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5 We can also change the order in which the qi ’s compose matrix Q, but this fact is not relevant for the present discussion. 28 Chapter 2 Fundamentals of Adaptive Filtering An important result is that any Hermitian matrix R can be diagonalized by a suitable unitary matrix Q, even if the eigenvalues of R are not distinct. The proof is omitted here and can be found in [19]. Therefore, for Hermitian matrices with repeated eigenvalues it is always possible to find a complete set of orthonormal eigenvectors.

The desired gain is usually f = 1. In the case the desired signal impinges the array at an angle θ with respect to the array, the incoming θ signal reaches the ith antenna delayed by i d cos with respect to the 0th antenna [23]. Let’s consider c the case of a narrowband array such that all antennas detect the impinging signal with the same amplitude when measured taking into consideration their relative delays, which are multiples of d cos θ c . 103) for i = 0, 1, . . , N , in order to add coherently the delays of the desired incoming signal at a given direction θ.

5 We can also change the order in which the qi ’s compose matrix Q, but this fact is not relevant for the present discussion. 28 Chapter 2 Fundamentals of Adaptive Filtering An important result is that any Hermitian matrix R can be diagonalized by a suitable unitary matrix Q, even if the eigenvalues of R are not distinct. The proof is omitted here and can be found in [19]. Therefore, for Hermitian matrices with repeated eigenvalues it is always possible to find a complete set of orthonormal eigenvectors.