Paper V: Cyclic Cayley coordinates

E14 Cyclic Cayley coordinates

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Keep the rotation and its chart separate

For the phase factorization below, \(M\) and \(N\) are positive phase orders. The reflection coordinate rotates in the unit disk. The Cayley map expresses that same motion as a fractional linear impedance action. A pole of the chart does not mean that the projective action fails. This realization is independent of the separated jump detector.

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This section records a secondary realization from the original signal manuscript. It is logically independent of the recovery theorems. The classical Smith chart uses the Cayley transform between normalized impedance and a reflection coordinate [Smith1969].

Let

\[ \mathbb H = \{z\in{\mathbb C}:\Re z>0\} \]

and define

\[ C(z)=\frac{z-1}{z+1}. \]

It maps \(\mathbb H\) conformally onto the unit disk \(\mathbb D\), with inverse

\[ C^{-1}(w)=\frac{1+w}{1-w}. \]

For \(\theta\in{\mathbb R}\), let

\[ R_\theta(w)={\mathrm e}^{{\mathrm i}\theta}w. \]

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Theorem E14.1: Cyclic Mobius action

The pullback

\[ \Phi_\theta =C^{-1}R_\theta C \]

is the fractional linear map

\[ \boxed{ \Phi_\theta(z) = \frac{(1+{\mathrm e}^{{\mathrm i}\theta})z+(1-{\mathrm e}^{{\mathrm i}\theta})} {(1-{\mathrm e}^{{\mathrm i}\theta})z+(1+{\mathrm e}^{{\mathrm i}\theta})}. } \]

It preserves \(\mathbb H\) and satisfies

\[ \Phi_\theta\Phi_\varphi = \Phi_{\theta+\varphi}. \]

For an integer \(N\geq1\) and \(\theta=2\pi/N\), the map has order \(N\).

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Proof of Theorem E14.1

Substitute \({\mathrm e}^{{\mathrm i}\theta}(z-1)/(z+1)\) into \(C^{-1}\) and simplify. Preservation of \(\mathbb H\) follows because \(C\) maps it to \(\mathbb D\), rotation preserves \(\mathbb D\), and \(C^{-1}\) maps back. The group law follows by conjugating the rotation group law. When \(\theta=2\pi/N\), the \(N\)th power is the identity. Conversely, if its \(k\)th power is the identity, conjugation by \(C\) and evaluation at \(w=1/2\) give \({\mathrm e}^{2\pi{\mathrm i} k/N}=1\), so \(N\) divides \(k\). Hence the order is exactly \(N\).

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Corollary E14.2: Impedance and admittance

For \(N=2\), \(\theta=\pi\) and

\[ \Phi_\pi(z)=\frac1z. \]

Thus the two phase action is normalized impedance and admittance inversion.

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Proof of Corollary E14.2

Insert \({\mathrm e}^{{\mathrm i}\pi}=-1\) in the formula of Theorem E14.1.

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Proposition E14.3: Radial and phase coordinates

Write

\[ C(z)=\rho{\mathrm e}^{{\mathrm i}\varphi}, \qquad 0\leq\rho<1. \]

Then

\[ C(\Phi_\theta(z)) = \rho{\mathrm e}^{{\mathrm i}(\varphi+\theta)}. \]

The radial mismatch \(\rho\) is invariant and the phase coordinate advances by \(\theta\).

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Proof of Proposition E14.3

This is the defining conjugacy \(C\Phi_\theta=R_\theta C\).

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Proposition E14.4: Coprime phase factorization

Let \(M\) and \(N\) be coprime. The cyclic phase labels modulo \(MN\) are equivalent to pairs of labels modulo \(M\) and modulo \(N\):

\[ {\mathbb Z}/(MN){\mathbb Z} \cong {\mathbb Z}/M{\mathbb Z}\times{\mathbb Z}/N{\mathbb Z}. \]

Under this identification, the rotation by \(2\pi/(MN)\) is determined by its two projected cyclic actions.

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Proof of Proposition E14.4

This is the Chinese remainder theorem applied to the additive phase label. A label modulo \(MN\) is uniquely determined by its two residues, and addition of labels is preserved.

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Phase rotation in the reflection disk and the corresponding fractional-linear action in impedance coordinates. — The cyclic action is an ordinary phase rotation in the reflection disk and a fractional linear action in the impedance half plane.

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