A.6  Matched filter and Cauchy-Schwarz’s inequality

Proving the matched filter is analogous to the following problem: given a vector $\mathbf{\text{x}}$, what is the vector $\mathbf{\text{y}}$ that maximizes the inner product $⟨\mathbf{\text{x}},\mathbf{\text{y}}⟩$? The Cauchy-Schwarz’s inequality states that:

Because $⟨\mathbf{\text{x}},\mathbf{\text{x}}⟩=\parallel \mathbf{\text{x}}{\parallel }^2$, taking the square root of Eq. (A.19), leads to $|⟨\mathbf{\text{x}},\mathbf{\text{y}}⟩|\le \parallel \mathbf{\text{x}}\parallel \parallel \mathbf{\text{y}}\parallel$. Hence, the inner product $⟨\mathbf{\text{x}},\mathbf{\text{y}}⟩=\parallel \mathbf{\text{x}}\parallel \parallel \mathbf{\text{y}}\parallel \text{cos}<!--\; FUNCTION\; APPLICATION\; -->(𝜃)$ assumes its maximum value $\parallel \mathbf{\text{x}}\parallel \parallel \mathbf{\text{y}}\parallel$ when $𝜃=0$ or $180$ degrees, which corresponds to the vector $\mathbf{\text{y}}$ and $\mathbf{\text{x}}$ being colinear.