Account for the entire input
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Exact filtering removes energy as well as modes. The selected and complementary components are orthogonal. With the unnormalized squared norm \(\|x\|^2=\sum_{n=0}^{11}|x(n)|^2\),
\[
\|x\|^2=150,\qquad\|Px\|^2=48,\qquad\|(I-P)x\|^2=102.
\]
On the normalized state \(x/\sqrt{150}\), the ideal Lüders projective measurement \(\{P,I-P\}\) has selected-outcome probability \(8/25\). Conditional on that outcome, the resulting normalized state is \(c_{12}/\sqrt{48}\). This conditional measurement is not deterministic unitary denoising. The transform \(F\) is reversible; the projection \(P\) discards the complementary subspace.