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Phase-Only Zero-Forcing for Secure Wireless Communication in Multi-User Systems

Jordan Hong, Safwan Jamal, Ashish Khisti

TL;DR

A novel successive partition zero-forcing (SPZF) scheme is proposed, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability and can attain arbitrarily low outage probability in the limit of large number of transmit antenna.

Abstract

Artificial noise (AN) transmission is a physical layer security technique in multi-antenna wireless communication systems. Synthetic noise is broadcast to all receivers except designated legitimate users via beamforming in the legitimate users' null space. We consider AN transmission employing a single RF chain and analog beamforming, where beamforming vectors maintain constant magnitude while allowing arbitrary phases. Our primary objective is to design a constant-magnitude vector capable of nullifying multiple users' channel vectors simultaneously. To tackle this zero-forcing problem, we propose a novel successive partition zero-forcing (SPZF) scheme, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability. The SPZF scheme can be generalized to any number of users, but our analysis focuses on the two-user case. Theoretical analysis reveals that our proposed SPZF scheme can attain arbitrarily low outage probability in the limit of large number of transmit antenna. We present three partition algorithms (random, iterative, and genetic) to minimize the outage probability. The outage probabilities and secrecy rates of the three partition algorithms are compared via numerical simulations. We find that the more advanced partition algorithms (iterative and genetic) achieve higher secrecy rates than the random algorithm, particularly under conditions of high signal-to-noise ratio (SNR), large number of eavesdroppers, or small number of transmit antennas.

Phase-Only Zero-Forcing for Secure Wireless Communication in Multi-User Systems

TL;DR

A novel successive partition zero-forcing (SPZF) scheme is proposed, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability and can attain arbitrarily low outage probability in the limit of large number of transmit antenna.

Abstract

Artificial noise (AN) transmission is a physical layer security technique in multi-antenna wireless communication systems. Synthetic noise is broadcast to all receivers except designated legitimate users via beamforming in the legitimate users' null space. We consider AN transmission employing a single RF chain and analog beamforming, where beamforming vectors maintain constant magnitude while allowing arbitrary phases. Our primary objective is to design a constant-magnitude vector capable of nullifying multiple users' channel vectors simultaneously. To tackle this zero-forcing problem, we propose a novel successive partition zero-forcing (SPZF) scheme, which transforms the multi-user zero-forcing task into optimizing channel partitioning to minimize outage probability. The SPZF scheme can be generalized to any number of users, but our analysis focuses on the two-user case. Theoretical analysis reveals that our proposed SPZF scheme can attain arbitrarily low outage probability in the limit of large number of transmit antenna. We present three partition algorithms (random, iterative, and genetic) to minimize the outage probability. The outage probabilities and secrecy rates of the three partition algorithms are compared via numerical simulations. We find that the more advanced partition algorithms (iterative and genetic) achieve higher secrecy rates than the random algorithm, particularly under conditions of high signal-to-noise ratio (SNR), large number of eavesdroppers, or small number of transmit antennas.

Paper Structure

This paper contains 42 sections, 10 theorems, 51 equations, 10 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. System Model
  3. Problem Formation and Related Work
  4. Phase-only zero-forcing: the general $K$ case
  5. Phase-only zero-forcing: special case of $K = 1$
  6. Phase-only zero-forcing for the special case of $K = 2$
  7. Successive Partition Zero-forcing
  8. Motivational Example for the special case of $K = 2$
  9. Formalized SPZF algorithm for $K = 2$
  10. Theoretical Analysis for the special case of $K = 2$
  11. Problem Transformation for the i.i.d. Rayleigh channel
  12. Objective functions
  13. Proposed Partition Algorithm
  14. Random partition
  15. Iterative Partition Algorithm
  16. ...and 27 more sections

Key Result

Lemma 1

Given any $\mathbf{h} \in \mathbb{C} ^N$ with $N \geq 3$, there exists a PZF vector $\mathbf{w}$ that satisfies Equation eq:pzf if and only if:

Figures (10)

  • Figure 1: Schematics of the phased-array transmission structure
  • Figure 2: Geometric interpretation of Equation \ref{['eq:pzf-K1']} when N = 3: rotating each $h_i$ to form a triangle.
  • Figure 3: Flowchart of the genetic algorithm.
  • Figure 4: Example of chromosome of a solution. We encode each potential partition solution as a $N$-vector. The $i$-th element takes value from $\{1, \cdots, m\}$, which denotes the partition index that $h_i$ belongs to.
  • Figure 5: Comparison of $\Pr[\mathtt{outage}]$ of the SPZF using a random partition v.s. $f_\text{Ray}(m)$ outage probability of a Rayleigh i.i.d. fading channel with $m$ terms. The two-channel outage when $N = 50$ matches the empirical $f_\text{Ray}(m)$.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Lemma 1: Polygon inequality
  • Lemma 2
  • Definition 1: Partition
  • Definition 2: Partition matrix
  • Definition 3
  • Definition 4
  • Remark 1
  • Definition 5: Error events
  • Definition 6
  • Proposition 1
  • ...and 8 more