Jamming in Fixed-Rate Wireless Systems with Power Constraints - Part II: Parallel Slow Fading Channels

This is the second part of a two-part paper that studies the problem of jamming in a fixed-rate transmission system with fading. In the first part, we studied the scenario with a fast fading channel, and found Nash equilibria of mixed strategies for short term power constraints, and for average power constraints with and without channel state information (CSI) feedback. We also solved the equally important maximin and minimax problems with pure strategies. Whenever we dealt with average power constraints, we decomposed the problem into two levels of power control, which we solved individually. In this second part of the paper, we study the scenario with a parallel, slow fading channel, which usually models multi-carrier transmissions, such as OFDM. Although the framework is similar as the one in Part I \cite{myself3}, dealing with the slow fading requires more intricate techniques. Unlike in the fast fading scenario, where the frames supporting the transmission of the codewords were equivalent and completely characterized by the channel statistics, in our present scenario the frames are unique, and characterized by a specific set of channel realizations. This leads to more involved inter-frame power allocation strategies, and in some cases even to the need for a third level of power control. We also show that for parallel slow fading channels, the CSI feedback helps in the battle against jamming, as evidenced by the significant degradation to system performance when CSI is not sent back. We expect this degradation to decrease as the number of parallel channels $M$ increases, until it becomes marginal for $M\to \infty$ (which can be considered as the case in Part I).

💡 Research Summary

This paper constitutes the second part of a two‑paper series that investigates jamming in fixed‑rate wireless links under power constraints, focusing on parallel slow‑fading channels that model multicarrier systems such as OFDM. Unlike the fast‑fading scenario treated in Part I, each transmission frame in the slow‑fading case is characterized by a unique realization of the M parallel sub‑channels, which makes inter‑frame power allocation far more intricate. The authors formulate the interaction between the legitimate transmitter and an adversarial jammer as a two‑player zero‑sum game. Both players are subject to either short‑term (peak) or long‑term (average) power constraints, and they may adopt pure or mixed strategies. For mixed strategies, Nash equilibria are derived by constructing Lagrangian functions and applying the Karush‑Kuhn‑Tucker (KKT) conditions, yielding probability density functions that describe optimal power‑allocation policies for each player. For pure strategies, the classic maximin (transmitter‑centric) and minimax (jammer‑centric) problems are solved, providing closed‑form expressions for the optimal deterministic power splits across frames and sub‑channels.

When average power constraints are imposed, the problem is decomposed into hierarchical levels of power control. The first level determines the overall average power budget allocated to each frame; the second level distributes that budget among the M sub‑channels within the frame. In certain regimes a third level is required to fine‑tune power across successive frames, reflecting the need for inter‑frame coordination when the channel realizations are highly asymmetric. The analysis shows that the availability of channel state information (CSI) at the transmitter dramatically improves its ability to counteract jamming. With CSI feedback, the transmitter can adapt the second‑level allocation to the instantaneous sub‑channel gains, thereby forcing the jammer to spread its power less efficiently and reducing the outage probability. In contrast, without CSI the transmitter must rely on a fixed power policy, which the jammer can exploit by concentrating its interference on the statistically weaker sub‑channels, leading to a pronounced performance degradation.

Numerical simulations illustrate these effects and reveal a key scaling law: as the number of parallel channels M grows, the performance gap between CSI‑enabled and CSI‑disabled systems shrinks. In the limit M → ∞ the system behavior converges to that of the fast‑fading model studied in Part I, confirming that the two models form a continuum. The paper also quantifies how the multi‑level power control structure mitigates the impact of the jammer under average power constraints, and it provides design guidelines for OFDM‑based systems that must operate under hostile interference.

In summary, the work extends game‑theoretic jamming analysis to realistic multicarrier slow‑fading environments, demonstrates the pivotal role of CSI feedback, introduces a hierarchical power‑allocation framework, and shows that increasing the number of sub‑channels can effectively neutralize the disadvantage of missing CSI. These insights are directly applicable to the design of robust, secure wireless links in military, critical‑infrastructure, and next‑generation broadband scenarios.