On the Design of an Optimal Multi-Tone Jammer Against the Wiener Interpolation Filter

In the context of civilian and military communications, anti-jamming techniques are essential to ensure information integrity in the presence of malicious interference. A conventional time-domain approach relies on computing the Wiener interpolation filter to estimate and suppress the jamming waveform from the received samples. It is widely acknowledged that this method is effective for protecting wideband systems against narrowband interference. In this work, this paradigm is questioned through the design of a $K$-tone jamming waveform that is intrinsically difficult to estimate assuming a $L$-tap Wiener interpolation filter. This design relies on an optimization procedure that maximizes the analytical Bayesian mean squared error associated with the jamming waveform estimate. Additionally, an analytical proof is provided showing that a multi-tone jamming waveform composed of $L/2+1$ tones is sufficient to render the Wiener-filter-based anti-jamming module completely ineffective. The analytical results are validated through Monte Carlo simulations assuming both perfect knowledge and practical estimates of the correlation functions of the received signal.

💡 Research Summary

The paper investigates the vulnerability of Wiener interpolation filtering, a widely adopted anti‑jamming technique in direct‑sequence spread‑spectrum (DSSS) communications, and proposes an optimal multi‑tone jammer that maximally degrades its performance. The authors model the received signal as the sum of a spread‑spectrum useful signal, additive white Gaussian noise, and a K‑tone jamming waveform composed of complex sinusoids with amplitudes α_k, frequencies ω_k, and random phases. They derive a closed‑form expression for the Wiener filter coefficients w* in terms of the received‑signal covariance matrix C_rr and the cross‑covariance vector C_rθ, both of which depend explicitly on the jammer parameters. Using these expressions, they obtain an analytical formula for the Bayesian mean‑square error (BMSE) of the interference estimate, which can be written as BMSE = 1_K^T Cε(α,ω) 1_K. The matrix Cε is a function of the amplitude diagonal matrix J(α), the Dirichlet‑kernel‑based matrix Γ(ω), and the signal‑to‑noise ratio (SNR) and jammer‑to‑signal ratio (JSR).

The core design problem is to choose α and ω that maximize this BMSE, i.e., make the Wiener filter’s estimate as inaccurate as possible. The authors solve the problem analytically for a single tone (recovering known results) and for two tones, showing that the worst case occurs when the two frequencies are separated by approximately π/(L/2 + 1), where L is the filter length, and the amplitudes are equal. For 3 ≤ K < L/2 + 1 tones they propose a low‑complexity algorithm that spreads the tones uniformly over the band, again with equal amplitudes, to minimize inter‑tone error correlation. Crucially, they prove that when K ≥ L/2 + 1, the error‑covariance matrix C*ε approaches the identity, meaning the BMSE reaches its theoretical maximum and the Wiener filter becomes essentially ineffective regardless of the exact tone frequencies.

Monte‑Carlo simulations validate the theory under both perfect knowledge of C_rr and C_rθ and realistic blind estimation of these matrices from the received data. Results show a dramatic increase in BMSE and a corresponding drop in output SINR for the proposed multi‑tone jammer, even as the filter length L grows. The paper thus demonstrates that a carefully crafted multi‑tone jamming signal, with as few as L/2 + 1 tones, can completely neutralize Wiener‑filter‑based anti‑jamming modules, highlighting a previously under‑explored attack vector against spread‑spectrum systems.