= Mixed peak and RMS gain-calibration error
A <mixed peak and RMS gain-calibration error> occurs when input and output amplitude reports use different reconstruction scales but their numerical ratio is treated as the physical gain. Let the true <peak amplitude phasors> obey $\widehat V_{\rm out}=G\widehat V_{\rm in}$. If a reported amplitude $A$ reconstructs the signal with scale $a$, so that $\widehat V=aA$, then
$$
G=\frac{a_{\rm out}}{a_{\rm in}}\frac{A_{\rm out}}{A_{\rm in}}.
$$
The scale is $1$ for a <peak phasor> and $\sqrt2$ for an <RMS phasor>. Equal physical units do not establish equal amplitude conventions. The sinusoidal RMS-to-peak conversion follows from the period average of the squared signal.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 7, printed pp. 4–7, equations 7-2 and 7-5]{ref}
For a peak-normalized input and RMS-normalized output, the raw ratio is $G/\sqrt2$; for the reverse pairing it is $\sqrt2G$. The amplitude-gain error in decibels is therefore minus or plus
$$
20\log_{10}\sqrt2\approx3.0103\,\mathrm{dB}.
$$
These positive real normalization factors do not themselves change the relative phase. They change the measured gain magnitude.
For an original calibration example, let a sinusoidal input be reported as $0.2\,\mathrm V$ peak and its output as $1.0\,\mathrm V$ RMS. The raw amplitude ratio is $5$, or about $13.9794\,\mathrm{dB}$. Converting the output to peak amplitude gives actual gain $5\sqrt2\approx7.0711$, or about $16.9897\,\mathrm{dB}$. A source-to-receiver comparison that ignores the conventions would falsely report a $3.0103\,\mathrm{dB}$ gain deficit.
A further factor-of-two issue occurs when one report is a positive-frequency <Fourier coefficient> rather than a <peak phasor>; <peak phasor versus positive-frequency Fourier coefficient> derives that distinction. It produces a different $6.0206\,\mathrm{dB}$ amplitude factor and must not be treated as an RMS conversion. Likewise, <peak-versus-RMS normalization in complex electrical power> uses a product of reconstruction scales rather than the ratio used here. A calibration record should therefore state the sinusoidal amplitude convention and, for sampled estimates, the FFT normalization before numerical gains are compared.
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