# Maxmin Strategy for a Dual Radar and Communication OFDM Waveforms System Facing Uncertainty About the Background Noise

## Abstract

The paper considers the problem of designing the maxmin strategy for a dual-purpose communication and radar system that employs multicarrier OFDM style waveforms, but faces an uncertain level of background noise. As the payoff for the system, we consider the weighted sum of the communication throughput and the radar’s SINR. The problem is formulated as a zero-sum game between the system and a rival, which may be thought of as the environment or nature. Since the payoff for such a system combines different type of metrics (SINR and throughput), this makes underlying problem associated with jamming such a systems different from the typical jamming problem arising in communication scenarios, where the payoff usually involves only one of these metrics. In this paper, the existence and uniqueness of the equilibrium strategies are proven as well as water-filling equations to design the equilibrium are derived. Finally, using Nash product the optimal value of weights are found to optimize tradeoff of radar and communication objectives.

## Keywords

Dual-purpose communication and radar system Maxmin Background noise## References

- 1.Aubry, A., De Maio, A., Huang, Y., Piezzo, M., Farina, A.: A new radar waveform design algorithm with improved feasibility for spectral coexistence. IEEE Trans. Aerosp. Electron. Syst.
**51**, 1029–1038 (2015)CrossRefGoogle Scholar - 2.Bica, M., Huang, K.W., Koivunen, V., Mitra, U.: Mutual information based radar waveform design for joint radar and cellular communication systems. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3671–3675 (2016)Google Scholar
- 3.Bica, M., Koivunen, V.: Delay estimation method for coexisting radar and wireless communication systems. In: IEEE Radar Conference, pp. 1557–1561 (2017)Google Scholar
- 4.Garnaev, A., Liu, Y., Trappe, W.: Anti-jamming strategy versus a low-power jamming attack when intelligence of adversary’s attack type is unknown. IEEE Trans. Sig. Inf. Process. Netw.
**2**, 49–56 (2016)MathSciNetGoogle Scholar - 5.Garnaev, A., Trappe, W.: To eavesdrop or jam, that is the question. In: Mellouk, A., Sherif, M.H., Li, J., Bellavista, P. (eds.) ADHOCNETS 2013. LNICST, vol. 129, pp. 146–161. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-04105-6_10CrossRefGoogle Scholar
- 6.Garnaev, A., Trappe, W.: Bargaining over the fair trade-off between secrecy and throughput in OFDM communications. IEEE Trans. Inf. Forensics Secur.
**12**, 242–251 (2017)CrossRefGoogle Scholar - 7.Garnaev, A., Trappe, W.: The rival might be not smart: revising a CDMA jamming game. In: IEEE Wireless Communications and Networking Conference (WCNC). IEEE (2018)Google Scholar
- 8.Garnaev, A., Trappe, W., Petropulu, A.: Bargaining over fair performing dual radar and communication task. In: 50th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, pp. 47–51, November 2016Google Scholar
- 9.Garnaev, A., Trappe, W., Petropulu, A.: Optimal design of a dual-purpose communication-radar system in the presence of a Jammer. In: IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp. 1–5 (2018)Google Scholar
- 10.Garnaev, A., Trappe, W.: Fair scheduling of two-hop transmission with energy harvesting. In: Zhou, Y., Kunz, T. (eds.) Ad Hoc Networks. LNICST, vol. 223, pp. 189–198. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74439-1_17CrossRefGoogle Scholar
- 11.Gogineni, S., Rangaswamy, M., Nehorai, A.: Multi-modal OFDM waveform design. In: IEEE Radar Conference, pp. 1–5 (2013)Google Scholar
- 12.Gohary, R.H., Huang, Y., Luo, Z.-Q., Pang, J.-S.: A generalized iterative water-filling algorithm for distributed power control in the presence of a Jammer. In: IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2373–2376 (2009)Google Scholar
- 13.Han, Z., Niyato, D., Saad, W., Basar, T., Hjrungnes, A.: Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
- 14.Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim.
**9**, 112–147 (1998)MathSciNetCrossRefGoogle Scholar - 15.Lan, T., Kao, D., Chiang, M., Sabharwal, A.: An axiomatic theory of fairness in network resource allocation. In: IEEE INFOCOM, pp. 1–9 (2010)Google Scholar
- 16.Li, B., Petropulu, A.P., Trappe, W.: Optimum co-design for spectrum sharing between matrix completion based MIMO radars and a MIMO communication system. IEEE Trans. Sig. Process.
**64**, 4562–4575 (2016)MathSciNetCrossRefGoogle Scholar - 17.Liu, Y., Garnaev, A., Trappe, W.: Maintaining throughput network connectivity in ad hoc networks. In: 41st IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6380–6384 (2016)Google Scholar
- 18.Namvar, N., Saad, W., Bahadori, N., Kelleys, B.: Jamming in the Internet of Things: a game-theoretic perspective. In: IEEE Global Communications Conference (GLOBECOM), Washington, DC (2016)Google Scholar
- 19.Park, H., van der Schaar, M.: Bargaining strategies for networked multimedia resource management. IEEE Trans. Sig. Process.
**55**, 3496–3511 (2007)MathSciNetCrossRefGoogle Scholar - 20.Poor, H.V.: An Introduction to Signal Detection and Estimation. Springer, New York (1994). https://doi.org/10.1007/978-1-4757-2341-0CrossRefzbMATHGoogle Scholar
- 21.Federal Communications Commission (FCC): FCC proposes innovative small cell use in 3.5 GHz band, December 2012. https://apps.fcc.gov/edocs_public/attachmatch/DOC-317911A1.pdf
- 22.Slimeni, F., Scheers, B., Le Nir, V., Chtourou, Z., Attia, R.: Closed form expression of the saddle point in cognitive radio and Jammer power allocation game. In: Noguet, D., Moessner, K., Palicot, J. (eds.) CrownCom 2016. LNICST, vol. 172, pp. 29–40. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40352-6_3CrossRefGoogle Scholar
- 23.Song, T., Stark, W.E., Li, T., Tugnait, J.K.: Optimal multiband transmission under hostile jamming. IEEE Trans. Commun.
**64**, 4013–4027 (2016)CrossRefGoogle Scholar - 24.Thomson, W.: Bargaining and the Theory of Cooperative Games: John Nash and Beyond. Edward Elgar Pub., Cheltenham (2010)CrossRefGoogle Scholar
- 25.Turlapaty, A., Jin, Y.: A joint design of transmit waveforms for radar and communications systems in coexistence. In: IEEE Radar Conference, pp. 0315–0319 (2014)Google Scholar
- 26.Yang, D., Xue, G., Zhang, J., Richa, A., Fang, X.: Coping with a smart Jammer in wireless networks: a stackelberg game approach. IEEE Trans. Wirel. Commun.
**12**, 4038–4047 (2013)CrossRefGoogle Scholar