Full-Frequency Temporal Patching (FFTP)

• Full-Frequency Temporal Patching (FFTP) is a method that segments spectrogram data into temporal patches covering the entire frequency range to preserve harmonic structures.
• FFTP enhances computational efficiency by reducing patch count significantly compared to square patching, resulting in lower latency and improved accuracy in audio and anomaly detection tasks.
• FFTP supports targeted spectral analysis and augmentation through techniques like SpecMask and PFT, enabling robust performance in audio classification and multivariate anomaly detection.

Full-Frequency Temporal Patching (FFTP) is a methodology for segmenting time-series or spectrogram data in a manner that preserves global frequency information within localized temporal windows. Designed to address the intrinsic asymmetry of time-frequency data, FFTP creates patches that span the complete frequency dimension while limiting each patch's extent in the time domain. This technique enables models to maintain harmonic continuity and extract temporally-resolved spectral patterns for tasks such as audio classification and multivariate anomaly detection. FFTP also facilitates computational efficiency by reducing the overall number of patches, enabling targeted spectral analysis, and providing a structural basis for frequency-domain augmentation methods.

1. Foundations of Full-Frequency Temporal Patching

FFTP was developed in response to the limitations of square patching methods, which are widely used in computer vision and often transferred uncritically to audio spectrogram processing. In standard square patching, input spectrograms are divided into fixed-size blocks along both time and frequency axes, resulting in fragmentation of harmonic or spectral structures and an inflated count of input tokens. FFTP remedies this by imposing patches that span the entire set of frequency bins while confining the patch to a short temporal segment. In technical terms, given a log-mel spectrogram $X \in \mathbb{R}^{B \times 1 \times F \times T}$, FFTP applies a 2D convolution with kernel size $(F, T_p)$ and stride $(F, s_t)$ along frequency and time, respectively, to yield a sequence of $$N = \left\lfloor \frac{T - T_p}{s_t} + 1 \right\rfloor$$ full-frequency temporal embeddings per input example (Makineni et al., 28 Aug 2025).

This patch definition highlights the fundamental principle of FFTP: harmonics and global spectral features are best preserved when frequency-wise continuity is maintained across each input patch. This leads to richer representations and obviates the need for the excessive number of patches required by square patching approaches.

2. Formal Algorithmic Structure and Computation

FFTP is implemented by mapping the input audio waveform to a spectrogram, then applying a full-frequency convolution as the initial transformation. The convolutional patch extraction is described by

$$Z = \text{Conv2D}(X; W_c, s) \in \mathbb{R}^{B \times D \times 1 \times N},$$

where $$W_c \in \mathbb{R}^{D \times 1 \times F \times T_p}$$ is the learned kernel and $N$ is the total number of temporal patches. The resulting tensor is flattened and transposed to produce a sequence of patch tokens for subsequent transformer or state-space model processing

$$Z' = \text{Transpose}(\text{Flatten}(Z)) \in \mathbb{R}^{B \times N \times D}.$$

Each token corresponds to a localized interval in time but encodes the entire frequency signature, ensuring that downstream models process semantically coherent input features.

In the context of frequency-domain anomaly detection, FFTP may also refer to the partitioning of the spectrum into frequency patches following discrete Fourier transformation. This operation is formalized by computing $X(\omega) = \sum_t X(t) e^{-i \omega t}$ and "unfolding" the frequency axis into contiguous bands via tensor decomposition (Wu et al., 16 Oct 2024).

3. Computational Efficiency and Performance Implications

A central advantage of FFTP lies in its substantial reduction of patch count and computational cost. Benchmark results demonstrate that for typical audio spectrograms, square patching yields patch counts in excess of 1,200, whereas FFTP can reduce this to as few as 96 patches. This translates to improvements in GFLOPs from 103.35 (square patch) to 4.15 (FFTP-based), and up to 83.26% reductions in latency and training time (Makineni et al., 28 Aug 2025).

Performance metrics across standard datasets underline the efficacy of FFTP. On AudioSet-18k, the Audio Spectrogram Transformer (AST) achieved 11.25% mAP with square patching; FFTP improved this to 15.38%, and, when combined with SpecMask augmentation, reached 18.32% mAP. On SpeechCommandsV2, accuracy rose from 85.27% (square patch) to 93.73% (FFTP), and further to 95.94% with SpecMask. Similar gains are documented for the Audio Mamba (AuM) model.

In anomaly detection tasks, partitioning the frequency domain into patches allows precise localization of spectral anomalies and enhances detection accuracy, especially for complex subsequence anomalies (Wu et al., 16 Oct 2024).

