Adversarial Signal Processing

• Adversarial signal processing is an interdisciplinary field combining signal processing, machine learning, and game theory to develop defenses against adaptive adversaries.
• The approach models adversary behaviors using Bayesian inverse filtering and optimization, enabling resilient inference and robust signal representations.
• Applications span radio communications, radar, audio, and generative AI, guiding the design of both attack algorithms and defensive countermeasures.

Adversarial signal processing is a field at the intersection of classical signal processing, machine learning, and game theory, which analyzes, designs, and defends systems subject to deliberate, adaptive adversarial interventions. The domain encompasses the modeling of sophisticated attackers, the derivation of robust inference and decision rules, the synthesis of resilient signal representations, and the development of attack-agnostic defenses across application modalities—spanning radio-frequency communications, sensor networks, audio, biomedical signals, and generative AI. Key themes include Bayesian inverse filtering, adversarially-constrained optimization, game-theoretic analysis, and adversarially driven learning and augmentation schemes.

1. Foundational Model Classes and Bayesian Adversarial Inference

The canonical adversarial Bayesian model defines two interacting agents: a "defender" with an evolving signal or state process $\{x_k\}$, and an "adversary" observing the process through noise, updating a Bayesian posterior $\pi_k(x)$, and acting based on this belief. The central task for the defender is to estimate the adversary’s latent beliefs and capabilities from observed adversarial actions, and to design defensive probing signals that optimally calibrate and mitigate adversarial inference. This requires solving an inverse filtering problem on the space of distributions ("random measures"), where the defender computes the so-called “posterior over posteriors”: $$\alpha_k(d\pi) = p(\pi_k \in d\pi | x_{0:k}, a_{1:k}),$$ with recursions defined on the evolution space of possible adversary posteriors conditioned on the defender's own state and noisy observations of adversarial actions. For linear-Gaussian specializations, these recursions yield finite-dimensional inverse Kalman filters, while for finite-state spaces, one obtains an inverse HMM filter (Krishnamurthy et al., 2019).

Estimation of the adversary’s observation quality then proceeds via maximum likelihood methods, employing log-likelihoods over action sequences and defender states: $$L(\theta) = \log p(a_{1:N} | x_{0:N}; \theta) = \sum_{k=1}^N \log \left[ \int G_{\pi, a_k} \, [P^\top \alpha_{k-1}^\theta \star B^\theta_{x_k, \cdot}](\pi) \, d\pi \right],$$ where $B^\theta$ is a parametric sensor model and $\alpha_k^\theta$ is the unnormalized measure over adversarial beliefs given parameter $\theta$. Input (probe signal) design is then cast as minimization of the covariance of $\hat \theta$ by search over state-sequence distributions or Markov transition kernels, using stochastic dominance or stochastic-gradient (SPSA) optimization (Krishnamurthy et al., 2019).

2. Algorithms for Adversarial Attack and Defense in Signal Domains

Signal-domain adversarial attack algorithms transpose core techniques from adversarial machine learning to physical signals, respecting measurement constraints, propagation physics, and spectral limits:

• Gradient-based attacks (FGSM/PGD) add perturbations $\delta$ to input signals $x$ to maximize classifier loss:

$$\delta = \epsilon \cdot \mathrm{sign}(\nabla_x L(f(x), y)),$$

with projections onto $\ell_p$ balls to enforce bounded power or amplitude (Sadeghi et al., 2018, Sandler et al., 2022).

• Universal adversarial perturbations (UAPs) are synthesized over batches (e.g., via PCA on loss gradients) to generalize across inputs and time, enabling real-time, emitter-agnostic deployment (Sadeghi et al., 2018, Sandler et al., 2022).
• Dual-domain attacks enforce both time- and frequency-domain constraints:

$$\begin{aligned} & \text{maximize} \quad L_{\text{adv}}(x+\delta, y) \ & \text{subject to} \quad \|\delta\|_p \leq \epsilon_t, \quad \|F(\delta) \cdot M_{\text{band}}\|_2 \leq \epsilon_f, \end{aligned}$$

making adversarial examples that are robust to receiver filtering and maintain in-band spectral content (Wen et al., 28 Oct 2025).

• PDE-constrained signal attacks design physically realizable waveforms (e.g., for sonar, radar) that, via wave propagation, yield adversarial spectrograms at a remote detector, requiring efficient adjoint- and bandlimited-optimization methods (Bassett et al., 27 Feb 2024).
• Adversarially-aware GANs generate perturbations that are distributionally indistinguishable from noise over channel statistics (CDI-aware) and train discriminators to enforce channel realism, enabling both strong attacks and robustified training (Sinha et al., 2023).
• Adversarial denoising leverages encoder–decoder architectures with adversarial losses in latent space, aligning the representation distribution of clean and noisy signals and improving denoising beyond MSE-based criteria (Casas et al., 2018).

