[Paper Review] On Optimal Jamming Over an Additive Noise Channel

Emrah Akyol, Kenneth Rose+1 more|arXiv (Cornell University)|Mar 12, 2013

Wireless Communication Security Techniques5 references4 citations

TL;DR

This paper derives the optimal zero-delay jamming strategy over an additive noise channel for general source and channel distributions, showing that the jammer's optimal strategy is to enforce linearity in transmitter and receiver mappings by satisfying a 'matching condition' on the jamming noise density. The approach generalizes prior Gaussian-only results and provides a numerical procedure to approximate optimal jamming when exact linearity cannot be imposed.

ABSTRACT

This paper considers the problem of optimal zero-delay jamming over an additive noise channel. Early work had already solved this problem for a Gaussian source and channel. Building on a sequence of recent results on conditions for linearity of optimal estimation, and of optimal mappings in source-channel coding, we derive the saddle-point solution to the jamming problem for general sources and channels, without recourse to Gaussian assumptions. We show that linearity conditions play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both transmitter and receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and receiver cannot benefit from non-linear strategies. This result is shown to subsume the known result for Gaussian source and channel. We analyze conditions and general settings where such unbeatable strategy can indeed be achieved by the jammer. Moreover, we provide the procedure to approximate optimal jamming in the remaining (source-channel) cases where the jammer cannot impose linearity on the transmitter and the receiver.

Motivation & Objective

• To extend the known optimal jamming solution from Gaussian to general source and channel distributions.
• To identify conditions under which the jammer can force linear transmitter and receiver mappings, thereby maximizing distortion.
• To derive a general 'matching condition' on the jamming noise density that ensures optimal linearity in estimation and communication.
• To provide a numerical procedure for approximating optimal jamming in cases where the matching condition does not hold.
• To unify and generalize prior results on optimal jamming under zero-delay, power-constrained, and zero-sum game settings.

Proposed method

• Leverages recent results on linearity in optimal estimation and source-channel coding to analyze the saddle-point solution of the jamming game.
• Derives a 'matching condition' on the jamming noise density that ensures the optimal transmitter and receiver mappings are linear.
• Uses orthonormal polynomial expansions and characteristic functions to express the mean squared error (MSE) in terms of coefficients, minimizing the sum of squared coefficients to maximize distortion.
• Applies Fourier transforms to convert the estimation and decoding equations into differential equations that characterize the optimal jamming noise.
• Develops a numerical method to approximate the optimal jamming density when the matching condition is not satisfied.
• Analyzes asymptotic behavior in high channel SNR to derive optimal jamming strategies under power constraints.

Experimental results

Research questions

• RQ1 Under what conditions can a jammer force the transmitter and receiver to use linear mappings in a zero-delay communication system?
• RQ2 What is the necessary and sufficient condition on the jamming noise density to ensure linearity of the optimal transmitter and receiver mappings?
• RQ3 How does the optimal jamming strategy generalize beyond the Gaussian source-channel case?
• RQ4 What procedure can be used to approximate optimal jamming when the jamming noise cannot enforce linearity?
• RQ5 How does the jammer's strategy depend on the higher-order statistical properties of the source and noise?

Key findings

• The optimal jamming strategy is to select a jamming noise density that satisfies a 'matching condition' ensuring that the optimal transmitter and receiver mappings are linear.
• The matching condition generalizes the known Gaussian jamming result, where the jammer generates independent Gaussian noise.
• When the matching condition holds, the jammer can force the system into a linear regime, making non-linear strategies suboptimal for the transmitter and receiver.
• In cases where the matching condition does not hold, the paper provides a numerical procedure to approximate the optimal jamming density by minimizing the sum of squared coefficients in a polynomial expansion of the decoder.
• The optimal jamming noise is not necessarily Gaussian in non-Gaussian settings, even though it still aims to enforce linearity.
• The analysis shows that linearity is pivotal in jamming, as the jammer's goal is to prevent the transmitter and receiver from benefiting from non-linear strategies.