Noise Aliasing Strategy

• Noise aliasing strategy is a rigorously developed set of techniques aimed at preserving signal fidelity by mitigating the folding of out-of-band noise into desired frequency ranges.
• It integrates optimal filter design, discrete wavelet transforms, and adaptive Wiener filtering to address aliasing in diverse sampling scenarios.
• Applications span digital signal processing, machine learning, and adaptive optics, yielding measurable gains in SNR, accuracy, and robustness across scientific domains.

Noise aliasing strategy encompasses a rigorously developed suite of mathematical, computational, and hardware-algorithmic methods for suppressing, quantifying, or mitigating the folding of out-of-band noise and signal content into the frequency range of interest during sampling, decimation, or sub-sampling. This class of strategies is foundational in digital signal processing, computational physics, machine learning, acoustics, flow analysis, and astronomical instrumentation. The rationale is to maximize information preservation and minimize spurious artifacts due to under-sampling, spectral folding, and numerical grid coarseness—effects that can critically influence noise robustness, adversarial vulnerability, and physical accuracy in scientific computing and machine learning.

1. Theoretical Framework and Information-Theoretic Foundations

A canonical model for noise aliasing in multirate systems is the $M$-fold decimator: given a relevant data process $S_n$ and additive noise $N_n$, with input $X_n=S_n+N_n$, the decimator output is $Y_n=\left(H*X\right)_{nM}$, where $H$ is the pre-decimation anti-aliasing filter. The principal objective is to maximize the mutual information between the output $Y_n$ and the underlying $S_n$ after aliasing suppression. The relevant information-loss rate is

$$\ell_S\bigl(X^{(M)}\to Y\bigr) = I\bigl(S^{(M)};X^{(M)}\bigr) - I\bigl(S^{(M)};Y\bigr)$$

For stationary Gaussian $S_n$, $N_n$ with known spectra, the optimal filter $H_{\rm opt}$ maximizes the post-decimation SNR at each frequency by passing, for each downsampled frequency bin, only the alias branch with the highest SNR. The procedure follows:

1. Estimate power spectra $S(\omega)$ and $N(\omega)$.
2. For $\omega \in [-\pi/M, \pi/M]$, compute aliased SNRs $\mathrm{SNR}_k(\omega)=S(\omega_k)/N(\omega_k)$, $\omega_k = (\omega - 2\pi k)/M$.
3. Form the filter $H(e^{j\omega_k})$ by selecting the maximal SNR branch.
4. Approximate by FIR to ensure implementability if required.

For non-Gaussian $S_n$ with Gaussian $N_n$, this strategy provides an upper bound on information loss, justifying filter design based on second-order statistics (Geiger et al., 2013).

2. Anti-Aliasing and De-Aliasing in Machine Learning Architectures

Common CNN down-sampling layers such as max-pooling, strided convolution, and average-pooling violate the Nyquist criterion by omitting precise anti-aliasing low-pass filtering. The result is that high-frequency noise—random or adversarial—aliases into coarser feature maps, undermining structure preservation and catastrophic to robust classification. A principled noise aliasing strategy replaces these layers with discrete wavelet transform (DWT) modules:

• At each down-sampling, only the low-frequency $X_{ll}$ component is propagated, and all detail ($X_{lh}$, $X_{hl}$, $X_{hh}$) subbands are dropped.
• This completely eliminates aliasing of high-frequency noise and structurally enforces noise-robustness and adversarial stability across layers.
• No additional learnable parameters are introduced; DWT/IDWT strictly inverts in encoder–decoder architectures.

Empirically, these modifications, when integrated into VGG, ResNet, or DenseNet backbones, yield 0.6–1.7 percentage point top-1 accuracy improvements on ImageNet, 5–10 point mean corruption error (mCE) reductions on ImageNet-C, and consistent increases in robustness to adversarial attacks (3–4 percentage point gains under PGD for ResNet101). In object detection, similar improvements in AP across all scales are obtained (Li et al., 2021).

