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The fundamentals of Fourier Transform are presented, with analytical solutions derived for Continuous Fourier Transform (CFT) of truncated signals, to benchmark against Fast Fourier Transform (FFT). Certain artifacts from FFT were identified for decay curves. An existing method for Infrared Thermography, Pulse Phase Thermography (PPT), was benchmarked against a proposed method using polynomial fitting with CFT, to analyse cooling curves for defect identification in Non-Destructive Testing (NDT). Existing FFT methods used in PPT were shown to be dependent on sampling rates, with inherent artifacts and inconsistencies in both amplitude and phase. It was shown that the proposed method produced consistent amplitude and phase, with no artifacts, as long as the start of the cooling curves are sufficiently represented. It is hoped that a collaborative approach will be adopted to unify data in Thermography for machine learning models to thrive, in order to facilitate automated geometry and defect recognition and move the field forward.
In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimensional space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensiona
The Quantum Fourier Transformation ($QFT$) is a key building block for a whole wealth of quantum algorithms. Despite its proven efficiency, only a few proof-of-principle demonstrations have been reported. Here we utilize $QFT$ to enhance the perf
It is well known that matched filtering techniques cannot be applied for searching extensive parameter space volumes for continuous gravitational wave signals. This is the reason why alternative strategies are being pursued. Hierarchical strategies a
Given two intervals $I, J subset mathbb{R}$, we ask whether it is possible to reconstruct a real-valued function $f in L^2(I)$ from knowing its Hilbert transform $Hf$ on $J$. When neither interval is fully contained in the other, this problem has a u
We define an involution on the space of compact tempered unipotent representations of inner twists of a split simple $p$-adic group $G$ and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroup