ULA Arrays and Their Array Response

 

The array response vectors for uniform linear arrays (ULA) with elements, N are represented as

 

 

 

Similarly, for Uniform Planer arrays (UPA), the array response vectors can be represented as, 

 

 

 

Where 0 ≤ x ≤ W₁ -1 and 0 ≤  y ≤  W₂ -1

 

 

 

 

Fig : Directional cosine vectors at each antenna element  

 

 

In figure above, it is illustrated that when transmitted signal reaches at receiver it creates different path lengths at different antenna element. Here in figure the second ray creates path-length difference of dcosθ with the first element. As there will be phase difference for same angle of departure (AOD) rays transmitted from Tx side. The corresponding calculation of phase difference is shown in figure.