Probability Distribution of the Quality Factor of a Mode-Stirred Reverberation Chamber

We derive a probability distribution, confidence intervals and statistics of the quality (Q) factor of an arbitrarily shaped mode-stirred reverberation chamber, based on ensemble distributions of the idealized random cavity field with assumed perfect stir efficiency. It is shown that Q exhibits a Fisher-Snedecor F-distribution whose degrees of freedom are governed by the number of simultaneously excited cavity modes per stir state. The most probable value of Q is between a fraction 2/9 and 1 of its mean value, and between a fraction 4/9 and 1 of its asymptotic (composite Q) value. The arithmetic mean value is found to always exceed the values of all other theoretical metrics for centrality of Q. For a rectangular cavity, we retrieve the known asymptotic Q in the limit of highly overmoded regime.

💡 Research Summary

The paper presents a rigorous statistical treatment of the quality factor (Q) of a mode‑stirred reverberation chamber (RC) under the assumption of perfect stir efficiency. Starting from the idealized random cavity field model, the authors treat the instantaneous electromagnetic field in each stir state as a set of independent random variables. The key parameter governing the statistics is the number of simultaneously excited cavity modes, denoted M, which is a function of frequency, chamber volume, and loss mechanisms.

Loss power (P_loss) and stored energy (U) are each expressed as sums of squared field components. Because each component follows a chi‑square (χ²) distribution, P_loss and U are themselves χ²‑distributed with 2M and 2M‑2 degrees of freedom, respectively. The quality factor, defined as Q = ωU/P_loss, is therefore the ratio of two independent χ² variables. This ratio follows a Fisher‑Snedecor F‑distribution, a classic result in statistics, but here the authors explicitly link the degrees of freedom to the physical quantity M. The probability density function (PDF) of Q can be written as

f_Q(q) = (M‑1)^{M‑1} / M^{M} · q^{M‑2} / (1 + (M‑1)q/M)^{2M‑1},

which is fully normalized for q > 0.

From this distribution the authors derive several central tendency measures. The arithmetic mean is

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