Resolution and noise in Fourier-transform spectroscopy.
Resolution and noise in Fourier transform spectrometer, considering mechanical limitations and diffraction effects
FOURIER TRANSFORM↗
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Geometric interpretations of the Discrete Fourier Transform (DFT)
One, two, and three dimensional Discrete Fourier Transforms (DFT) and geometric interpretations of their periodicities are presented. These operators are examined for their relationship with the two sided, continuous Fourier transform. Discrete or continuous transforms of real functions have certain symmetry properties. The symmetries are examined for the one, two, and three dimensional cases. Extension to higher dimension is straight forward.
Campbell, C. W.↗
Multiplexing and Demultiplexing Signals for Radiography Application Using the Discrete Fourier Transform
Our goal is to develop an X-ray phase-contrast imaging system that can provide excellent soft tissue contrast of phase, attenuation, and small-angle scatter. We propose to replace the common system of G0, G1, and G2 gradings with a biprism array to replace the G1 grading and introduce a novel X-ray tube designed to replace the motion of the phase stepping grading G2. The proposed X-ray tube uses temporal multiplexing to provide simultaneous virtual “electronic phase stepping.” In this work the discrete Fourier transform is used to separate from the composite measurement individual X-ray phase contrast measurements sampled at different frequencies. The method performs a discrete Fourier transform of a composite refence sequence to obtain using the frequency amplitudes calibration factors needed to extract the X-ray phase contrast measurement amplitudes from the composite image. The composite reference sequence is the sum of the individual sequences, at different frequencies, with amplitudes of one. The method takes the discrete Fourier transform of this composite reference sequence; whereby, the amplitude of each frequency component is compared with the total sum of its stand-alone sequence amplitude. A calibration factor is determined so that the amplitude of this composite reference frequency times the calibration factor must equal the total sum of the sequence amplitude—the zero-frequency amplitude of the discrete Fourier transform of its stand-alone sequence. To demultiplex the composite measured signal these calibration factors are multiplied by the amplitudes of the frequency components of the discrete Fourier transform of the composite X-phase-contrast measurement to obtain the amplitude of each frequency encoded measurement. Using these calibration factors, we demonstrate with the discrete Fourier transform in Mathematica the extraction of individual images from a composite image that one would expect obtaining from our proposed new X-ray phase contrast imaging system. We then demonstrate as an example how using images from X-ray phase contrast data one can calculate phase, attenuation and the dark field images using grading phase step data supplied to use from Microworks, GmbH in Karlsruhe, Germany.
42 ENGINEERING↗
IR spectroscopic characteristics of cell cycle and cell death probed by synchrotron radiation based Fourier transform IR spectromicroscopy
Synchrotron radiation based Fourier transform IR (SR-FTIR) spectromicroscopy allows the study of individual living cells with a high signal to noise ratio. Here we report the use of the SR-FTIR technique to investigate changes in IR spectral features from individual human lung fibroblast (IMR-90) cells in vitro at different points in their cell cycle. Clear changes are observed in the spectral regions corresponding to proteins, DNA, and RNA as a cell changes from the G(1)-phase to the S-phase and finally into mitosis. These spectral changes include markers for the changing secondary structure of proteins in the cell, as well as variations in DNA/RNA content and packing as the cell cycle progresses. We also observe spectral features that indicate that occasional cells are undergoing various steps in the process of cell death. The dying or dead cell has a shift in the protein amide I and II bands corresponding to changing protein morphologies, and a significant increase in the intensity of an ester carbonyl C===O peak at 1743 cm(-1) is observed. Copyright John Wiley & Sons, Inc. Biopolymers (Biospectroscopy) 57: 329-335, 2000.
Non-NASA Center↗
Structure in the 3D Galaxy Distribution. III. Fourier Transforming the Universe: Phase and Power Spectra
We demonstrate the effectiveness of a relatively straightforward analysis of the complex 3D Fourier transform of galaxy coordinates derived from redshift surveys. Numerical demonstrations of this approach are carried out on a volume-limited sample of the Sloan Digital Sky Survey redshift survey. The direct unbinned transform yields a complex 3D data cube quite similar to that from the Fast Fourier Transform of finely binned galaxy positions. In both cases, deconvolution of the sampling window function yields estimates of the true transform. Simple power spectrum estimates from these transforms are roughly consistent with those using more elaborate methods. The complex Fourier transform characterizes spatial distributional properties beyond the power spectrum in a manner different from (and we argue is more easily interpreted than) the conventional multipoint hierarchy. We identify some threads of modern large-scale inference methodology that will presumably yield detections in new wider and deeper surveys.
