= Scattering parameters
{title2=$b=Sa$}
<Scattering parameters> describe a linear network by relating waves incident on its ports to waves leaving them at a specified <frequency>. For incident-wave vector $a$ and outgoing-wave vector $b$,
$$
b=Sa.
$$
The entry $S_{ij}$ is outgoing wave $b_i$ divided by the incident wave $a_j$ when every other incident wave is zero. The first index is the receiving port; the second is the excited port. Keysight's primary analyzer documentation makes this order explicit and identifies $S_{11},S_{22}$ as reflection measurements and $S_{21},S_{12}$ as transmission measurements.<Measurement Parameters in Keysight E5055A documentation>[S-Parameters and Common Measurements with S-Parameters]{ref}
For a two-port network,
$$
\begin{pmatrix}b_1\\b_2\end{pmatrix}
=\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}
\begin{pmatrix}a_1\\a_2\end{pmatrix}.
$$
A matched termination at port 2 gives $a_2=0$ for a port-1 measurement. Leaving that port open generally produces a reflected incident wave instead and changes the observed input response. Port reference impedances and calibration reference planes belong to the definition of the measured network.
An original normalization calculation uses positive real reference <characteristic impedance> $Z_0$ and complex RMS port quantities $V,I$, with current into the network:
$$
a=\frac{V+Z_0I}{2\sqrt{Z_0}},\qquad
b=\frac{V-Z_0I}{2\sqrt{Z_0}}.
$$
Subtraction of their squared magnitudes gives
$$
|a|^2-|b|^2=\operatorname{Re}(VI^*),
$$
so these wave magnitudes have squared units of <power>. The real-positive reference assumption is essential to this simple normalization; other reference conventions require their own definitions.
For an illustrative matched-output test with incident <power> $|a_1|^2=1\,\mathrm W$, suppose $S_{11}=0.1$ and $S_{21}=0.8e^{-i\pi/3}$. Reflected power is $0.01\,\mathrm W$ and transmitted power is $0.64\,\mathrm W$. Net power entering the device through the two ports is $0.35\,\mathrm W$. The transmission phase is $-\pi/3$; magnitude alone would omit it. These are stipulated network values, not a particular analyzer measurement.
For a one-port load, the same normalization gives <reflection coefficient>
$$
\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0}.
$$
A $150\,\Omega$ resistive load at a $50\,\Omega$ reference gives $\Gamma=0.5$, returning one quarter of incident <power>. Matching requires the stated reference, not merely a small resistance.
A <vector network analyzer> separates reference and test receiver signals and forms the relevant complex ratios. NIST's optical-receiver metrology apparatus used electrical network and mismatch characterization alongside optical beating, illustrating why interconnecting hardware cannot always be treated as transparent.<Frequency response metrology for high-speed optical receivers by Paul D. Hale and others>[PDF pp.1–2, Heterodyne measurements and network corrections]{ref} A proposed measurement therefore records termination, reference plane, <frequency> and amplitude regime; an S-matrix measured for a linear operating point does not by itself characterize nonlinear large-signal behavior.
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