Diversity vs Degrees of Freedom in Gaussian Fading Channels
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2026
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| _version_ | 1866915799793401856 |
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| author | Godavarti, Mahesh |
| author_facet | Godavarti, Mahesh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Diversity vs Degrees of Freedom in Gaussian Fading Channels Godavarti, Mahesh Information Theory 94A15, 94A05, 62B10, 15A03, 52C17 E.4; G.3; H.1.1; C.4 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. |
| title | Diversity vs Degrees of Freedom in Gaussian Fading Channels |
| topic | Information Theory 94A15, 94A05, 62B10, 15A03, 52C17 E.4; G.3; H.1.1; C.4 |
| url | https://arxiv.org/abs/2602.14371 |