4. Augmentation and Spectral Robustness: The SpecMask Approach

Spectrogram augmentation is critical for improving model robustness to temporal variations and preserving the integrity of spectral features. SpecMask is a patch-aligned augmentation methodology designed in concert with FFTP. Unlike standard SpecAugment, SpecMask synchronizes its masking operations to FFTP token boundaries. Its fixed masking budget is split such that approximately 70% of the masked area comprises full-frequency temporal strips, with the remaining 30% constituted by smaller local time-frequency regions (Makineni et al., 28 Aug 2025).

Algorithmic implementation iteratively selects regions for masking (with or without full-frequency constraint) and substitutes their values with the spectrogram mean, thereby neutralizing potential bias. SpecMask exhibits improvements in test metrics by enhancing temporal robustness while maintaining key spectral structures.

5. Frequency Patching for Anomaly Detection: Multichannel Extensions

In multivariate time-series contexts, as explored in CATCH (Wu et al., 16 Oct 2024), FFTP is extended through frequency-domain patching and channel-aware fusion. Each multichannel time series is transformed to the frequency domain and then partitioned into frequency bands ("frequency patches"). The Channel Fusion Module (CFM) further refines the representation by discovering correlations between input channels for each frequency patch.

The CFM leverages a patch-wise mask generator and masked-attention mechanism: attention scores $\mathcal{S} = (QK^\top) \odot \mathcal{M}$ incorporate learned binary masks $\mathcal{M}$, which are trained via bi-level multi-objective optimization. At the lower level, standard loss functions update base parameters, while at the upper level, clustering and regularization losses ensure relevant channel correlations are learned efficiently.

Extensive evaluations across synthetic and real-world datasets (feature dimensions: 3–72, sequence lengths: $\sim$ thousands to $>10^6$) demonstrate state-of-the-art detection performance. The full-frequency patching mechanism is especially advantageous in managing heterogeneous anomaly types, including contextual, shapelet, seasonal, trend, and mixed subsequence anomalies.

6. Fast Partial Fourier Transform (PFT) and Selective Frequency Computation

PFT is a computational refinement relevant to FFTP where only a specified consecutive range of Fourier coefficients is computed for each temporal patch, rather than the entire frequency spectrum. PFT modifies the classical Cooley–Tukey FFT by "centering" the summation indices and approximating slowly oscillating twiddle factors with low-degree polynomials. The transform is defined as

$$\hat{a}_m = \sum_{k=0}^{p-1}\sum_{l=0}^{q-1} a_{qk+l}\, e^{-2\pi i m (l - q/2)/N}\, e^{-2\pi i m k / p}\, e^{-\pi i m/p},$$

with polynomial approximations for $e^{-2\pi i m (l-q/2)/N}$ and arbitrary output range selection. Target frequency bands $R_{\mu,M} = [\mu-M, \mu+M]$ can be specified directly. The computational complexity is $O(N + M\log M)$ for $M \ll N$, with empirical results showing speedups up to 13–21× over existing FFT libraries when few output coefficients are needed and accuracy maintained at relative $\ell_2$ error $< 10^{-6}$ (Park et al., 2020).

A plausible implication is that PFT can be "plugged into" any FFTP application to achieve selective spectral analysis with reduced cost, supporting variable patch or segment lengths—beneficial in both audio and multivariate temporal applications.

7. Applications, Implications, and Future Directions

FFTP and its associated methodologies are directly applicable to domains with temporally and spectrally complex signals:

• Audio Classification: FFTP enhances transformer and SSM performance by providing harmonically rich, temporally localized input tokens; SpecMask further improves generalization and robustness.
• Anomaly Detection: In channel-rich, multivariate time series, frequency patching and fusion modules detect subtle anomalies. CATCH demonstrates SOTA performance across diverse datasets and anomaly types (Wu et al., 16 Oct 2024).
• Efficient Spectral Processing: FFTP combined with PFT enables selective computation of Fourier coefficients, optimizing feature extraction where only part of the spectrum is necessary, with theoretical and empirically verified bounds on accuracy and computation (Park et al., 2020).

Potential future directions include optimizing patch configurations, developing domain-specific augmentation strategies, extending FFTP and PFT to non-audio time series modalities, and integrating these techniques with self-supervised representation learning. The modularity of patch extraction and augmentation supports adaptation to broader signal processing tasks beyond audio, including sensor telemetry, financial time series, and healthcare monitoring.

FFTP represents a structural reimagining of patch-based data processing, grounded in frequency-domain principles and reinforced by empirical evidence from contemporary transformer-based and anomaly detection systems.