La trasformata veloce di Fourier (FFT): analisi e implementazione in C++

La trasformata di Fourier (FT) è uno strumento molto potente implementato, oggi, in un enorme numero di tecnologie. Il suo primo esempio di applicazione fu proprio il campionamento e la digitalizzazione di segnali analogici. Nel tempo l'utilizzo della FT è stato ampliato a più orizzonti in ambito digitale, basti pensare che il formato di compressione '.jpg' utilizza una FT bidimensionale, mentre uno degli ultimi esempi di applicazione si ha nell'imaging digitale in ambito medico (risonanza magnetica nucleare, tomografia assiale computerizzata TAC ecc...). Nonostante gli utilizzi della FT siano molto diversificati il suo basilare funzionamento non è mai cambiato: essa non fa altro che modificare il dominio di una funzione del tempo (un segnale) in un dominio delle frequenze, permettendo così lo studio della composizione in termini di frequenza, ampiezza e fase del segnale stesso. Parallelamente all'evoluzione in termini di applicazioni si è sviluppato uno studio volto a migliorare e ottimizzare la computazione della stessa, data l'esponenziale crescita del suo utilizzo. In questa trattazione si vuole analizzare uno degli algoritmi di ottimizzazione più celebri e utilizzati in tal senso: la trasformata veloce di Fourier (Fast Fourier Transformation o FFT). Si delineeranno quindi le caratteristiche salienti della FT, e verrà introdotto l'algoritmo di computazione tramite linguaggio C++ dedicando particolare attenzione ai limiti di applicazione di tale algoritmo e a come poter modificare opportunamente la nostra funzione in modo da ricondurci entro i limiti di validità.

Broadband Fourier transform rotational spectroscopy for structure determination: The water heptamer

Over the recent years chirped-pulse, Fourier-transform microwave (CP-FTMW) spectrometers have chan- ged the scope of rotational spectroscopy. The broad frequency and large dynamic range make possible structural determinations in molecular systems of increasingly larger size from measurements of heavy atom (13C, 15N, 18O) isotopes recorded in natural abundance in the same spectrum as that of the parent isotopic species. The design of a broadband spectrometer operating in the 2–8 GHz frequency range with further improvements in sensitivity is presented. The current CP-FTMW spectrometer performance is benchmarked in the analyses of the rotational spectrum of the water heptamer, (H2O)7, in both 2– 8 GHz and 6–18 GHz frequency ranges. Two isomers of the water heptamer have been observed in a pulsed supersonic molecular expansion. High level ab initio structural searches were performed to pro- vide plausible low-energy candidates which were directly compared with accurate structures provided from broadband rotational spectra. The full substitution structure of the most stable species has been obtained through the analysis of all possible singly-substituted isotopologues (H218O and HDO), and a least-squares rm(1) geometry of the oxygen framework determined from 16 different isotopic species compares with the calculated O–O equilibrium distances at the 0.01 Å level.

Broadband Fourier transform rotational spectroscopy for structure determination: The water heptamer

Over the recent years chirped-pulse, Fourier-transform microwave (CP-FTMW) spectrometers have changed the scope of rotational spectroscopy. The broad frequency and large dynamic range make possible structural determinations in molecular systems of increasingly larger size from measurements of heavy atom (C-13, N-15, O-18) isotopes recorded in natural abundance in the same spectrum as that of the parent isotopic species. The design of a broadband spectrometer operating in the 2-8 GHz frequency range with further improvements in sensitivity is presented. The current CP-FTMW spectrometer performance is benchmarked in the analyses of the rotational spectrum of the water heptamer, (H2O)(7), in both 2-8 GHz and 6-18 GHz frequency ranges. Two isomers of the water heptamer have been observed in a pulsed supersonic molecular expansion. High level ab initio structural searches were performed to provide plausible low-energy candidates which were directly compared with accurate structures provided from broadband rotational spectra. The full substitution structure of the most stable species has been obtained through the analysis of all possible singly-substituted isotopologues ((H2O)-O-18 and HDO), and a least-squares r(m)((1)) geometry of the oxygen framework determined from 16 different isotopic species compares with the calculated O-O equilibrium distances at the 0.01 angstrom level. (C) 2013 Elsevier B.V. All rights reserved.

