A diagonal selector becomes a rational filter
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The filter is diagonal in one basis and a rational convolution in the other. Let \(F_{na}=\zeta^{na}/\sqrt{12}\) and \(D_{aa}=\mathbf1_{a\in U_{12}}\). With the positive-sign unitary matrix \(F\), the negative-sign DFT satisfies \(F^\dagger x=\widehat x/\sqrt{12}\).
\[
P=FDF^\dagger,\qquad P_{nm}=\frac{c_{12}(n-m)}{12}.
\]
Because \(D^2=D\) and \(F^\dagger F=I\), we obtain \(P^2=P\). Because \(D^\dagger=D\), we also obtain \(P^\dagger=P\). This is an orthogonal projector of rank four.