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Diversity vs Degrees of Freedom in Gaussian Fading Channels

Mahesh Godavarti

Abstract

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.

Diversity vs Degrees of Freedom in Gaussian Fading Channels

Abstract

The standard definitions of degrees of freedom (DOF) and diversity both normalize by . 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- MIMO, DOF and diversity are both ranks of the bilinear map~ with different variables free: -covering the image of~ gives DOF on the gauge; expanding across all dimensions of the fading map gives diversity on the linear~ 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: , , and , . The main result is a cross-gauge tradeoff for noncoherent fast fading: capacity lives on , but B-diversity lives on , 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.
Paper Structure (39 sections, 16 theorems, 42 equations, 8 tables)

This paper contains 39 sections, 16 theorems, 42 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Fixed-$H$ MIMO: Where Gauges Come From
  3. The Duality
  4. The Cross-Gauge Observation
  5. Bridge Lemma
  6. Main Result: The Cross-Gauge Tradeoff
  7. Fast Fading Model
  8. Minimal Definitions
  9. Scale-Family Reduction
  10. Cross-Gauge Tradeoff
  11. Interpretation
  12. Multipath Extension
  13. The Bhattacharyya Packing Framework
  14. General Channel Model
  15. Layer 1: Gauge-Free Primitives
  16. ...and 24 more sections

Key Result

Lemma 1

Let $D = X_1 - X_2$ have singular values $\sigma_1,\dots,\sigma_r$ ($r = \mathrm{rank}(D)$). (a) Known channel (fixed $H$). a sum of $N\!\cdot\!\mathrm{rank}(D)$ terms, each proportional to $\rho$. (b) Rayleigh fading ($H$ i.i.d. $\mathcal{CN}(0,1)$). a sum of $N\!\cdot\!\mathrm{rank}(D)$ terms, each $\sim\log\rho$.

Theorems & Definitions (53)

  • Lemma 1: Bridge Lemma
  • proof
  • Remark 1: Fading costs one gauge level
  • Remark 2: Rank without SNR asymptotics
  • Lemma 2: Gauge class under peak vs. average power
  • Lemma 3: Energy-only reduction
  • Theorem 1: Cross-gauge tradeoff
  • proof
  • Remark 3: B-diversity vs. classical diversity: fast fading
  • Remark 4: Operational status of B-diversity
  • ...and 43 more