Enhanced (1)G(4) emission in NaLaF4: Pr3+, Yb3+ and charge transfer in NaLaF4: Ce3+, Yb3+ studied by fourier transform luminescence spectroscopy

A high resolution luminescence study of NaLaF4: 1%Pr3+, 5%Yb3+ and NaLaF4: 1%Ce3+, 5%Yb3+ in the UV to NIR spectral range using a InGaAs detector and a fourier transform interferometer is reported. Although the Pr3+(P-3(0) -> (1)G(4), Yb3+(F-2(7/2) -> F-2(5/2)) energy transfer step takes place, significant Pr3+ (1)G(4) emission around 993, 1330 and 1850 nm is observed. No experimental proof for the second energy transfer step in the down-conversion process between Pr3+ and Yb3+ can be given. In the case of NaLaF4: Ce3+, Yb3+ it is concluded that the observed Yb3+ emission upon Ce3+ 5d excitation is the result of a charge transfer process instead of down-conversion. (C) 2010 Elsevier B.V. All rights reserved.

Characterization of water-soluble organic compounds in ambient aerosol using ultrahigh-resolution elctrospray ionization fourier transform ion cyclotron resonance mass spectrometry.

Atmospheric aerosol water-soluble organic compounds (WSOC) exist in a complex mixture of thousands of organic compounds which may have a significant influence on the climate-relevant properties of the atmospheric aerosol. To understand the potential influences, the ambient aerosol was collected at a nonurban mountainous site near Steamboat Springs, CO. The WSOC fraction was analyzed using positive and negative electrospray ionization Fourier transform ion cyclotron resonance mass spectrometry. Approximately 2400 and 4000 molecular formulas were identified from the detected positive and negative ions, respectively. The formulas contained carbon (C), hydrogen (H), oxygen (O), nitrogen (N), and sulfur (S) atoms over the mass range of 100-800 Da in both ionization modes. The number range of double bond equivalents (DBE), the mean O:C, H:C, and oxidation state of carbon for the positive ions were 0 – 18, 0.25 ± 0.15, 1.39 ± 0.29, and -0.89 ± 0.23, respectively. Comparatively, the negative ion values were 0 – 14, 0.53 ± 0.20, 1.48 ± 0.30, and -0.41 ± 0.45, respectively. Overall, the positive ion molecular formulas were less oxygenated than negative ions as seen with the lower O:C and OSc values. Molecular formulas of the positive ions classified as aliphatic, olefinic, and aromatic compound classes based on the aromaticity index values. Aliphatic compounds were the CHNO and CHO formulas that had mean DBE values of about 5 and 3, respectively. However, a majority of the CHOS, CHNOS, and CHS formulas were defined as olefinic compounds and had mean DBE values of about 12, 13, and 10, respectively. Overall, more than half of the assigned molecular formulas contained sulfur and were olefinic to aromatic compounds with a DBE range of 7-18. Source of the unsaturated sulfur containing compounds is currently unknown. Several nitrogen containing compounds were in common with the field and laboratory studies of the biomass burning aerosol and aged secondary organic aerosol products of the limonene ozonolysis.

Universally Applicable Model for the Quantitative Determination of Lake Sediment Composition Using Fourier Transform Infrared Spectroscopy

ABSTRACT: Fourier transform infrared spectroscopy (FTIRS) can provide detailed information on organic and minerogenic constituents of sediment records. Based on a large number of sediment samples of varying age (0�340 000 yrs) and from very diverse lake settings in Antarctica, Argentina, Canada, Macedonia/Albania, Siberia, and Sweden, we have developed universally applicable calibration models for the quantitative determination of biogenic silica (BSi; n = 816), total inorganic carbon (TIC; n = 879), and total organic carbon (TOC; n = 3164) using FTIRS. These models are based on the differential absorbance of infrared radiation at specific wavelengths with varying concentrations of individual parameters, due to molecular vibrations associated with each parameter. The calibration models have low prediction errors and the predicted values are highly correlated with conventionally measured values (R = 0.94�0.99). Robustness tests indicate the accuracy of the newly developed FTIRS calibration models is similar to that of conventional geochemical analyses. Consequently FTIRS offers a useful and rapid alternative to conventional analyses for the quantitative determination of BSi, TIC, and TOC. The rapidity, cost-effectiveness, and small sample size required enables FTIRS determination of geochemical properties to be undertaken at higher resolutions than would otherwise be possible with the same resource allocation, thus providing crucial sedimentological information for climatic and environmental reconstructions.

