DISCRETE AND CONTINUOUS FOURIER TRANSFORMS ANAL... EXCLUSIVE
The frequency-domain spectrum obtained by Fourier transformation (FT) of a time-domain signal is accurate only for a continuous noiseless time-domain signal of infinite duration. For discrete noisy truncated time-domain signals, non-FT (e.g., Bayesian analysis) methods may provide more accurate spectral estimates of time-domain signal frequencies, relaxation time(s), and relative abundances. In this paper, we show that Bayesian analysis of simulated and experimental ion cyclotron resonance (ICR) time-domain noisy signals can produce a spectrum with mass accuracy improved by a factor of 10 or more over that obtained from a magnitude-mode discrete fast Fourier transform (FFT) spectrum. Moreover, Bayesian analysis offers the useful advantage that it automatically estimates the precision of its iteratively determined spectral parameters. The main disadvantage of Bayesian analysis is its lengthy computation time compared to that of FFT (hours vs seconds on the same hardware for approximately 4K time-domain data points); the Bayesian computation time increases rapidly with the number of spectral peaks and (less rapidly) with the number of time-domain data points. Bayesian analysis should thus prove useful for those FT/ICR applications involving relatively few data points and/or requiring high mass accuracy.
DISCRETE AND CONTINUOUS FOURIER TRANSFORMS ANAL...
This looks like a continuous curve, but is actually (half a) million discrete points. Notice that because \(f\) was discontinuous on \(\mathbbR\), \(\hatf(k)\) decays slowly like \(k^-1\). How do we know to trust the answer? A convergence study in N shows that200 nodes was indeed enough to reduce the quadrature error to below the\(10^-10\) NUFFT tolerance:
In research and development laboratories, chemical or pharmaceutical analysis has been carried out by evaluating sample signals obtained from instruments. However, the qualitative and quantitative determination based on raw signals may not be always possible due to sample complexity. In such cases, there is a need for powerful signal processing methodologies that can effectively process raw signals to get correct results. Wavelet transform is one of the most indispensable and popular signal processing methods currently used for noise removal, background correction, differentiation, data smoothing and filtering, data compression and separation of overlapping signals etc. This review article describes the theoretical aspects of wavelet transform (i.e., discrete, continuous and fractional) and its characteristic applications in UV spectroscopic analysis of pharmaceuticals.
Abstract:The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.Keywords: Fourier Theory; DFT in polar coordinates; polar coordinates; multidimensional DFT; discrete Hankel Transform; discrete Fourier Transform; Orthogonality 041b061a72