Diversity vs Degrees of Freedom in Gaussian Fading Channels

The classical definitions extract degrees of freedom (DOF) via $C(\snr)/\log\snr$ and diversity (DIV) via $-\log P_e(\snr)/\log\snr$, using $\log\snr$ as the common gauge for both. These ratios hide a two-step process: first, identify the gauge on which capacity or reliability actually grows; second, normalize the coefficient on that gauge by the appropriate atom. For coherent multiple-input multiple-output (MIMO) both gauges happen to be $\log\snr$ and both atom coefficients happen to be one. This paper shows that the two-step process is necessary outside this calibration case and makes it explicit using a Bhattacharyya-frontier construction. A capacity--packing sandwich theorem shows that fixed-resolution output-law packing and covering recover the capacity gauge, and a binary-endpoint theorem shows that the two-message Bhattacharyya frontier identifies the zero-rate diversity gauge. Endpoint DOF and endpoint DIV are obtained by dividing the raw coefficient on the identified gauge by the corresponding atom coefficient. For fixed deterministic channel matrix~$H$, the capacity gauge is $\log\snr$ with endpoint DOF $T\,\mathrm{rank}(H)$, while the zero-rate diversity gauge is $\snr$ with endpoint DIV $T\sigma_1^2(H)$, making fixed-$H$ a cross-gauge channel. For noncoherent scalar fast fading with $N$~receive antennas, the capacity gauge is $\log\log\snr$ with DOF~$1$, while the zero-rate diversity gauge is $\log\snr$ with endpoint DIV~$N$; the exact load-$r$ frontier gauge is $(\log\snr)^{1-r}$. The framework recovers the same-gauge cases, coherent Rayleigh MIMO and noncoherent block fading, with zero-rate DIV~$MN$. Audit tables separate exact results from lower bounds and open problems.