A convolution model for energy transport in a therapeutic fast neutron beam

A three-dimensional model has been proposed that uses Monte Carlo and fast Fourier transform convolution techniques to calculate the dose distribution from a fast neutron beam. This method transports scattered neutrons and photons in the forward, lateral, and backward directions and protons, electrons, and positrons in the forward and lateral directions by convolving energy spread kernels with initial interaction available energy distributions. The primary neutron and photon spectrums have been derived from narrow beam attenuation measurements. The positions and strengths of the effective primary neutron, scattered neutron, and photon sources have been derived from dual ion chamber measurements. The size of the effective primary neutron source has been measured using a copper activation technique. Heterogeneous tissue calculations require a weighted sum of two convolutions for each component since the kernels must be invariant for FFT convolution. Comparisons between calculations and measurements were performed for several water and heterogeneous phantom geometries. ^

Fourier transform infrared spectroscopy reveals a rigid α-helical assembly for the tetrameric Streptomyces lividans K+ channel

The structure of the tetrameric K+ channel from Streptomyces lividans in a lipid bilayer environment was studied by polarized attenuated total reflection Fourier transform infrared spectroscopy. The channel displays approximately 43% α-helical and 25% β-sheet content. In addition, H/D exchange experiments show that only 43% of the backbone amide protons are exchangeable with solvent. On average, the α-helices are tilted 33° normal to the membrane surface. The results are discussed in relationship to the lactose permease of Escherichia coli, a membrane transport protein.

Fourier Transform Infrared Microspectroscopy Detects Changes in Protein Secondary Structure Associated with Desiccation Tolerance in Developing Maize Embryos1

Isolated immature maize (Zea mays L.) embryos have been shown to acquire tolerance to rapid drying between 22 and 25 d after pollination (DAP) and to slow drying from 18 DAP onward. To investigate adaptations in protein profile in association with the acquisition of desiccation tolerance in isolated, immature maize embryos, we applied in situ Fourier transform infrared microspectroscopy. In fresh, viable, 20- and 25-DAP embryo axes, the shapes of the different amide-I bands were identical, and this was maintained after flash drying. On rapid drying, the 20-DAP axes had a reduced relative proportion of α-helical protein structure and lost viability. Rapidly dried 25-DAP embryos germinated (74%) and had a protein profile similar to the fresh control axes. On slow drying, the α-helical contribution in both the 20- and 25-DAP embryo axes increased compared with that in the fresh control axes, and survival of desiccation was high. The protein profile in dry, mature axes resembled that after slow drying of the immature axes. Rapid drying resulted in an almost complete loss of membrane integrity in the 20-DAP embryo axes and much less so in the 25-DAP axes. After slow drying, low plasma membrane permeability ensued in both the 20- and 25-DAP axes. We conclude that slow drying of excised, immature embryos leads to an increased proportion of α-helical protein structures in their axes, which coincides with additional tolerance of desiccation stress.

Integração da equação de movimento através da Transformada de Fourier com o uso de ponderadores de ordem elevada

Baseando-se em um sistema com um grau de liberdade, é apresentada neste trabalho a equação de movimento, bem como a sua resolução através das Transformadas de Fourier e da Transformada Rápida de Fourier (FFT). Através da análise da forma como são feitas as integrações nas transformadas, foram estudados e aplicados os ponderadores de Newton-Cotes na resolução da equação de movimento, de forma a aumentar substancialmente a precisão dos resultados em comparação com a forma convencional da Transformada de Fourier.