Jamming in Fixed-Rate Wireless Systems with Power Constraints - Part I: Fast Fading Channels

This is the first part of a two-part paper that studies the problem of jamming in a fixed-rate transmission system with fading. Both transmitter and jammer are subject to power constraints which can be enforced over each codeword short-term / peak) or over all codewords (long-term / average), hence generating different scenarios. All our jamming problems are formulated as zero-sum games, having the probability of outage as pay-off function and power control functions as strategies. The paper aims at providing a comprehensive coverage of these problems, under fast and slow fading, peak and average power constraints, pure and mixed strategies, with and without channel state information (CSI) feedback. In this first part we study the fast fading scenario. We first assume full CSI to be available to all parties. For peak power constraints, a Nash equilibrium of pure strategies is found. For average power constraints, both pure and mixed strategies are investigated. With pure strategies, we derive the optimal power control functions for both intra-frame and inter-frame power allocation. Maximin and minimax solutions are found and shown to be different, which implies the non-existence of a saddle point. In addition we provide alternative perspectives in obtaining the optimal intra-frame power control functions under the long-term power constraints. With mixed strategies, the Nash equilibrium is found by solving the generalized form of an older problem dating back to Bell and Cover \cite{bell}. Finally, we derive a Nash equilibrium of the game in which no CSI is fed back from the receiver. We show that full channel state information brings only a very slight improvement in the system’s performance.

💡 Research Summary

This paper tackles the problem of jamming in a fixed‑rate wireless communication system operating over fast‑fading channels, formulating the interaction between the legitimate transmitter and a malicious jammer as a zero‑sum game. Both parties are subject to power constraints that can be enforced either on a per‑codeword (peak) basis or over the long term (average). The authors consider four main scenarios: (1) peak‑power constraints with full channel state information (CSI) at all nodes, (2) average‑power constraints with full CSI, (3) average‑power constraints with mixed (probabilistic) strategies, and (4) the case where no CSI is fed back to the transmitter.

Under peak‑power constraints, the game admits a pure‑strategy Nash equilibrium. The optimal policies are intuitive: the transmitter allocates power only when the instantaneous channel gain exceeds a threshold, while the jammer concentrates its power on the complementary low‑gain intervals. This “reverse water‑filling” behavior is the exact counterpart of the classic water‑filling solution for capacity maximization, but now aimed at minimizing the opponent’s signal‑to‑noise ratio.

When only average power limits are imposed, the situation becomes more intricate. The authors first derive the optimal intra‑frame (within a codeword) power control functions for both players, which resemble a dynamic water‑filling rule that adapts to the instantaneous fading realization. They then address the inter‑frame (across codewords) allocation needed to satisfy the long‑term power budget, effectively smoothing the instantaneous allocations over time. By solving the maximin and minimax problems separately, they show that the two solutions differ, proving that no pure‑strategy saddle point exists. Consequently, a pure‑strategy equilibrium cannot be guaranteed.

To overcome this limitation, the paper introduces mixed strategies. Building on the classic Bell‑Cover problem—where two players choose continuous probability distributions to minimize/maximize an expected payoff—the authors formulate a generalized version appropriate for the fast‑fading jamming game. The resulting equilibrium consists of specific cumulative distribution functions for the transmitter’s and jammer’s power levels. These distributions implement a probabilistic version of the reverse water‑filling principle, ensuring that each player’s expected outage probability is minimized against any opponent’s response. Numerical results demonstrate that the mixed‑strategy equilibrium yields a substantially lower outage probability than any pure‑strategy solution under the same average power constraints.

Finally, the authors examine the scenario where the transmitter receives no CSI feedback. In this case, both players can only rely on statistical knowledge of the fading process and must select fixed power levels (or fixed distributions) a priori. Surprisingly, the performance loss relative to the full‑CSI case is minimal, especially under average‑power constraints. This finding suggests that, for fast‑fading environments, the benefit of instantaneous CSI feedback is modest, and system designers may opt to forgo the overhead associated with CSI acquisition without sacrificing much reliability.

Overall, the paper provides a comprehensive game‑theoretic treatment of jamming in fixed‑rate systems, covering peak and average power constraints, pure and mixed strategies, and both CSI‑aware and CSI‑agnostic settings. The key contributions are: (i) identification of a pure‑strategy Nash equilibrium under peak constraints; (ii) proof of the non‑existence of a pure‑strategy saddle point under average constraints; (iii) derivation of the optimal mixed‑strategy equilibrium via a generalized Bell‑Cover solution; and (iv) quantitative evidence that full CSI yields only marginal gains in fast‑fading regimes. These results have practical implications for the design of power‑efficient, robust wireless links in adversarial environments, especially for battery‑limited devices and IoT applications where average power budgets dominate and CSI feedback may be costly or unreliable.