12 Nonlinear signal packets
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Linear time invariance is not required for exact packet coordinates. Let \(A\) and \(B\) be formal power series over a commutative coefficient ring (in particular \({\mathbb C}\)). Set
All phase indices range from \(0\) to \(N-1\). For another representation \(B(z)\), multiplication gives a nonlinear or bilinear interaction.
Theorem 12.1: Cyclic product law
For \(q\in{\mathbb N}\), the \(q\)th-power identity is
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Proof of Theorem 12.1
Every coefficient in \(A_iB_j\) has degree congruent to \(i+j\) modulo \(N\). At any fixed degree \(n\), the Cauchy coefficient sum is finite. Sorting these terms by the residues of their two indices gives the first identity coefficient by coefficient. For the power identity, use induction on \(q\) and the finite product law. At \(q=0\), the index set contains the single empty tuple, its product is \(1\), and its residue sum is \(0\); hence the right side is \(1\) for \(k=0\) and \(0\) otherwise, as required. No convergence is needed for these formal identities. For analytic power series, the same identities hold wherever the series are absolutely convergent and their Cauchy products may be evaluated.
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For a finite nonlinear shift operator
the exact packet coordinates are simply
Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.