Linear Decorrelating Detector

Linear Decorrelating Detector

The linear decorrelating detector given by (2.13) is characterized by the following results.

Lemma 2.1: The linear decorrelating detector d₁ in (2.13) is the unique weight vector w range(U_s), such that w^Hs₁ = 1, and w^Hs_k = 0 for k = 2, ..., K.

Proof: Since rank(U_s) = K, the vector w that satisfies the foregoing conditions exists and is unique. Moreover, these conditions have been verified in the proof of Proposition 1 in Section 2.2.2.

Lemma 2.2: The decorrelating detector d₁ in (2.13) is the unique weight vector w range(U_s) that minimizes subject to w^Hs₁ = 1.

Proof: Since

Equation 2.73

it then follows that for w range(U_s) = range(s), f(w) is minimized if and only if w^Hs_k = 0 for k = 2, ..., K. By Lemma 2.1 the unique solution is w = d₁.

Proposition 2.3: The linear decorrelating detector d₁ in (2.13) is given in terms of the signal subspace parameters by

Equation 2.74

Equation 2.75

Proof: A vector w range(U_s) if and only if it can be written as w = U_sx, for some . Then by Lemma 2.2 the linear decorrelating detector d₁ has the form d₁ = U_sx₁, where

Equation 2.76

which in turn follows directly from (2.27) and (2.72). The optimization problem (2.76) can be solved by the method of Lagrange multipliers. Let

Since the matrix L_s - s²I_K is positive definite, L(x) is a strictly convex function of x. Therefore, the unique global minimum of L(x) is achieved at x₁, where L(x₁) = 0, or

Equation 2.77