update several posts, add matlab code to figs
This commit is contained in:
@@ -75,6 +75,51 @@ $$\begin{equation}
|
||||
|
||||
{width=1000px}
|
||||
|
||||
{% note primary MATLAB Code %}
|
||||
|
||||
``` matlab fig2_1516.m
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Figure 2.15 & 2.16
|
||||
% Beampattern of uniformly weighted linear array
|
||||
% Xiaomin Lu
|
||||
% Updated 1/5/99
|
||||
% Last updated by K. Bell 7/22/01, 10/4/01
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
clear all
|
||||
close all
|
||||
|
||||
N = 11; % 11 elements
|
||||
% Note, Eq. (2.92) is in fact the standard Dirichlet function
|
||||
% beam = diric(psi, N);
|
||||
% if you don't have the signal processing toolbox, this does the same thing
|
||||
psi = (-5:1/400:5)*pi;
|
||||
beam = zeros(1,length(psi(:)));
|
||||
y=sin(.5*psi);
|
||||
i=find(abs(y)>1e-12); % set where x is not divisible by 2 pi
|
||||
j=1:length(psi(:));
|
||||
j(i)=[]; % complement set
|
||||
beam(i)=sin((N/2)*psi(i))./(N*y(i));
|
||||
beam(j)=sign(cos(psi(j)*((N+1)/2)));
|
||||
|
||||
plot(psi/pi, beam)
|
||||
grid
|
||||
xlabel('$\psi/\pi$','Interpreter','latex')
|
||||
ylabel('Frequency wavenumber response function')
|
||||
axis([-5 5 -0.4 1])
|
||||
set(gcf,'Position',[0 0 1000 600])
|
||||
figure
|
||||
beam = 20*log10(abs(beam));
|
||||
plot(psi/pi,beam);
|
||||
xlabel('$\psi/\pi$','Interpreter','latex')
|
||||
ylabel('Frequency wavenumber response function (dB)')
|
||||
axis([-5 5 -25 0])
|
||||
grid
|
||||
set(gcf,'Position',[0 0 1000 600])
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
若不指定 $\boldsymbol{w}$ 的类型，则 $\boldsymbol{\varUpsilon}_\psi(\psi)$ 是复数，所以相位也应该画出来。但是，均匀加权线阵具有对称性，因此得到的频率-波数响应是实函数。
|
||||
|
||||
  也可以用 $k_z$ 来表示频率-波数响应
|
||||
@@ -121,10 +166,138 @@ $$\begin{equation}
|
||||
|
||||
{width=800px}
|
||||
|
||||
{% note primary MATLAB Code %}
|
||||
|
||||
``` matlab fig2_17.m
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Figure 2.17
|
||||
% Polar plot of B (Theta)
|
||||
% Lillian Xu 1/5/99
|
||||
% Updated by K. Bell 6/25/01
|
||||
% Functions called: polardb
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
|
||||
clear all
|
||||
close all
|
||||
|
||||
N = 11; % Elements in array
|
||||
d = 0.5; % spacing wrt wavelength
|
||||
beamwidth = 2/(N*d); % null-to-null (LL BW is half this)
|
||||
D=d*(-(N-1)/2:1:(N-1)/2); % element locations
|
||||
ang = pi*(-1:0.001:1);
|
||||
u = cos(ang);
|
||||
n2=size(u,2);
|
||||
AS = exp(-1i*2*pi*cos(90/180*pi)*D'); % BP points to 90 (broadside)
|
||||
Au = exp(-1i*2*pi*D'*u);
|
||||
B = real(AS'*Au)/N;
|
||||
G = 20*log10((abs(B)));
|
||||
|
||||
h=polardb(ang,G,-40);
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
<a id="fig.1-3-4"></a>
|
||||
|
||||
{width=1000px}
|
||||
|
||||
{% note primary MATLAB Code %}
|
||||
|
||||
``` matlab fig2_18.m
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Figure 2.18
|
||||
% beam patterns in different spaces
|
||||
% Lillian Xiaolan Xu
|
||||
% Last updated 09/07/2000
|
||||
% updated by K. Bell 7/22/01, 10/4/01
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
clear all
|
||||
close all
|
||||
|
||||
N = 10;
|
||||
n = (-(N-1)/2:(N-1)/2).';
|
||||
psi = pi*(-3:0.001:3);
|
||||
w = 1/N*[ones(N,1)];
|
||||
d = 1/2;
|
||||
vv = exp(1i*2*d*n*psi);
|
||||
B = (abs(w'*vv));
|
||||
|
||||
u1=-1;
|
||||
u2=3;
|
||||
y1=0;
|
||||
y2=1;
|
||||
|
||||
subplot(4,1,1)
|
||||
plot(-psi/d,B);
|
||||
hold on
|
||||
axis([u1*pi/d u2*pi/d y1 y2])
|
||||
set(gca,'XTick',[-pi/d 0 pi/d 2*pi/d 3*pi/d])
|
||||
set(gca,'YTick',[0 0.5 1])
|
||||
set(gca,'XTickLabel',{'$-\pi/d$';'$0$';'$\pi/d$';'$2\pi/d$';'$3\pi/d$'}, ...
