In this appendix, (8) and (14) are proved. Consider a URA consisting of elements. The array receives one signal from a known direction and one interferer (this proof can be extended similarly). From (1), let the received signal at the array in the presence of mutual coupling for each element be

where , are the received signal and jammer at the th element, expressed as

By taking mutual coupling into account, from (11) for each column

(a) Absence of the Mutual Coupling

If the one row from each column is multiplied by and subtracted from the next row and then the result of each column is multiplied by and subtracted from the next column, in the absence of mutual coupling, this will cancel out all the signals and only noise and interferer will be left

The weight vectors should be in a way that produces zero output; therefore, a reduced rank matrix is formed in which the weighted sum of all its elements would be zero. In order to make the matrix not singular, the additional equation is introduced through the constraint that the same weights when operating on the signal produced a gain factor , which is the first equation. Therefore, (5) will be

Then, performing the matrix multiplication in (A.6) for the first row of the matrix will give

With performing the matrix multiplication in (A.6) for the second row of the matrix the following is obtained:

So

, , and , (A.9) will be true for all if and only if each summation in the parenthesis is equal to zero. Therefore, the first summation will be used

Similarly, the same can be done for the third row of the matrix (A.5), and so forth. In the absence of mutual coupling . From (A.3) and (A.10)

Then, (A.11) will be as simple as

Therefore, the desired signal can be recovered by

(b) Presence of the Mutual Coupling

When there is mutual coupling, the matrix (A.5) can be formed and the (A.3) and (A.10) can be written in a similar way

Similar to (A.11), the following can be presented

The recovered signal will be as follows: