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 ${\hat{V}}_{\mathrm{out}}=G{\hat{V}}_{\mathrm{in}}$. If a reported amplitude $A$ reconstructs the signal with scale $a$, so that $\hat{V}=aA$, thenThe scale is $1$ for a peak phasor and $\sqrt{2}$ 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.^[ref]

For a peak-normalized input and RMS-normalized output, the raw ratio is $G/\sqrt{2}$; for the reverse pairing it is $\sqrt{2}G$. The amplitude-gain error in decibels is therefore minus or plusThese 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 $5\sqrt{2}\approx 7.0711$, 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.