Beam pattern analysis based on spectral weighting

Spectral weighting

    This article mainly analyzes the linear array , The following formula gives $u$ The derivation formula of spatial beam pattern
$$B_u(u)={\omega }^H{v}_u(u)=e^{-j\frac{N-1}{2}\frac{2\pi d}{\lambda }u}\sum _{n=0}^{N-1}{\omega }_n^{\ast }{e}^{jn\frac{2\pi d}{\lambda }u}$$
    We can see , The beam pattern is the product of weight and manifold vector , This paper mainly analyzes the different effects of different weights on the beam pattern , Because the weights considered below are all real symmetric , So you can put n The positions of the array elements are replaced by the following labels $$\tilde{n}=n-\frac{N-1}{2},n=0,1,\cdots ,N-1$$

cosine weighting

    consider N In the case of an odd number ,cosine A weight of $$\omega (\tilde{n})=sin(\frac{\pi }{2N})cos(\pi \frac{\tilde{n}}{N}),-\frac{N-1}{2}\le \tilde{n}\le \frac{N-1}{2}$$
among $sin(\frac{\pi }{2N})$ It's a constant , The purpose is to yes $B_u(0)=1,holdcosineWritebecomefingerCountshapetype,ByStepchangeshapemostendcanWithhave\; toTo$
$$B_u(u)=\frac{1}{2}sin(\frac{\pi }{2N})\{\frac{sin(\frac{N\pi }{2}(u-\frac{1}{N}))}{sin(\frac{\pi }{2}(u-\frac{1}{N}))}+\frac{sin(\frac{N\pi }{2}(u+\frac{1}{N}))}{sin(\frac{\pi }{2}(u+\frac{1}{N}))}\}$$
We can make direct use of the computing power of the computer , Let it help us deal with the product of weights and manifold vectors , The results are given below

We can see that the side lobe becomes lower , And the main lobe widened , Let's take a look at the comparison of various data

matlab Code

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(-j*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');
set(h,'Fontsize',12)
xlabel('\it u','Fontsize',14)
ylabel('Beam pattern (dB)','Fontsize',14)

Raised cosine weighting

    A weight $$\omega (\tilde{n})=c(p)(p+(1-p)cos(\pi \frac{\tilde{n}}{N})),\tilde{n}=-\frac{N-1}{2},\cdots ,\frac{N-1}{2}$$
among $c(p)$ It's a constant , bring $B_u(0)=1$$$c(p)=\frac{p}{N}+\frac{(1-p)}{2}sin(\frac{\pi }{2N})$$
The figure below shows $p=0.31,0.17,0$ Beam pattern at


When $p$ Less hours , The height of the first side lobe decreases , The main beam width increases , So we can already make HPBW Narrowing , And the first side lobe is much lower than that in the case of uniform distribution

matlab Code

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(-j*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('{\it p}=0.31','{\it p}=0.17','{\it p}=0');
set(h,'Fontsize',12)
xlabel('\it u','Fontsize',14)
ylabel('Beam pattern (dB)','Fontsize',14)

cosine^m weighting

A series of cosine weighting , formation $co{s}^m(\frac{\pi \tilde{n}}{N})$,z The weight of the array is
$${\omega }_m(\tilde{n})=\{\begin{array}{ll}{c}_{2}co{s}^2(\frac{\pi \tilde{n}}{N}),& m=2\\ c_{3}co{s}^3(\frac{\pi \tilde{n}}{N}),& m=3\\ c_{4}co{s}^4(\frac{\pi \tilde{n}}{N}),& m=4\end{array}$$
among $c_2,c_3,c_4$ It is still a normalized constant
The following figure shows the beam pattern , When m increases , Sidelobe reduction , But the main beam becomes wider

matlab Code

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(-j*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('{\it m}=2','{\it m}=3','{\it m}=4');
set(h,'Fontsize',12)
xlabel('\it u','Fontsize',14)
ylabel('Beam pattern (dB)','Fontsize',14)

summary

The above analysis cosine Weighting and its several deformations , Overall speaking ,cosine Weighting widens the width of the main lobe but lowers the height of the first side lobe , And in the raised cosine weighting ,p The smaller the size, the more obvious the trend , Also in cosine Of m Power weighted ,m The bigger, the more obvious .