The standard definitions of degrees of freedom (DOF) and diversity both normalize by $\logρ$. When this ruler is wrong, both measurements give zero or become undefined, yet intuitively DOF and diversity ought to be channel properties, not artifacts of a normalization choice. We formalize this for Gaussian fading channels. For fixed-$H$ MIMO, DOF and diversity are both ranks of the bilinear map~$HX$ with different variables free: $\varepsilon$-covering the image of~$X\!\mapsto\!HX$ gives DOF on the $\logρ$ gauge; expanding across all dimensions of the fading map gives diversity on the linear~$ρ$ gauge. Covering produces logs; expansion produces linear growth; so in every model studied here the two gauges differ. These geometric definitions do not yield tradeoff curves. We bridge the gap with Bhattacharyya packing, obtaining gauge-DOF and B-diversity as workable proxies -- finite and informative on every gauge, including those where the classical diversity order is zero. Three gauge classes emerge: $\logρ$, $\log\logρ$, and $(\logρ)^β$, $β\in(0,1)$. The main result is a cross-gauge tradeoff for noncoherent fast fading: capacity lives on $\log\logρ$, but B-diversity lives on $\logρ$, exponentially larger, with matching upper and lower bounds. For coherent MIMO, block fading, and irregular-spectrum channels, the same approach recovers or extends known scaling laws.

翻译：自由度和分集度的标准定义均通过$\logρ$进行归一化。当该标度选择不当时，两种度量结果将归零或失去定义，但直觉上自由度和分集度应是信道固有属性而非归一化方式的人为产物。本文针对高斯衰落信道对此进行形式化阐释。对于固定$H$的MIMO系统，自由度和分集度实质是双线性映射$HX$在不同变量自由时的秩：对映射$X\!\mapsto\!HX$的像进行$\varepsilon$覆盖可在$\logρ$标度上得到自由度；在衰落映射的所有维度上进行扩展则在线性$ρ$标度上获得分集度。覆盖产生对数增长，扩展产生线性增长，因此在本文研究的所有模型中两种标度始终存在差异。这些几何定义并不直接生成折衷曲线。我们通过Bhattacharyya填充法弥合此间隙，得到标度-自由度和B-分集度作为实用代理指标——它们在所有标度（包括经典分集阶数为零的情形）上均保持有限且具有信息量。研究揭示了三类标度：$\logρ$、$\log\logρ$以及$(\logρ)^β$（$β\in(0,1)$）。主要成果是针对非相干快衰落的跨标度折衷关系：容量存在于$\log\logρ$标度，而B-分集度存在于指数级更大的$\logρ$标度，且上下界完全匹配。对于相干MIMO、块衰落及不规则频谱信道，该方法同样恢复或扩展了已知的标度律。