Manuscript En01-B@2 · En01-B-v2.tex · proposition · f226d1eb

Proposition: Separation-error propagation

Preparation and detector conditions for the control › Residual ledger · lines 1977–2007 · proof included or linked

En01-B@2#prop:separation

If $∣ d_{i} - d ˉ_{i} ∣ \leq δ_{i}$ and $∣ Γ_{r_{c}}^{'} (d) ∣ \leq L_{i}$ on the corresponding windows, then

i \sum w_{i} Γ_{r_{c}} (d_{i}) - i \sum w_{i} Γ_{r_{c}} (d ˉ_{i}) \leq i \sum ∣ w_{i} ∣ L_{i} δ_{i} .

For the homogeneous sphere, Gaussian smearing gives $A_{r_{c}} \leq A_{0}$ , and

∣ Γ_{r_{c}}^{'} (d) ∣ \leq 2 A_{r_{c}} ∣ d ∣ \leq 2 A_{0} ∣ d ∣.

For nominal weights cancelling $d_{i} ˉ^{2}$ , the quadratic nuisance leakage is at most

∣ β_{2} ∣ i \sum ∣ w_{i} ∣ (2∣ d ˉ_{i} ∣ δ_{i} + δ_{i}^{2}) .

Proof. Apply the mean-value theorem termwise. Differentiate (eq:rate) and use $∣ sin (k_{e} d) ∣ \leq ∣ k_{e} d ∣$ . Smearing multiplies the positive moment integral by a factor at most one. The last bound expands $d_{i}^{2} - d_{i} ˉ^{2}$ .

Displayed formulas in this passage

unlabeled · line 1981
$i \sum w_{i} Γ_{r_{c}} (d_{i}) - i \sum w_{i} Γ_{r_{c}} (d ˉ_{i}) \leq i \sum ∣ w_{i} ∣ L_{i} δ_{i} .$
unlabeled · line 1990
$∣ Γ_{r_{c}}^{'} (d) ∣ \leq 2 A_{r_{c}} ∣ d ∣ \leq 2 A_{0} ∣ d ∣.$
unlabeled · line 1995
$∣ β_{2} ∣ i \sum ∣ w_{i} ∣ (2∣ d ˉ_{i} ∣ δ_{i} + δ_{i}^{2}) .$

Proposition: Separation-error propagation

Displayed formulas in this passage

Nearby in this section