3. Signal Processing and Adaptive Optics: Optimal Anti-Aliasing Filtering

In astronomical adaptive optics, aliasing is intrinsic to the wave-front sensing process due to the finite pitch of sub-apertures (Shack–Hartmann sampling). The anti-aliasing Wiener filter generalizes the standard Wiener phase reconstructor by accounting for spectral replicas in the measurement operator: $$W_{AA}(\mathbf{k}) = \frac{H^\dagger(\mathbf{k}) S_\Phi(\mathbf{k})}{|H(\mathbf{k})|^2 S_\Phi(\mathbf{k}) + W_\alpha(\mathbf{k}) + \gamma S_\eta}$$ where $W_\alpha(\mathbf{k})$ is the aliased-phase PSD and $S_\eta$ is noise PSD. The anti-aliasing filter achieves a propagation coefficient $a_p=0.035$ (vs.\ $0.073$ for LSQ) and noise coefficient $n_p=0.80$ (vs.\ $1.00$) at $D/d=32$. Residual RMS error and Strehl are improved; pre-coronagraphic raw contrast is up to $2\times$ higher at small angles (Correia et al., 2014).

Post-facto digital solutions must be combined with optical spatial pre-filters (e.g., pinholes) to achieve absolute suppression, since digital reconstruction cannot reverse physical aliasing upstream.

4. Advanced Computational Physics and Nonlinear De-Aliasing

In computationally intensive simulations such as quantum kinetic equations or time-resolved flow data, finite discretization in momentum, time, or spatial grids causes unresolved phase oscillations and subsequent aliasing in correlators or power spectra. Recent strategies leverage intrinsic properties of the governing equations:

• Non-Markovian quantum kinetics: A grid-based simulation introduces memory-induced phase oscillations in two-particle correlators $\mathcal G$. Aliasing sets in when these phases shift by more than $\pi$ between adjacent grid points. The solution is to apply a discrete diffusion operator in grid space: $$\widetilde{\mathcal G} = (1 + \alpha\Delta^{\text{dis}}_k + \alpha\Delta^{\text{dis}}_p)\mathcal{G}$$ with energy-conserving stencil, tuned via parameter $\Gamma$, yielding full suppression of spurious high-frequency oscillations and numerical instabilities (Makait et al., 21 Jan 2025).
• Flow simulations: For sub-Nyquist time resolution, a suite of detection and de-aliasing techniques is available:
  • Time-derivative detection: Use discrete Fourier transforms (DFTs) of $y$ and $z = \frac{d}{dt}y$ to estimate and subtract alias contributions at each frequency.
  • Spatial filtering: In convective systems, project out-of-band content in wavenumber space by tailored FIR kernels in $x$.
  • Integral-based de-aliasing: Anticipate high-frequency contamination in nonlinear forcing by integrating in time, storing $q$, $q_{\rm int}$ at the downsampled rate, and reconstructing the original forcing in the frequency domain, leveraging the geometric attenuation of aliasing by $|\omega_s/\omega|$ per integration stage.

These techniques enable spectral fidelity at a fraction of the storage and computational cost of brute-force oversampling or classical, high-order filtering (Karban et al., 2022).

5. Nonlinear and Neural Approaches to Spatial Aliasing in Array Processing

Spatial aliasing in uniform linear or cross-shaped microphone arrays becomes critical for $d > c/(2f_{\max})$, causing ambiguity above $f_a = c/(2d)$. Conventional fixed linear beamformers (e.g. MVDR) cannot resolve target directions once phase-wrapping occurs, leading to directionally aliased noise.

Deep learning–based strategies, including U-Net–predicted M × M complex-valued de-aliasing filters, now permit signal-dependent, tile-wise suppression of spatial aliases:

• The filter $H(t, f) = \text{diag}(h_1, ..., h_M)$ or fully cross-connected variant is predicted from the multichannel STFT.
• Training uses the mean PHASEN perceptual loss or time-frequency domain MSE against reference (alias-free) beamformer outputs.
• On stereo and FOA spatial capture, objective C-Si-SNR gains of +7–20 dB over conventional beamforming, and subjective MUSHRA tests showing statistically insignificant degradation compared to references, are realized (Guzik et al., 26 May 2025).

