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Asymptotic analysis of sum-rate under SIC

Andrea Baiocchi, Asmad Razzaque

TL;DR

This paper explores the large-scale performance of SIC in a theoretical framework, and the probability distribution of the number of correctly decoded packets is shown to tend to a deterministic distribution asymptotically for large values of $n.

Abstract

Limitation of the cost of coordination and contention among a large number of nodes calls for grant-free approaches, exploiting physical layer techniques to solve collisions. Successive Interference Cancellation (SIC) is becoming a key building block of multiple access channel receiver, in an effort to support massive Internet of Things (IoT). In this paper, we explore the large-scale performance of SIC in a theoretical framework. A general model of a SIC receiver is stated for a shared channel with $n$ transmitters. The asymptotic sum-rate performance is characterized as $n \rightarrow \infty$, for a suitably scaled target Signal to Noise Interference Ratio (SNIR). The probability distribution of the number of correctly decoded packets is shown to tend to a deterministic distribution asymptotically for large values of $n$. The asymptotic analysis is carried out for any probability distribution of the wireless channel gain, assuming that the average received power level is same for all nodes, through power control.

Asymptotic analysis of sum-rate under SIC

TL;DR

This paper explores the large-scale performance of SIC in a theoretical framework, and the probability distribution of the number of correctly decoded packets is shown to tend to a deterministic distribution asymptotically for large values of $n.

Abstract

Limitation of the cost of coordination and contention among a large number of nodes calls for grant-free approaches, exploiting physical layer techniques to solve collisions. Successive Interference Cancellation (SIC) is becoming a key building block of multiple access channel receiver, in an effort to support massive Internet of Things (IoT). In this paper, we explore the large-scale performance of SIC in a theoretical framework. A general model of a SIC receiver is stated for a shared channel with transmitters. The asymptotic sum-rate performance is characterized as , for a suitably scaled target Signal to Noise Interference Ratio (SNIR). The probability distribution of the number of correctly decoded packets is shown to tend to a deterministic distribution asymptotically for large values of . The asymptotic analysis is carried out for any probability distribution of the wireless channel gain, assuming that the average received power level is same for all nodes, through power control.
Paper Structure (14 sections, 4 theorems, 107 equations, 9 figures)

This paper contains 14 sections, 4 theorems, 107 equations, 9 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Scaling of multiple access parameters with the number of nodes
  4. Asymptotic Analysis
  5. Generalization
  6. Conclusions
  7. Proof of bounds on $m_k(\gamma)$
  8. Proof of instability with fixed parameters
  9. Proof of scaling of $p_n$ and $\gamma_n$
  10. Proof of \ref{['theo:introVj']}
  11. Proof of \ref{['theo:asymptoticpropertiesofmomentsofVj']}
  12. Asymptotic regime for large $n$: mean value of $V_j$
  13. Asymptotic regime for large $n$: variance of $V_j$
  14. Proof of \ref{['theo:mainresult']}

Key Result

Theorem 1

Let $V_j, \, j = 1,\dots,n$, be a non-negative random variables defined by where $X_k, \, k = 1,\dots,n$, are i.i.d. negative exponential random variables with mean 1 and for $j = 1,\dots,n$. The packet carried by the $m$-th strongest signal is decoded successfully if and only if $V_j \ge c$ for $j = 1,\dots,m$.

Figures (9)

  • Figure 1: Probability that the random variable $V_j$ exceeds the threshold $c$ for $j = 1,\dots,n$. The abscissa is normalized as $j/n$. The required SNIR threshold is set to $\gamma = 1/( \alpha n )$, with $\alpha = 0.32$ in the plot on the left, $\alpha = 0.38$ in the plot on the right.
  • Figure 2: Sequence of mean values $\mu_j(n)$ (discrete stems ending with a circle) compared with function $f_{\alpha,\xi}(x)$ (thick red line) as a function of $x = j/n$ for several values of $n$ and $\xi = 0$ and $\alpha = 0.32$.
  • Figure 3: Sequence of mean values $\mu_j(n)$ as a function of $x = j/n$ for several values of $n$ and $\xi = 0$. The shaded area corresponds to one standard deviation, i.e., $\mu_j(n) \pm \sigma_j(n)$. The dashed line is the function $f_{\alpha,\xi}(x)$ defined in \ref{['eq:definitionfofx']}. The required SNIR threshold is set to $\gamma = 1/( \alpha n )$ with $\alpha = 0.32$
  • Figure 4: Sequence of mean values $\mu_j(n)$ as a function of $x = j/n$ for several values of $n$ and $\xi = 0$. The shaded area corresponds to one standard deviation, i.e., $\mu_j(n) \pm \sigma_j(n)$. The dashed line is the function $f_{\alpha,\xi}(x)$ defined in \ref{['eq:definitionfofx']}. The required SNIR threshold is set to $\gamma = 1/( \alpha n )$ with $\alpha = 0.38$
  • Figure 5: Sequence of mean values $\mu_j(n)$ as a function of $x = j/n$ for several values of $n$ and $\xi = 0$. The shaded area corresponds to one standard deviation, i.e., $\mu_j(n) \pm \sigma_j(n)$. The dashed line is the function $f_{\alpha,\xi}(x)$ defined in \ref{['eq:definitionfofx']}. The required SNIR threshold is set to $\gamma = 1/( \alpha n )$ with $\alpha = 0.32$
  • ...and 4 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4