	Y	Z	S	H	G	A	B	T
Y		Y→Z	Y→S	Y→H	Y→G	Y→A	Y→B	Y→T
Z	Z→Y		Z→S	Z→H	Z→G	Z→A	Z→B	Z→T
S	S→Y	S→Z		S→H	S→G	S→A	S→B	S→T
H	H→Y	H→Z	H→S		H→G	H→A	H→B	H→T
G	G→Y	G→Z	G→S	G→H		G→A	G→B	G→T
A	A→Y	A→Z	A→S	A→H	A→G		A→B	A→T
B	B→Y	B→Z	B→S	B→H	B→G	B→A		B→T
T	T→Y	T→Z	T→S	T→H	T→G	T→A	T→B

From S-parameters to Y-parameters

In matrix form, the formula is

$\mathbf{Y} = Z_0^{-1} K^{-1}\left(I_N-\mathbf{S}\right)\left(I_N+\mathbf{S}\right)^{-1} K$

When dealing with 2 ports, we obtain

$\begin{align*}Y_{11} &=\frac{S_{22}-S_{11}-S_{11}\,S_{22}+S_{12}\,S_{21}+1}{Z_{1}\,\left(S_{11}+S_{22}+S_{11}\,S_{22}-S_{12}\,S_{21}+1\right)}\\Y_{12} &=-\frac{2\,S_{12}\,k_{2}}{Z_{1}\,k_{1}\,\left(S_{11}+S_{22}+S_{11}\,S_{22}-S_{12}\,S_{21}+1\right)}\\Y_{21} &=-\frac{2\,S_{21}\,k_{1}}{Z_{2}\,k_{2}\,\left(S_{11}+S_{22}+S_{11}\,S_{22}-S_{12}\,S_{21}+1\right)}\\Y_{22} &=\frac{S_{11}-S_{22}-S_{11}\,S_{22}+S_{12}\,S_{21}+1}{Z_{2}\,\left(S_{11}+S_{22}+S_{11}\,S_{22}-S_{12}\,S_{21}+1\right)}\\\end{align*}$

The formulas are obtained with the methods explained here. The MATLAB implementation can be found in circuitconversions on Gitlab.

Definitions

$\underbrace{ \begin{bmatrix}B_1 \\ \vdots \\ B_N \end{bmatrix} }_{\mathbf{B}} = \mathbf{S} \underbrace{ \begin{bmatrix}A_1 \\ \vdots \\ A_N \end{bmatrix} }_{\mathbf{A}}$

$\underbrace{ \begin{bmatrix}I_1 \\ \vdots \\ I_N \end{bmatrix} }_{\mathbf{I}} = \mathbf{Y} \underbrace{ \begin{bmatrix}V_1 \\ \vdots \\ V_N \end{bmatrix} }_{\mathbf{V}}$

The incident and reflected waves are defined as

$\mathbf{A} = K \left( \mathbf{V} + Z_0 \mathbf{I} \right)\qquad \mathbf{B} = K \left( \mathbf{V} - Z_0 \mathbf{I} \right)$

where $K$ and $Z_0$ are defined as

$K = \begin{bmatrix}k_1 & & \\ & \ddots & \\ & & k_N \end{bmatrix}\qquad Z_0 = \begin{bmatrix}Z_{0,1} & & \\ & \ddots & \\ & & Z_{0,N} \end{bmatrix}$

with $Z_{0,i}$ the reference impedance for port $i$ and $k_i$ defined as

$k_i=\frac{1}{2\sqrt{\Re\left( Z_{0,i}\right) }}\qquad k_i=\alpha\frac{\sqrt{\Re\left( Z_{0,i} \right) }}{2\left|Z_{0,i}\right|}$

depending on whether you are using power- or pseudowaves.