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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');

David Goodmanson
on 8 May 2021

Edited: David Goodmanson
on 8 May 2021

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');

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