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bw(num1)=acos(r)-acos(cos(theta)-0.443/Nd);
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end
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end
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loglog(Nd_set(1:size(bw,2)),abs(bw*180/pi));
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hold on
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rightpoint=abs(bw(1)*180/pi);
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text(1040,rightpoint,[num2str(theta_deg(num)),'$^\circ$'],Interpreter='latex'); % used to be 7
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end
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axis([1 1000 0.04 100])
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xlabel('$Nd/\lambda$','Interpreter','latex')
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ylabel('3-dB beamwidth in degrees')
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scanlimit=-acos(1-2*0.443./Nd_set);
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loglog(Nd_set,abs(scanlimit*180/pi),'--')
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Endfire = 2*acos(1-0.433./Nd_set);
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loglog(Nd_set,abs(Endfire*180/pi))
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rightpoint=abs(Endfire(1)*180/pi);
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text(1040,rightpoint,'$\theta=0^\circ$',Interpreter='latex'); % used to be 7
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%title('HPBW versus steering angle: standard linear array with uniform weighting')
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text(12,35,'Endfire')
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text(12,14.4,'Scan limit')
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text(12,1.8,'Broadside')
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grid on;
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set(gcf,'Position',[0 0 800 500])
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```
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{% endnote %}
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  当波束调向到正侧向的附近（$\bar{\theta}_\mathrm{T}$ 的值很小）且 $Nd \gg \lambda$ 时，$\theta_\mathrm{H}$ 很小，利用小角度展开形式得到
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$$\begin{equation} \label{HPBWWithSmallAngle}
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@@ -220,6 +349,38 @@ $$\begin{equation} \label{HPBWOfEndfireUWLA}
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{width=800px}
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{% note primary MATLAB Code %}
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``` matlab fig2_26.m
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Figure 2.26
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% Beam pattern of a standard 10-element linear array
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% with uniform amplitude weighting at end-fire
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% Xin Zhang 1/20/99
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% Last updated by K. Bell 6/25/01
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% Functions called: polardb
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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clear all
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close all
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N = 10;
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n = (-(N-1)/2:(N-1)/2).';
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theta = pi*(-1:0.001:1);
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u = cos(theta);
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d = 1/2;
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vv = exp(1i*2*pi*d*n*u);
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theta_T = 0/180*pi;
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w = 1/N*ones(N,1).*exp(1i*n*pi*cos(theta_T));
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B = w'*vv;
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B = 10*log10(abs(B).^2);
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figure
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h=polardb(theta,B,-40);
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hold off
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```
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{% endnote %}
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可以把式 $\eqref{HPBWOfEndfireUWLA}$ 写成
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$$\begin{equation}
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