|
||||
'TickLabelInterpreter','latex')
|
||||
|
||||
grid on
|
||||
point=0.8;
|
||||
point_up=0.84;
|
||||
point_down=0.76;
|
||||
plot([-pi/d pi/d],[point point],'--')
|
||||
plot([-1*pi/d -0.9*pi/d],[point point_up])
|
||||
plot([-1*pi/d -0.9*pi/d],[point point_down])
|
||||
plot([0.9*pi/d pi/d],[point_up point])
|
||||
plot([0.9*pi/d pi/d],[point_down point])
|
||||
text(-0.3*pi/d,1.2,'Visible region')
|
||||
|
||||
plot([pi/d 3*pi/d],[point point],'--')
|
||||
plot([pi/d 1.1*pi/d],[point point_up])
|
||||
plot([pi/d 1.1*pi/d],[point point_down])
|
||||
plot([2.9*pi/d 3*pi/d],[point_up point])
|
||||
plot([2.9*pi/d 3*pi/d],[point_down point])
|
||||
text(1.7*pi/d,1.2,'Virtual region')
|
||||
legend({'$k_z$-space'},'Interpreter','latex')
|
||||
|
||||
subplot(4,1,2)
|
||||
plot(psi,B);
|
||||
hold on
|
||||
axis([u1*pi u2*pi y1 y2])
|
||||
set(gca,'XTick',[-pi 0 pi 2*pi 3*pi])
|
||||
set(gca,'YTick',[0 0.5 1])
|
||||
set(gca,'XTickLabel',{'$-\pi$';'$0$';'$\pi$';'$2\pi$';'$3\pi$'}, ...
|
||||
'TickLabelInterpreter','latex')
|
||||
|
||||
grid on
|
||||
legend({'$\psi$-space'},'Interpreter','latex')
|
||||
|
||||
subplot(4,1,3)
|
||||
plot(psi/pi,B);
|
||||
hold on
|
||||
axis([u1 u2 y1 y2])
|
||||
set(gca,'XTick',[-1 0 1 2 3])
|
||||
set(gca,'YTick',[0 0.5 1])
|
||||
grid on
|
||||
legend({'$u$-space'},'Interpreter','latex')
|
||||
|
||||
theta = -180:0.1:360;
|
||||
vv = exp(1i*2*d*n*pi*cos(theta/180*pi));
|
||||
B = (abs(w'*vv));
|
||||
subplot(4,1,4)
|
||||
plot(theta,B);
|
||||
hold on
|
||||
axis([-180 180 y1 y2])
|
||||
set(gca,'XTick',[-180 -90 0 90 180])
|
||||
set(gca,'YTick',[0 0.5 1])
|
||||
set(gca,'XTickLabel',{'$180^\circ$';'$90^\circ$';'$0^\circ$';'$-90^\circ$'; ...
|
||||
'$-180^\circ$'},'TickLabelInterpreter','latex')
|
||||
|
||||
grid on
|
||||
legend({'$\theta$-space'},'Interpreter','latex')
|
||||
|
||||
set(gcf,'Position',[0 0 800 800])
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
{% note info %}
|
||||
|
||||
虽然以上例子比较简单，但是可以用来展示线性阵列的一些重要特征。
|
||||
@@ -151,6 +324,76 @@ $$\begin{equation}
|
||||
|
||||
{width=700px}
|
||||
|
||||
{% note primary MATLAB Code %}
|
||||
|
||||
``` matlab fig2_19.m
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Figure 2.19
|
||||
% Main lobe of beam pattern
|
||||
% Kristine Bell
|
||||
% Last updated 6/4/01, 10/4/01
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
% bp_sect - plots conventional bp with sector
|
||||
|
||||
clear all
|
||||
% close all
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Uniform Linear Array
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
N = 10; % Elements in array
|
||||
d = 0.5; % spacing wrt wavelength
|
||||
beamwidth = 2/(N*d); % null-to-null (LL BW is half this)
|
||||
D=d*(-(N-1)/2:1:(N-1)/2); % element locations
|
||||
u = -0.45:0.01:0.45;
|
||||
AS = exp(-1i*2*pi*D'*0); % BP points to 0
|
||||
Au = exp(-1i*2*pi*D'*u);
|
||||
B = real(AS'*Au)/N; % BP
|
||||
|
||||
h=plot(u,B,'-');
|
||||
set(h,'LineWidth',1.5)
|
||||
hold on
|
||||
% HPBW
|
||||
I=find(B>=0.707);
|
||||
plot([u(min(I)-1) u(max(I)+1)],[B(min(I)-1) B(max(I)+1)],'-')
|
||||
plot(u(min(I)-1)*[1 1],B(min(I)-1)*[1 1]+0.03*[-1 1],'-')
