I want to implement these equations in MATLAB but how?

The equations are given in the attachment. Assume lambda=1. How to implement the given equations in MATLAB?

[Catalytic]

What is ?

[Sadiq Akbar]

Thanks for your response. It is symbol of Kronecker product

Answer 1

[Matt J]

fn=@(theta,k) exp((0:k-1).' * (2i*pi*d/lambda*sin(theta)));

A=fn(theta,M);
B=fn(theta,N);

y=reshape(  A*B.', [],1);

[Sadiq Akbar]

Thank you dear Matt J for your response. I ran it like this:

clear all
clc
theta=[10 20 30];
M=10; 
N=4;  
K=3;  
lambda=1;
d=0.5;

fn=@(theta,k) exp((2i*pi*d/lambda*sin(theta))*(0:k-1).');

a=fn(theta,M);
b=fn(theta,N);

y=sum(kron(a,b))

But it gives the following error:

Error using *

Inner matrix dimensions must agree.

Error in MattJ>@(theta,k)exp((2i*pi*d/lambda*sin(theta))*(0:k-1).')

Error in MattJ (line 12)

a=fn(theta,M);

[Matt J]

Try this instead.

theta=[10 20 30];
M=10; 
N=4;  
K=3;  
lambda=1;
d=0.5;

fn=@(theta,k) exp((0:k-1).' * (2i*pi*d/lambda*sin(theta)));

A=fn(theta,M);
B=fn(theta,N);

y=reshape(  A*B.', [],1)

y = 
   3.0000 + 0.0000i
  -2.1000 - 0.7579i
   0.8893 - 0.1719i
  -1.2724 + 1.5341i
   2.2985 - 1.2640i
  -1.8220 + 0.0222i
   0.2295 - 0.0361i
   0.1951 + 1.2483i
   0.8243 - 1.4133i
  -1.1131 - 0.0234i
  -2.1000 - 0.7579i
   0.8893 - 0.1719i
  -1.2724 + 1.5341i
   2.2985 - 1.2640i
  -1.8220 + 0.0222i
   0.2295 - 0.0361i
   0.1951 + 1.2483i
   0.8243 - 1.4133i
  -1.1131 - 0.0234i
  -0.1750 + 0.9540i
   0.8893 - 0.1719i
  -1.2724 + 1.5341i
   2.2985 - 1.2640i
  -1.8220 + 0.0222i
   0.2295 - 0.0361i
   0.1951 + 1.2483i
   0.8243 - 1.4133i
  -1.1131 - 0.0234i
  -0.1750 + 0.9540i
   1.0742 - 0.2196i
  -1.2724 + 1.5341i
   2.2985 - 1.2640i
  -1.8220 + 0.0222i
   0.2295 - 0.0361i
   0.1951 + 1.2483i
   0.8243 - 1.4133i
  -1.1131 - 0.0234i
  -0.1750 + 0.9540i
   1.0742 - 0.2196i
  -0.1789 - 0.4202i

[Sadiq Akbar]

Thanks for your response. It works now but the size of y must be 40 x 3

40 is due to product of M and N and 3 is due to three values in theta.. But yours give y as 40 x 1.

[Matt J]

I don't see how a 40x3 matrix can come out of your equations. The summation is over vector quantities, not matrices.

[Sadiq Akbar]

Thanks for your response but may be the Kronecker product serves this purpose and I don't see any such command here.

[Matt J]

No, the Kronecker product of two vectors is a vector, e.g.,

a=[1;2;3];
b=[4;5];

kron(a,b)

ans = 6×1

4 5 8 10 12 15

[Sadiq Akbar]

I made changes to yours as:

theta=[10 20 30];
M=10; 
N=4;  
K=3;  
lambda=1;
d=0.5;

fn=@(theta,k) exp((0:k-1).' * (2i*pi*d/lambda*sin(theta)));

A=fn(theta,0:M-1);
B=fn(theta,0:N-1);

But now it gives A and B as empty matrices