Complex relative gain error ($|H/H_{\mathrm{ref}}-1|$)

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.