Galaxies↗
Fast Fourier Transform Spectral Analysis Program
Fast Fourier Transform Spectral Analysis Program is used in frequency spectrum analysis of postflight, space vehicle telemetered trajectory data. This computer program with a digital algorithm can calculate power spectrum rms amplitudes and cross spectrum of sampled parameters at even time increments.
Daniel, J. A., Jr.↗
Calculation of Fourier transforms on finite Abelian groups
Fourier transform calculations on finite Abelian groups
Apple, G. G.↗
Predicting the thermal/structural performance of the atmospheric trace molecules spectroscopy /ATMOS/ Fourier transform spectrometer
ATMOS is a Fourier transform spectrometer to measure atmospheric trace molecules over a spectral range of 2-16 microns. Assessment of the system performance of ATMOS includes evaluations of optical system errors induced by thermal and structural effects. In order to assess the optical system errors induced from thermal and structural effects, error budgets are assembled during system engineering tasks and line of sight and wavefront deformations predictions (using operational thermal and vibration environments and computer models) are subsequently compared to the error budgets. This paper discusses the thermal/structural error budgets, modelling and analysis methods used to predict thermal/structural induced errors and the comparisons that show that predictions are within the error budgets.
Miller, J. M.↗
Optimal Padding for the Two-Dimensional Fast Fourier Transform
One-dimensional Fast Fourier Transform (FFT) operations work fastest on grids whose size is divisible by a power of two. Because of this, padding grids (that are not already sized to a power of two) so that their size is the next highest power of two can speed up operations. While this works well for one-dimensional grids, it does not work well for two-dimensional grids. For a two-dimensional grid, there are certain pad sizes that work better than others. Therefore, the need exists to generalize a strategy for determining optimal pad sizes. There are three steps in the FFT algorithm. The first is to perform a one-dimensional transform on each row in the grid. The second step is to transpose the resulting matrix. The third step is to perform a one-dimensional transform on each row in the resulting grid. Steps one and three both benefit from padding the row to the next highest power of two, but the second step needs a novel approach. An algorithm was developed that struck a balance between optimizing the grid pad size with prime factors that are small (which are optimal for one-dimensional operations), and with prime factors that are large (which are optimal for two-dimensional operations). This algorithm optimizes based on average run times, and is not fine-tuned for any specific application. It increases the amount of times that processor-requested data is found in the set-associative processor cache. Cache retrievals are 4-10 times faster than conventional memory retrievals. The tested implementation of the algorithm resulted in faster execution times on all platforms tested, but with varying sized grids. This is because various computer architectures process commands differently. The test grid was 512 512. Using a 540 540 grid on a Pentium V processor, the code ran 30 percent faster. On a PowerPC, a 256x256 grid worked best. A Core2Duo computer preferred either a 1040x1040 (15 percent faster) or a 1008x1008 (30 percent faster) grid. There are many industries that can benefit from this algorithm, including optics, image-processing, signal-processing, and engineering applications.
Dean, Bruce H.↗
Structure in the 3D Galaxy Distribution: III. Fourier Transforming the Universe: Phase and Power Spectra
We demonstrate the effectiveness of a relatively straightforward analysis of the complex 3D Fourier transform of galaxy coordinates derived from redshift surveys. Numerical demonstrations of this approach are carried out on a volume-limited sample of the Sloan Digital Sky Survey redshift survey. The direct unbinned transform yields a complex 3D data cube quite similar to that from the Fast Fourier Transform (FFT) of finely binned galaxy positions. In both cases deconvolution of the sampling window function yields estimates of the true transform. Simple power spectrum estimates from these transforms are roughly consistent with those using more elaborate methods. The complex Fourier transform characterizes spatial distributional properties beyond the power spectrum in a manner different from (and we argue is more easily interpreted than) the conventional multi-point hierarchy. We identify some threads of modern large scale inference methodology that will presumably yield detections in new wider and deeper surveys.
Galactic clusters↗
Analysis and application of Fourier transform spectroscopy in atmospheric remote sensing
An analysis method for Fourier transform spectroscopy is summarized with applications to various types of distortion in atmospheric absorption spectra. This analysis method includes the fast Fourier transform method for simulating the interferometric spectrum and the nonlinear least-squares method for retrieving the information from a measured spectrum. It is shown that spectral distortions can be simulated quite well and that the correct information can be retrieved from a distorted spectrum by this analysis technique.