Extension to joint nonlinear spatial–tempo-spectral filtering via LSTM + TCN (“JNF-SSF”) architecture produces further improvement: consistent ΔSI-SDR of 11 dB at large microphone spacings, with robust suppression of spatially aliased energy above $f_a$ via learned cross-frequency cues and entirely data-driven adaptation (Mannanova et al., 30 Sep 2025).

6. Domain-Specific Applications and Algorithmic Guidelines

Best practices for noise aliasing strategy are now codified across disciplines:

• Sampling and Pre-filtering: Ensure input sampling $f_s \geq 2B$ (bandwidth), employ hardware (analog) and software (digital FIR) low-pass filters to block out-of-band noise.
• Spectral Cross-Checking: Characterize both signal and noise spectra, determine SNR for relevant components, and design filters to pass only information-rich alias branches.
• Multi-stage Filtering: Prefer cascaded low-order FIR stages to reduce computational/memory footprint in large databases.
• Detection and Correction: Employ time-derivative or integral-based methods for alias prediction and removal when governing equations permit.
• Hybrid Physical/Digital Mitigation: Combine upfront (physical) spatial or optical pre-filtering with digital (post-sampling) anti-aliasing/wiener reconstructors in high-performance imaging or sensor applications.
• Reporting and Reproducibility: Explicitly report all aliasing-bandwidth parameters and filter design criteria with the analysis results to ensure reproducibility and interpretability (Calosso et al., 2015, Geiger et al., 2013).

7. Quantitative Performance and Limitations

The following table summarizes representative numerical outcomes from diverse application domains, highlighting objective and subjective metrics improvements:

Application Domain	Strategy	Aliasing Suppression / Metric	Reference
CNN for Image Classification	DWT-based downsampling	+0.6–1.7 pt accuracy; −5~10 pt mCE	(Li et al., 2021)
High-Order AO (Astronomy)	AA Wiener Filter + Pre-filter	$a_p=0.035$ vs $0.073$ (LSQ); $2\times$ raw contrast gain	(Correia et al., 2014)
Quantum Kinetic Sim. (Plasma)	Discrete-space diffusion	Remove numerical instability; $<10^{-6}$ energy error	(Makait et al., 21 Jan 2025)
CFD/LES Database	Deriv./integral alias removal	1–5 dB error, $\sim 10\times$ storage reduction	(Karban et al., 2022)
Microphone Array Beamforming	U-Net de-aliasing filter	+16–20 dB Si-SNR; subjective parity	(Guzik et al., 26 May 2025)
Speech Enhancement (Large Array)	JNF-SSF DNN	+8 dB SI-SDR vs. beamformer	(Mannanova et al., 30 Sep 2025)

Energy-conservation and information bound guarantees apply in several frameworks, but the physical limits of anti-aliasing and possible over-smoothing must be carefully balanced (as in the plasma case, where diffusion parameter $\Gamma$ must be chosen to avoid erasing physically sharp features).

• (Geiger et al., 2013) Information Loss and Anti-Aliasing Filters in Multirate Systems
• (Correia et al., 2014) Anti-aliasing Wiener filtering for wave-front reconstruction in the spatial-frequency domain for high-order astronomical adaptive-optics systems
• (Calosso et al., 2015) Avoiding Aliasing in Allan Variance: an Application to Fiber Link Data Analysis
• (Li et al., 2021) WaveCNet: Wavelet Integrated CNNs to Suppress Aliasing Effect for Noise-Robust Image Classification
• (Karban et al., 2022) Solutions to aliasing in time-resolved flow data
• (Makait et al., 21 Jan 2025) Non-Markovian quantum kinetic simulations of uniform dense plasmas: mitigating the aliasing problem
• (Guzik et al., 26 May 2025) Deep learning based spatial aliasing reduction in beamforming for audio capture
• (Mannanova et al., 30 Sep 2025) An Analysis of Joint Nonlinear Spatial Filtering for Spatial Aliasing Reduction