Array signal processing simulation part IV -- Z-transform analysis array polynomial

Z Transformation

    For linear equally spaced arrays , The beam pattern can be expressed by an array polynomial . The beam pattern is $\psi$ Space can be written as $$B_{\psi }(\psi )=e^{-j(\frac{N-1}{2})\psi }(\sum _{n=0}^{N-1}{\omega }_n{e}^{-jn\psi })$$
    We define $$z=e^{j\psi }$$
    You can write $$B_z(z)=\sum _{n=0}^{N-1}{\omega }_n{z}^{-n}$$
    This expression is similar to z Transformation is similar , Put the real variable $\psi$ Mapping to a complex variable with unit amplitude z, Variable $\psi$ It's a complex variable z The aspect of . The beam pattern can be written as $$B_{\psi }(\psi )=[z^{-\frac{N-1}{2}}{B}_z(z){]}_{z=e^{j\psi }}$$

Real array weights

    For symmetric weighting , Yes $$\omega (n)=\omega (N-1-n)$$
   z Transformation for $$B_z(z)=\sum _{n=0}^{N-1}\omega (n)z^{-n}$$
    Can define $$M=\frac{N-1}{2}$$
    And write $$Bz(z)=z^{-M}\{\omega (M)+\omega (M-1)[z+z^{-1}]+\omega (M-2)[z^2+z^{-2}]+\cdots +\omega (0)[z^M+z^{-M}]\}$$
    According to the symmetry of the coefficient $$B_z(z^{-1})=z^{2M}{B}_z(z)$$
There is no further derivation here , Through the analysis of , Finally we can launch , A group of four zeros of a polynomial , Zero pairs which are conjugate and reciprocal to each other , Now we will analyze some actual cases

Case study

Uniformly weighted array

$$B_{u}i(u)=\frac{1}{N}\frac{sin(\frac{\pi Nd}{\lambda }u)}{sin(\frac{\pi d}{\lambda }u)},-1\le u\le 1$$
An array of patterns appears in $B_u(u)$ The molecule of is 0 The denominator is not 0 when $$sin(\frac{\pi Nd}{\lambda }u)=0$$
The position of zero point is $$u_n=\pm \frac{n}{N}\cdot \frac{\lambda }{d},n=1,2,\cdots ,\frac{N-1}{2}$$

We remember that the position of the first zero point determines the width of the main beam . therefore , We can study such technology , That is, the first zero point is constrained to be on a specific point , And adjust the position of other zeros to get an ideal pattern shape . Many commonly used patterns are developed in this way .

matlab Code

clear all;
close all;

N = 11; 
theta=2*pi*[0:0.01:1];

w = 1/N*ones(N,1);
z = roots(w);
figure
plot(real(z),imag(z),'o');
hold on;
plot(cos(theta),sin(theta),'-'); 
plot(1.5*[-1,1],0*[1,1],'-')
plot(0*[1,1],1.5*[-1,1],'-')
hold off;
set(gca,'YTick',[-1.5 -1 -0.5 0 0.5 1 1.5])
axis square
grid on;
xlabel('{\it x}','Fontsize',14)
ylabel('\it y','Fontsize',14)

cosine,cosine square

The beam pattern corresponding to the two zero point patterns has been mentioned in the previous article and will not be mentioned here .

matlab Code

clear all
close all

N=11; 
n=conj(-(N-1)/2:(N-1)/2)';

w2=cos(pi*n/N);
w3=w2.*w2;
z2=roots(w2);
z3=roots(w3);

% ---------------- Zero Plot.
figure
%%%%%%%%%%%% plot 1
subplot(1,2,1);
plot(real(z2),imag(z2),'o');
axis([-1.2 1.2 -1.2 1.2])
axis('square')
grid on;
title('Cosine','Fontsize',14);
xlabel('Real','Fontsize',14)
ylabel('Imaginary','Fontsize',14)
theta=2*pi*[0:0.01:1];
hold on
plot(cos(theta),sin(theta),'--'); 
plot([-0.9 -0.65],[0.03 0.17])
text(-0.6,0.2,'2 zeros','Fontsize',12)
hold off

%%%%%%%%%%% plot 2
subplot(1,2,2);
plot(real(z3),imag(z3),'o');
grid on;
axis([-1.2 1.2 -1.2 1.2])
axis('square')
title('Cosine-squared','Fontsize',14);
xlabel('Real','Fontsize',14)
ylabel('Imaginary','Fontsize',14)
theta=2*pi*[0:0.01:1];
hold on
plot(cos(theta),sin(theta),'--'); 
text(-1.2,-1.7,(['Another zero is at (',num2str(real(z3(1))),', ',num2str(imag(z3(1))),')']),'Fontsize',12)
%text(1.6,-0.5,(['another zero at :']));
%text(1.7,-1,(['( ',num2str(real(z3(1))),' , ',num2str(imag(z3(1))),' )']))
hold off

Hamming,Blackman-Harris

Hamming Window weight $$\omega (n)=0.54+0.46cos(\frac{2\pi \tilde{n}}{N})$$
Blackman-Harris A weight $$\omega (n)=0.42+0.5cos(\frac{2\pi \tilde{n}}{N})+0.08cos(\frac{4\pi \tilde{n}}{N})$$

matlab Code

N=11; 
n=conj(-(N-1)/2:(N-1)/2)';

w2=0.42+0.5*cos(2*pi*n/N)+0.08*cos(4*pi*n/N);
w3=0.54+0.46*cos(2*pi*n/N);
z2=roots(w2);
z3=roots(w3);

% ---------------- Zero Plot.
figure
%%%%%%%%%%%%	plot 1
subplot(1,2,1);
plot(real(z2),imag(z2),'o');
axis([-1.2 1.2 -1.2 1.2])
axis('square')
grid on;
title('blackman','Fontsize',14);
xlabel('Real','Fontsize',14)
ylabel('Imaginary','Fontsize',14)
theta=2*pi*[0:0.01:1];
hold on
plot(cos(theta),sin(theta),'--'); 
plot([-0.9 -0.65],[0.03 0.17])
text(-0.6,0.2,'2 zeros','Fontsize',12)
hold off

%%%%%%%%%%%	plot 2
subplot(1,2,2);
plot(real(z3),imag(z3),'o');
grid on;
axis([-1.2 1.2 -1.2 1.2])
axis('square')
title('Hamming','Fontsize',14);
xlabel('Real','Fontsize',14)
ylabel('Imaginary','Fontsize',14)
theta=2*pi*[0:0.01:1];
hold on
plot(cos(theta),sin(theta),'--'); 
text(-1.2,-1.7,(['Another zero is at (',num2str(real(z3(1))),', ',num2str(imag(z3(1))),')']),'Fontsize',12)
%text(1.6,-0.5,(['another zero at :']));
%text(1.7,-1,(['( ',num2str(real(z3(1))),' , ',num2str(imag(z3(1))),' )']))
hold off