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An Efficient Frequency Diversity Scheme for Ultra-Reliable Communications in Two-Path Fading Channels

Karl-Ludwig Besser, Eduard A. Jorswieck, Justin P. Coon

TL;DR

The frequency diversity scheme achieves a significant improvement in terms of reliability over using a single frequency and is demonstrated by a numerical simulation of a unmanned aerial vehicle (UAV) above terrain.

Abstract

We consider a two-ray ground reflection scenario with unknown distance between transmitter and receiver. By utilizing two frequencies in parallel, we can mitigate possible destructive interference and ensure ultra-reliability with only very limited knowledge at the transmitter. In order to achieve this ultra-reliability, we optimize the frequency spacing such that the worst-case receive power is maximized. Additionally, we provide an algorithm to calculate the optimal frequency spacing. Besides the receive power, we also analyze the achievable rate and outage probability. It is shown that the frequency diversity scheme achieves a significant improvement in terms of reliability over using a single frequency. In particular, we demonstrate the effectiveness of the proposed approach by a numerical simulation of an unmanned aerial vehicle (UAV) flying above flat terrain.

An Efficient Frequency Diversity Scheme for Ultra-Reliable Communications in Two-Path Fading Channels

TL;DR

The frequency diversity scheme achieves a significant improvement in terms of reliability over using a single frequency and is demonstrated by a numerical simulation of a unmanned aerial vehicle (UAV) above terrain.

Abstract

We consider a two-ray ground reflection scenario with unknown distance between transmitter and receiver. By utilizing two frequencies in parallel, we can mitigate possible destructive interference and ensure ultra-reliability with only very limited knowledge at the transmitter. In order to achieve this ultra-reliability, we optimize the frequency spacing such that the worst-case receive power is maximized. Additionally, we provide an algorithm to calculate the optimal frequency spacing. Besides the receive power, we also analyze the achievable rate and outage probability. It is shown that the frequency diversity scheme achieves a significant improvement in terms of reliability over using a single frequency. In particular, we demonstrate the effectiveness of the proposed approach by a numerical simulation of an unmanned aerial vehicle (UAV) flying above flat terrain.
Paper Structure (16 sections, 8 theorems, 42 equations, 13 figures, 1 table)

This paper contains 16 sections, 8 theorems, 42 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Problem Formulation
  4. Single Frequency
  5. Destructive Interference
  6. Worst-Case Receive Power
  7. Two Frequencies
  8. Optimal Frequency Spacing
  9. Achievable Rate Analysis
  10. Outage Probability
  11. Conclusion
  12. Proof of \ref{['prop:worst-case-rho']}
  13. Proof of \ref{['lem:rec-power-lower-bound-two-freq']}
  14. Proof of \ref{['lem:reformulation-opt-prob']}
  15. Proof of \ref{['thm:opt-dw-max-min']}
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the described two-ray ground reflection model with a single frequency $\omega=2\pi f$. The distance $d$ between transmitter and receiver is in the interval $[d_{\text{min}}, d_{\text{max}}]$. The minimal receive power is then given as

Figures (13)

  • Figure 1: Geometrical model of the considered two-ray ground reflection scenario. The transmitter is placed at height $h_{\text{Tx}, }$ above the ground. The receiver is located at height $h_{\text{Rx}, }$ at a (ground) distance $d$ away from the transmitter. The path and reflection path have lengths $\ell$ and $\tilde{\ell}$, respectively. The angle of reflection is $\alpha$.
  • Figure 2: Reflection coefficient $\Gamma$ from \ref{['eq:def-refl-coeff']} for different dielectric constants of the ground $\epsilon$.
  • Figure 3: Relative phase shift $\Delta\phi$ from \ref{['eq:condition-phase-shift-minimum-single-freq']} for $\omega/c=10$, $h_{\text{Tx}, } =10m$, and $h_{\text{Rx}, } =1.5m$. Additionally the distances $d_k$, $k=1, \dots{}, 4$, from \ref{['eq:distance-k-min-single-freq']} are indicated. (\ref{['ex:single-freq-local-min']})
  • Figure 4: Received power $P_{r}(d)$ from \ref{['eq:rec-power-single-freq']} when using a single frequency $f$ with system parameters $h_{\text{Tx}, } =10m$, $h_{\text{Rx}, } =1.5m$, $\rho=1$, $\Gamma=-1$, and $P_t=1$ for $f=f_1=477MHz$ and $f=f_2=2.4GHz$. Additionally, the distances $d_1(\omega_1)=46.7m$ and $d_2(\omega_1)=21.6m$ from \ref{['eq:distance-k-min-single-freq']} are indicated. (\ref{['ex:single-freq-local-min']} and \ref{['ex:single-freq-worst-case']})
  • Figure 5: Received power $P_{r}(d)$ from \ref{['eq:rec-power-single-freq']} when using a single frequency $f=2.4GHz$ with system parameters $h_{\text{Tx}, } =10m$, $h_{\text{Rx}, } =1.5m$, and $P_t=1$. (\ref{['ex:single-freq-rho']})
  • ...and 8 more figures

Theorems & Definitions (27)

  • Example 1
  • Theorem 1: Minimal Receive Power (Single Frequency)
  • Example 2: Single Frequency Worst-Case Receive Power
  • Example 3: Single Frequency Receive Power with Different Gains and Reflection Coefficients
  • Proposition 1: Worst-Case $\rho$ and $\Gamma$
  • proof
  • Lemma 1: Lower Bound on the Sum Power for Two Frequencies
  • proof
  • Proposition 2: Optimal Power Split for Two Frequencies
  • Example 4: Sum Power Lower Bound
  • ...and 17 more