Give the projector something to remove
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The input contains an arithmetic component and a deliberately complementary component. Take \(x(n)=c_{12}(n)+3\cos(\pi n/2)+2\). Its extra cosine uses modes \(3,9\) and its constant uses mode \(0\). All three are outside \(U_{12}\).
\[
Px=c_{12},\qquad (I-P)x=3\cos(\pi n/2)+2.
\]
The exact input is \((9,2,1,2,3,2,-5,2,3,2,1,2)\). Calling the complementary component “noise” is a modeling choice. Arbitrary noise need not lie in the kernel of \(P\).