Park, J. H.↗
A discrete Fourier transform for virtual memory machines
An algebraic theory of the Discrete Fourier Transform is developed in great detail. Examination of the details of the theory leads to a computationally efficient fast Fourier transform for the use on computers with virtual memory. Such an algorithm is of great use on modern desktop machines. A FORTRAN coded version of the algorithm is given for the case when the sequence of numbers to be transformed is a power of two.
Galant, David C.↗
Two Dimensional Symmetric Correlation Functions of the S Operator and Two Dimensional Fourier Transforms: Considering the Line Coupling for P and R Lines of Linear Molecules
The refinement of the Robert-Bonamy (RB) formalism by considering the line coupling for isotropic Raman Q lines of linear molecules developed in our previous study [Q. Ma, C. Boulet, and R. H. Tipping, J. Chem. Phys. 139, 034305 (2013)] has been extended to infrared P and R lines. In these calculations, the main task is to derive diagonal and off-diagonal matrix elements of the Liouville operator iS1 − S2 introduced in the formalism. When one considers the line coupling for isotropic Raman Q lines where their initial and final rotational quantum numbers are identical, the derivations of off-diagonal elements do not require extra correlation functions of the ^S operator and their Fourier transforms except for those used in deriving diagonal elements. In contrast, the derivations for infrared P and R lines become more difficult because they require a lot of new correlation functions and their Fourier transforms. By introducing two dimensional correlation functions labeled by two tensor ranks and making variable changes to become even functions, the derivations only require the latters' two dimensional Fourier transforms evaluated at two modulation frequencies characterizing the averaged energy gap and the frequency detuning between the two coupled transitions. With the coordinate representation, it is easy to accurately derive these two dimensional correlation functions. Meanwhile, by using the sampling theory one is able to effectively evaluate their two dimensional Fourier transforms. Thus, the obstacles in considering the line coupling for P and R lines have been overcome. Numerical calculations have been carried out for the half-widths of both the isotropic Raman Q lines and the infrared P and R lines of C2H2 broadened by N2. In comparison with values derived from the RB formalism, new calculated values are significantly reduced and become closer to measurements.
infrared radiation↗
Discrete fourier transform (DFT) analysis for applications using iterative transform methods
According to various embodiments, a method is provided for determining aberration data for an optical system. The method comprises collecting a data signal, and generating a pre-transformation algorithm. The data is pre-transformed by multiplying the data with the pre-transformation algorithm. A discrete Fourier transform of the pre-transformed data is performed in an iterative loop. The method further comprises back-transforming the data to generate aberration data.
Dean, Bruce H.↗
Interferogram analysis using Fourier transform techniques
A method of interferogram analysis is described in which Fourier transform techniques are used to map the complex fringe visibility in several types of interferograms. Algorithms are developed for estimation of both the amplitude and the phase of the fringes (yielding the modulus and the phase of the holographically recorded object Fourier transform). The algorithms were applied to the reduction of interferometric seeing measurements (i.e., the estimation of the fringe amplitude only), and the reduction of interferometric tests (i.e., estimation of the fringe phase only). The method was used to analyze scatter-plate interferograms obtained at NOAO.
Roddier, Claude↗
Modulated Fourier Transform Raman Fiber-Optic Spectroscopy
A modification to a commercial Fourier Transform (FT) Raman spectrometer is presented for the elimination of thermal backgrounds in the FT Raman spectra. The modification involves the use of a mechanical optical chopper to modulate the continuous wave laser, remote collection of the signal via fiber optics, and connection of a dual-phase digital-signal-processor (DSP) lock-in amplifier between the detector and the spectrometer's collection electronics to demodulate and filter the optical signals. The resulting Modulated Fourier Transform Raman Fiber-Optic Spectrometer is capable of completely eliminating thermal backgrounds at temperatures exceeding 300 C.
Jensen, Brian J.↗
A new hybrid algorithm for computing a fast discrete Fourier transform
For certain long transform lengths, Winograd's algorithm for computing the discrete Fourier transform is extended considerably. This is accomplished by performing the cyclic convolution, required by Winograd's method, with the Mersenne-prime number theoretic transform. This new algorithm requires fewer multiplications than either the standard fast Fourier transform or Winograd's more conventional algorithm.
Reed, I. S.↗
Feasibility Study of Velocity Synchronized Fourier Transform Hologram Camera System
Velocity synchronized Fourier transform hologram camera system
Dyes, W. A.↗