Manuscript Em03-C@1 · Em03-C-v1.tex · section · 47dd71ec

Matrix synthesis

A shared-port two-memory capture construction › Matrix synthesis · lines 1110–1126

Em03-C@1#eq:matrixcapture

Fix the orthonormal temporal tests $f_{1}, f_{2}$ just constructed, supported in $(a, b) = (- 2 L, 2 L)$ , and $0 < τ < 1$ . Set

q (t) D (t) G (t) λ (t) C (t) Ω (t) = (f_{1} (t), f_{2} (t))^{T}, = (1 - τ) I + τ \int_{a}^{t} q (s) q (s)^{†} d s, = D (t)^{1/2}, = τ G (t)^{- 1} q (t), = - G (t)^{- 1} G ˙ (t), = \frac{i}{2} (C (t) - C (t)^{†}) .

Orthonormality gives $D (a) = (1 - τ) I$ , $D (b) = I$ .

Displayed formulas in this passage

Em03-C@1#eq:matrixcapture · line 1114
$q (t) D (t) G (t) λ (t) C (t) Ω (t) = (f_{1} (t), f_{2} (t))^{T}, = (1 - τ) I + τ \int_{a}^{t} q (s) q (s)^{†} d s, = D (t)^{1/2}, = τ G (t)^{- 1} q (t), = - G (t)^{- 1} G ˙ (t), = \frac{i}{2} (C (t) - C (t)^{†}) .$