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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.

A vector network analyzer measures complex reflection and transmission of electromagnetic signals as a function of frequency. The word vector records that both magnitude and relative phase are measured. It is used in electrical measurement of devices such as transmission lines, filters and amplifiers; the measurement is a comparison of outgoing signals with a reference excitation.

A primary manufacturer explanation describes a tuned receiver that down-converts the measured signal and compares it with a reference split from the source. Its test set samples the reference signal automatically. A longer signal path produces a frequency-dependent phase delay, even if its magnitude is unchanged.^[ref]

For a two-port device, scattering parameters relate incident wave amplitudes $a_1,a_2$ to outgoing amplitudes $b_1,b_2$:When port 2 has no incident wave, $S_{11}=b_1/a_1$ and $S_{21}=b_2/a_1$. The termination and reference-plane conditions belong to the definition. For equal real reference impedances with power-normalized waves, $|S_{21}{|}^2$ is the transmitted-to-incident power ratio under that condition.

For example, $S_{21}=0.5e^{-i\pi /3}$ has magnitude $0.5$, phase $-60^{\circ }$ and power ratio $0.25$. Its logarithmic magnitude is $20{\log }_{10}(0.5)\approx -6.02\mathrm{dB}$. A magnitude-only measurement loses the phase. A pure delay $\tau$ contributes $e^{-2\pi if\tau }$; a $2\mathrm{ns}$ delay changes phase by $-36^{\circ }$ over a $50\mathrm{MHz}$ frequency increase, without changing magnitude. This is an original numerical example of the manufacturer's reference-delay principle.

Instrument calibration and a stated reference plane are consequently important: connecting cables and fixtures can contribute their own response. Phase wraps also require care because angles differing by $360^{\circ }$ describe the same complex value. The complex relative gain error can quantify a deviation that would be invisible if only transmitted magnitude were compared.