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For a nonzero reference transfer function $H_{\mathrm{ref}}$, the complex relative gain error is the relative errorIt measures the distance from the complex ratio to one, so it responds to both amplitude error and phase error. A vector vector network analyzer measures both quantities; replacing its complex result by a magnitude discards part of the measured information.^[ref]

Write $H/H_{\mathrm{ref}}=a{e}^{i\varphi }$, where $a\ge 0$ and $\varphi$ is the phase difference. Multiplying by the complex conjugate givesThus a pure five-degree phase error has $E=2\sin (2.5\pi /180)\approx 0.08724$, about $8.72\%$, even though the magnitude ratio is exactly one. For small $\delta =a-1$ and small $\varphi$ measured in radians, $E\approx \sqrt{{\delta }^2+{\varphi }^2}$. Adding the two unsigned errors is generally not the same operation.

The metric also controls signals. Suppose two linear time-invariant systems have frequency responses $H,H_{\mathrm{ref}}$ and act on the same input in L2 space. On its frequency support assume $H_{\mathrm{ref}}\ne 0$ almost everywhere and $|H/H_{\mathrm{ref}}-1|\le \epsilon$ almost everywhere. Applying the norm bound for a Fourier multiplier on L2 to the reference output yieldsThis is an output-relative bound, so input frequencies suppressed by a zero of $H_{\mathrm{ref}}$ require separate treatment. Near a reference zero, a small absolute error can produce an enormous complex relative error; an absolute or explicitly fixed-scale tolerance is then more informative.

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 $b$,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.^[ref]

For a two-port network,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:Subtraction of their squared magnitudes givesso 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 coefficientA $150\Omega$ 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.^[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.