Why is the radiation pattern does not show the required steering angle?

N = 16;%number of elements
wl = 1;%wavelength
k = 2.*pi./wl;
d = wl./2;%distance
f = zeros(1,180);%empty array for field

AF = zeros(1,100);%empty array for complec weights

thi = -45;%The scan angle
angle_radi = thi.*pi./180;%angle to radian
phi= (2.*pi.*d .*sin(angle_radi))./wl;%equation for phase shift

for a = 1:N %for loop to repeat until the Nth phase shift
  
    AF(a)= (a-1).*phi;%First number = (1-1)*phi 

  
  while AF(a) > 2*pi
           AF(a) = AF(a) - 2.*pi;
           if AF(a)<= 2*pi;break;end 
  end %minimize the data if it is larger than 2pi
  
  
  while AF(a) < -2*pi
           AF(a) =AF(a) + 2*pi;
           if AF(a) >= -2*pi;break;end 
    
  end %maximize the data if it is smaller than -2pi
      
end%does not affect the result but mitigate the calculation

as =  AF(2)+AF(3)+AF(4)+AF(5)+AF(6)+AF(7)+AF(8)+AF(9);%weights in total
 
for th =1:360 %angle from 1 to 360
    s = 0
    angle_1rad(th) = th.*pi/180;%angle to radian
    for i = 1:N
 
    s = s + exp((-1j).*k.*(i-1).*d.*sin(angle_1rad(th))+as)%equation for field
    
    end
 
         f(th)= abs(s);
end
 
M = max(f);
z  = f./M;%normalize the data
 
figure 
polarplot(angle_1rad+as,z);
%title('linear polar plot');
 

Answer 1

[David Goodmanson]

Hi ZW,

In the code below, the key line is

s = s + exp((-1j*k*d*(i-1).*sin(angle_1rad(th))) +1j*AF(i)); %equation for field

which is simply the sum of N phase angles due to propagation at the given angle, each multplied by the phase angle AF required to aim the beam at -45 degrees. I don't know the intent of the sum as = AF(2) + ...AF(9), but it is not necessary. I left out the while loops changing the values of angles by multiples of 2pi since it is unnecessary for accuracy (unless the number of antenna elements gets into maybe millions).

The polar plot shows a peak at -45 degrees and an equal peak at -135 degrees. With your geometry the line of emitters is the plus and minus vertical y axis, 90 degrees and 270 degrees, so there is equal emission on each side of the antenna as required.

The second code shortens things up by replacing some of the for loops with the vectorized version. (the code also uses exp(+1j*k*x)-type wave propagation rather than exp(-1j*k*x), since the former seems more intuitive).

N = 16; % number of elements
wl = 1; % wavelength
k = 2.*pi./wl;
d = wl./2; % distance
f = zeros(1,360);  % empty array for field 
AF = zeros(1,100); % empty array for complec weights 
thi = -45; % The scan angle
angle_radi = thi.*pi./180; % angle to radian
phi= (k.*d .*sin(angle_radi)); % eqn for phase shift [in terms of k]
for a = 1:N %for loop to repeat until the Nth phase shift
  AF(a)= (a-1).*phi; % First number = (1-1)*phi 
end
%as =  AF(2)+AF(3)+AF(4)+AF(5)+AF(6)+AF(7)+AF(8)+AF(9);%weights in total
for th =1:360 %angle from 1 to 360
    s = 0;
    angle_1rad(th) = th.*pi/180; % angle to radian
    for i = 1:N
      s = s + exp((-1j*k*d*(i-1).*sin(angle_1rad(th)))+1j*AF(i)); %equation for field
    end
    f(th)= abs(s);
end
%  
M = max(f);
z  = f./M;%normalize the data
figure(1) 
polarplot(angle_1rad,z);
title('linear polar plot');

N = 16; % number of elements
wl = 1; % wavelength
k = 2*pi/wl;
d = wl/2; % distance
thi = -45; % The scan angle
thirad = thi.*pi./180; % angle to radian
f = zeros(1,360);
anglerad = zeros(1,360);
phi= (k*d*sin(thirad)); % eqn for phase shift [in terms of k]
AF = (0:N-1)*phi;
for th =1:360 % angle from 1 to 360
  anglerad(th) = th.*pi/180; % angle to radian
  thvecN = k*d*(0:N-1)*sin(anglerad(th)); % phase angles due to propagation
  s = sum(exp(1j*(thvecN-AF)));
  f(th)= abs(s);
end
z  = f/max(f);  % normalize the data
figure(2) 
polarplot(anglerad,z);
title('linear polar plot');

[ZIYI WENG]

Thank you very much. I found a different way to solve it later that day, but thanks anyway. I will try your code and see what will happen.