3. Applications Across Signal Modalities

Wireless communications: Deep neural network demodulators and modulators are particularly susceptible to adversarial attacks, with physical-layer classifiers degraded by perturbations orders of magnitude smaller than additive jamming, often at PSR (perturbation-to-signal ratio) ≈ –10 dB (Sadeghi et al., 2018, Sandler et al., 2022). Dual-domain approaches maintain covert adversariality even under receiver filtering (Wen et al., 28 Oct 2025). Over-the-air deployment of universal perturbations has been validated with software-defined radios, confirming attack efficacy in real-world RF environments (Sandler et al., 2022). Attention-based and transformer architectures have introduced adversarial detection tokens and adversarial-robust distillation, elevating the resilience of low-power IoT devices (Zhang et al., 13 Jun 2025, Zhang et al., 13 Jun 2025).

Radar and sonar: Both GAN-based synthetic waveform generation for data augmentation and concealed electronic countermeasure generation via STFT-domain and time-frequency image attacks demonstrate the breadth of adversarial signal processing in radar settings. Attacks operating in the spectrogram domain can be inverted to physical waveforms (e.g., STDS), resulting in high transferability and stealth (Ma et al., 2023, Truong et al., 2020).

Audio/music: Deep networks for music and speech are vulnerable to very small, physically realizable spectral perturbations capable of changing content classification. Naïve adversarial training confers little robustness, further highlighting the signal-processing roots of model fragility tied to high-dimensionality and spectral aliasing (Kereliuk et al., 2015).

Distributed sensing and consensus: In adversarial sensor networks, Byzanine nodes can inject falsified decisions to bias detection. Game-theoretic fusion frameworks, soft isolation schemes, MAP-based dynamic programming, and belief-propagation on factor graphs realize near-optimal error rates even under correlated or Markovian attacks (Kallas, 28 Nov 2025).

4. Adversarial Signal Processing in Machine Learning and Generative AI

Adversarial signal-processing principles inform defenses and audits for frontier generative AI: computational safety is formally reduced to hypothesis testing on features φ(x) extracted from prompts or outputs. Signal-processing–driven detectors employ gradient-norms, loss landscapes, feature-space invariances, and adversarially paired training (e.g., robust paraphraser-versus-detector for text; autoencoder or DINO-based statistics for images) (Chen, 18 Feb 2025). Integration of these detectors directly leverages signal-theoretic concepts such as sensitivity, detection theory, and response to adversarially constructed signals.

Adversarial training and augmentation schemes, such as the injection of "booster signals" (external, universal perturbations outside the semantically active region of data), enhance the robustness and flatten loss surfaces of classifiers under strong $\ell_p$-bounded attacks, providing generalizable defenses with minimal test-time cost (Lee et al., 2023).

5. Theoretical Analysis, Vulnerabilities, and Design Principles

Signal-processing analysis clarifies that overparameterized models (e.g., Fourier-feature or random-feature linear interpolators) are structurally fragile to adversarial inputs due to localized Gibbs/aliasing phenomena: classification is nearly perfect, yet adversarial risk undergoes a sharp phase transition—infinitesimal perturbations localized near training data induce total loss of robustness, even in the absence of label or model noise (Narang et al., 2021). This suggests that adversarial vulnerability is a fundamental artifact of interpolation and high-frequency feature mixing.

Robust design principles derived from the above include: restricting adversarial energy via spectral masks, localizing perturbations with classifier attention maps (e.g., Grad-CAM), employing dual-domain (time-frequency) constraints, and using game-theoretic strategies to frame defense–attack tradeoffs. Certified robustness remains open, with current defenses relying on adversarial training with gradient-based augmentation and careful input preprocessing across modalities (Wen et al., 28 Oct 2025, Zhang et al., 13 Jun 2025).

6. Perspectives and Ongoing Challenges

Adversarial signal processing is a rapidly expanding domain, with unresolved challenges including designing black-box and channel-robust attacks, developing certified-defensible architectures for signal classifiers, scaling physical-domain (PDE-constrained) adversarial design, and generalizing theoretical insights on overparametrization-induced vulnerabilities to high-dimensional, real-world data (Bassett et al., 27 Feb 2024, Kallas, 28 Nov 2025). The unification of statistical signal processing, adversarial optimization, and robust machine learning is crucial for the future security and reliability of signal and systems engineering in adversarial environments.