The selected phases add and cancel
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The same four waves can reinforce, cancel, or reinforce with the opposite sign. At each sample \(n\), add the four numbers \(\zeta^{an}\) with \(a\in U_{12}\). Each has magnitude one. Their sum need not have magnitude one.
\[
c_{12}(0)=4,\qquad c_{12}(1)=0,\qquad c_{12}(6)=-4.
\]
Negative samples encode relative phase. They are part of the information that subsequent transforms use.