# Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward

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## Abstract

**:**

## 1. Introduction

#### 1.1. Main Contributions

- We propose a novel precoding technique to implement physical layer network coding using CF. Our precoder design is based on integer forcing and allows bypassing the self-noise limitation of existing formulations, thus providing higher achievable rates than existent relaying schemes.
- We developed a decoder for relays of CF and characterized it using the generalization of Bezout’s theorem. We also made an analytical derivation of the end-to-end probability of error for cubic lattices.
- We analyze the two phases of transmission in the CF scheme, thereby characterizing the end-to-end behavior of the CF based on different performance metrics and not only one-phase behavior, as in available studies. Compared to conventional designs, we obtain a gain of approximately 5 dB in terms of probability of error as compared to conventional time division multiplexing (TDM) transmission.

#### 1.2. Outline of the Paper

#### 1.3. Notations

## 2. System Model

#### 2.1. Phase I: Source to Relay Multiple Access Channel

#### 2.2. Phase II: Relay-to-Destination Point-to-Point Channel

## 3. Phase I : Proposed Precoding-Based Source to Relay Transmission

#### 3.1. Problem Formulation

#### 3.2. Proposed Integer Forcing Precoder

#### 3.3. Decoding at the Relay

## 4. Phase II: Proposed Point-to-Point Relay-to-Destination Transmission

#### 4.1. Problem Formulation

#### 4.2. Decoding Linear Combination at the Destination

#### 4.3. Decoding Original Signals at the Destination

## 5. Performance Analysis

#### 5.1. Probability of Error from Source to Relay Node

#### 5.2. Probability of Error from Relay-to-Destination Node

#### 5.3. Overall End-to-End Probability of Error

## 6. Numerical Results

#### 6.1. Probability of Error

#### 6.1.1. Probability of Error at the Relay

**Figure 2.**Comparison of the probability of error at the relay for the proposed scheme with varying signal to noise ratio (SNR) for $L=2$. The total signal power is adjusted for fair comparison with other schemes.

#### 6.1.2. Probability of Error at the Destination

**Figure 3.**Compute and forward (CF) with the integer forcing precoder (IFP) compared to Time Division Multiplexing (TDM) performance at the destination.

#### 6.2. Outage Probability

**Figure 4.**Outage probability curve. The horizontal axis shows the power available at the transmitter. The power available at the transmitter is assumed to be always greater than or equal to the signal power, hence ${\gamma}_{a}>1$. With the increasing channel variance, ${\sigma}_{h}$, the outage decreases.

#### 6.3. Achievable Rates

**h**= [${h}_{1}$ ${h}_{2}$], are approximated by integers,

**a**= [${a}_{1}$ ${a}_{2}$]. The $log$ is taken to the base of two. The rates are measured in bits per channel use (bpcu). For decode and forward, the relay decodes one of the signals, treating the other signal as noise. Hence:

**Figure 5.**Comparison of achievable rates for various schemes for $L=2$. The average value is taken over 10,000 different random channel realizations. The rates are measured in bits per channel use (bpcu).

## 7. Conclusions

## References

- Zhang, S.; Liew, S.; Lam, P. Physical Layer Network Coding. 2006. arXiv.org Website. Available online: http://arxiv.org/abs/0704.2475 (accessed on 12 August 2013).
- Nazer, B.; Gastpar, M. Compute and forward: Harnessing interference through structured codes. IEEE Trans. Inf. Theory
**2011**, 57, 6463–6486. [Google Scholar] [CrossRef] - Feng, C.; Silva, D.; Kschischang, F. An Algebraic Approach to Physical-Layer Network Coding. In Proceedings of 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), Austin, TX, USA, 13–18 June 2010.
- Niesen, U.; Whiting, P. The degrees of freedom of compute-and-forward. IEEE Trans. Inf. Theory
**2012**, 58, 5214–5232. [Google Scholar] [CrossRef] - Belfiore, J.-C. Lattice Codes for the Compute-and-Forward Protocol: The Flatness Factor. In Proceedings of the IEEE Information Theory Workshop (ITW), Paraty, Brazil, 16–20 October 2011; pp. 1–4.
- Nazer, B.; Gastpar, M. Reliable physical layer network coding. Proc. IEEE
**2011**, 99, 438–460. [Google Scholar] [CrossRef] - Chen, F.; Silva, D.; Kschischang, F.R. Blind Compute-and-Forward. In Proceedings of the IEEE International Symposium on Information Theory Proceedings (ISIT), Cambridge, MA, USA, 1–6 July 2012; pp. 403–407.
- Hong, S.-N.; Caire, G. Reverse Compute and Forward: A Low-Complexity Architecture for Downlink Distributed Antenna Systems. In Proceedings of the IEEE International Symposium on Information Theory Proceedings (ISIT), Cambridge, MA, USA, 1–6 July 2012; pp. 1147–1151.
- Yang, T.; Yuan, X.; Li, P.; Collings, I.B.; Yuan, J. A new physical-layer network coding scheme with eigen-direction alignment precoding for mimo two-way relaying. IEEE Trans. Commun.
**2013**, 61, 973–986. [Google Scholar] [CrossRef] - Gupta, S.; Vazquez-Castro, M.A. Physical-Layer Network Coding based on Integer-forcing Precoded Compute and Forward. In Proceedings of the IEEE 8th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Barcelona, Spain, 8–10 October 2012; pp. 600–605.
- Wiesel, A.; Eldar, Y.; Shamai, S. Zero-forcing precoding and generalized inverses. IEEE Trans. Signal Process.
**2008**, 56, 4409–4441. [Google Scholar] [CrossRef] - Hassibi, B.; Vikalo, H. On the sphere-decoding algorithm I. Expected complexity. IEEE Trans. Signal Process.
**2005**, 53, 2806–2818. [Google Scholar] [CrossRef] - Zamir, R. Lattices are Everywhere. In Proceedings of the Information Theory and Applications Workshop, San Diego, CA, USA, 8–13 February 2009; pp. 392–421.
- Forney, G.D., Jr. Coset codes-I: Introduction and geometrical classification. IEEE Trans. Inf. Theory
**1988**, 34, 1123–1151. [Google Scholar] [CrossRef] - Monteiro, F.A.; Wassell, I.J. Dual-Lattice-Aided MIMO Detection for Slow Fading Channel. In Proceedings of the IEEE International Symposium on Signal Processing and Information Technology, Bilbao, Spain, 14–17 December 2011; pp. 502–507.
- Tarokh, V.; Vardy, A.; Zeger, K. Universal bound on the performance of lattice codes. IEEE Trans. Inf. Theory
**1999**, 45, 670–681. [Google Scholar] [CrossRef] - Shub, M.; Smale, S. Complexity of Bezout’s theorem. I. Geometric aspects. J. Am. Math. Soc.
**1993**, 6, 459–501. [Google Scholar] [CrossRef] - Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed.; McGraw-Hill: New York, NY, USA, 1984; p. 104. [Google Scholar]

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**MDPI and ACS Style**

Gupta, S.; Vázquez-Castro, M.A.
Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward. *Future Internet* **2013**, *5*, 439-459.
https://doi.org/10.3390/fi5030439

**AMA Style**

Gupta S, Vázquez-Castro MA.
Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward. *Future Internet*. 2013; 5(3):439-459.
https://doi.org/10.3390/fi5030439

**Chicago/Turabian Style**

Gupta, Smrati, and M. A. Vázquez-Castro.
2013. "Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward" *Future Internet* 5, no. 3: 439-459.
https://doi.org/10.3390/fi5030439