|
||||
plot(u(max(I)+1)*[1 1],B(max(I)+1)*[1 1]+0.03*[-1 1],'-')
|
||||
plot([0.03 0.13],[0.72 0.845],'-')
|
||||
text(0.155,0.84,'$\Delta u_1 = $HPBW','Interpreter','latex');
|
||||
|
||||
% BW-NN
|
||||
plot([-0.2 0.2],[-0.3 -0.3],'-')
|
||||
plot(-0.2*[1 1],-0.3*[1 1]+0.03*[-1 1],'-')
|
||||
plot(0.2*[1 1],-0.3*[1 1]+0.03*[-1 1],'-')
|
||||
text(-0.05,-0.35,'$\Delta u_2 = BW_{NN}$','Interpreter','latex');
|
||||
|
||||
% axes
|
||||
plot([-1 1 ],[0 0],'-')
|
||||
plot([0 0],[-0 1.1],'-')
|
||||
|
||||
% tick marks
|
||||
plot(-0.4*[1 1],0.03*[-1 1])
|
||||
plot(-0.2*[1 1],0.03*[-1 1])
|
||||
plot(0.2*[1 1],0.03*[-1 1])
|
||||
plot(0.4*[1 1],0.03*[-1 1])
|
||||
|
||||
yy = -0.1;
|
||||
%tick labels
|
||||
text(-0.45,yy,'$-2\frac{\lambda}{ND}$','Interpreter','latex');
|
||||
text(-0.21,yy,'$-\frac{\lambda}{ND}$','Interpreter','latex');
|
||||
text(-0.005,yy,'0','Interpreter','latex');
|
||||
text(0.18,yy,'$\frac{\lambda}{ND}$','Interpreter','latex');
|
||||
text(0.38,yy,'$2\frac{\lambda}{ND}$','Interpreter','latex');
|
||||
|
||||
text(0.01,1.1,'$B(u)$','Interpreter','latex')
|
||||
hold off
|
||||
axis([-0.42 0.42 -0.4 1.2])
|
||||
|
||||
set(gca,'Visible','off')
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
  3dB 波束宽度是波束宽度的一个度量。定义为对应 $\left| B_u(u) \right|^2 = 0.5$ 或 $\left| B_u(u) \right| = 1/\sqrt{2}$ 的点。可以通过令波束方向图 $B_u(u)$（式 $\eqref{BeamPatternOfUWLAInDirectionCosineDomain}$）等于 $1/\sqrt{2}$，来求出 $u$ 空间的半功率点。对于 $N\geqslant 10$，通过解下面的方程可以得到一个很好的近似：
|
||||
|
||||
$$\begin{equation}
|
||||
@@ -295,6 +538,55 @@ $$\begin{equation}
|
||||
|
||||
{width=1000px}
|
||||
|
||||
{% note primary MATLAB Code %}
|
||||
|
||||
``` matlab fig2_20.m
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
% Figure 2.20
|
||||
% Effect of element spacing on beam pattern
|
||||
% Xin Zhang
|
||||
% Lillian Xu updated 09/2000
|
||||
% updated by K. Bell 7/22/2001, 10/4/01
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
clear all
|
||||
close all
|
||||
|
||||
N = 10;
|
||||
n = (-(N-1)/2:(N-1)/2).';
|
||||
psi = pi*(-3:0.001:3);
|
||||
w = 1/N*[ones(N,1)];
|
||||
d = [1/4 1/2 1];
|
||||
|
||||
figure
|
||||
for k =1:length(d)
|
||||
vv = exp(1i*2*d(k)*n*psi);
|
||||
B(k,:) = 20*log10(abs(w'*vv));
|
||||
subplot(3,1,k)
|
||||
plot(psi/pi,B(k,:));
|
||||
hold on
|
||||
plot([-1 -1],[-25 0],'--');
|
||||
plot([1 1],[-25 0],'--');
|
||||
axis([-3 3 -25 5])
|
||||
set(gca,'YTick',[-25 -20 -15 -10 -5 0])
|
||||
grid on
|
||||
xlabel('$u$','Interpreter','latex')
|
||||
if k == 1
|
||||
plot([-1 1],[2.5 2.5])
|
||||
plot([-1 -0.9],[2.5 0.5])
|
||||
plot([-1 -0.9],[2.5 4.5])
|
||||
plot([0.9 1],[0.5 2.5])
|
||||
plot([0.9 1],[4.5 2.5])
|
||||
text(-0.4,7,'Visible region')
|
||||
end
|
||||
hold off
|
||||
end
|
||||
|
||||
set(gcf,'Position',[0 0 800 600])
|
||||
```
|
||||
|
||||
{% endnote %}
|
||||
|
||||
  在[图 1-3-6](#fig.1-3-6) 中，针对多种 $d/\lambda$ 的值，画出了 $\left| \boldsymbol{\varUpsilon}_u(u) \right|$。图中说明了“栅瓣”这个重要的概念，即和主波束一样高的波瓣。栅瓣发生在当波束方向图 $B_\psi(\psi)$（式 $\eqref{BeamPatternOfUWLAInWaveNumberDomain}$）中的分子和分母均为零的时候。栅瓣出现的间隔为
|
||||
|
||||
$$\begin{equation}
|
||||
|
||||
Reference in New Issue
Block a user