QTAM: QTransform Amplitude Modulation

Abstract

We present Q-Transform Amplitude Modulation (QTAM), a novel, fully invertible implementation of the Constant-Q Transform algorithm, designed to enable robust signal denoising and the disentanglement of overlapping transient events in current and next generation gravitational wave (GW) observatories. Time-frequency (TF) analysis faces a fundamental dichotomy: critically sampled transforms are computationally efficient but lack time-shift invariance, limiting their efficacy for robust pattern recognition and Deep Learning applications. While alternative approaches such as the Dual-Tree Complex Wavelet Transform provide efficient approximate shift-invariance, their wavelet constructions remain tied to dyadic scale frequency tilings that are poorly matched to the simultaneous representation of GW chirps and instrumental glitches. Conversely, overcomplete transforms provide the necessary shift-invariance and tunable frequency resolution, but their implementations generate highly redundant data volumes that are prohibitive for low-latency (LL) processing. Furthermore, standard attempts to compress these dense representations rely on lossy interpolation, destroying the phase coherence required to reconstruct the signal. QTAM bridges this gap by employing a methodology inspired by Amplitude Modulation radio broadcasting. By modeling the Q-transform output as a slowly varying complex envelope carried by a deterministic high-frequency term, we achieve lossless data decimation via spectral shifting to baseband. We demonstrate that QTAM is linear and fully invertible, allowing exact reconstruction of the original signal with machine precision while retaining the shift-invariance of dense spectrograms. Leveraging native GPU acceleration, QTAM enables TF pipelines to operate within LL O(1s) bounds. We validate the method's potential for denoising and disentanglement on GW data and signal injections.

Figures (15)

• Figure 1: The TF plane on which the CQT is computed. Frequencies range from 20 to 2048 Hz over a 0.2s interval. The horizontal frequency bands are referred to as tiles, exhibiting varying temporal resolutions.
• Figure 2: Schematic of spectral demodulation in QTAM. Top: The standard CQT filter bank where windows are centered at their absolute frequencies $f_k$. The large spectral support necessitates a high sampling rate to resolve the carrier oscillations. Bottom: The same windows after being circularly shifted to baseband ($0 Hz$). By removing the carrier frequency offset, the spectral support is compressed to the signal's inherent bandwidth, allowing for lossless downsampling.
• Figure 3: Demonstration of information preservation under volume compression. Panels (a) and (b) show the standard resolution $(64 \times 410)$. Panels (c) and (d) show the compressed representation ($64 \times 33$). Panels (e) and (f) show the re-upsampled reconstruction. The residuals between the original and reconstructed matrices are negligible $\mathcal{O}(10^{-14})$, confirming that the compression discards only redundant data.
• Figure 4: (Top) Overlay of the original GW150914 (H1) whitened strain data (cyan) and the signal reconstructed via the inverse QTAM transformation (blue). The signals are visually indistinguishable. (Bottom) Histogram of the reconstruction residuals ($x_{\mathrm{inv}} - x_{\mathrm{orig}}$) overlaid with a Gaussian fit (black solid line) and the theoretical zero-mean hypothesis (blue dashed line). The residuals follow a Gaussian distribution with a standard deviation of $\sigma \approx 7.10 \times 10^{-8}$. A Z-test was performed to check for zero-mean compatibility ($Z \approx 1.19$), confirming that the reconstruction is accurate up to numerical precision.
• Figure 5: Demodulated signal features of (H1) GW150914. Left: Energy spectrogram in the compressed baseband representation ($64 \times 33$ bins). Right: Instantaneous frequency, computed as the time-derivative of the unwrapped phase. By removing carrier oscillations, the demodulated phase derivative explicitly reveals the signal's "chirp" morphology as a coherent track.