Fourier Analysis
Analysis based on the mathematical function first formulated by Jean-Baptiste-Joseph Fourier in 1807. The function, known as the Fourier transform, describes the sinusoidal pattern of any fluctuating pattern in the physical world in terms of its amplitude and its phase. It has broad applications in biomedicine, e.g., analysis of the x-ray crystallography data pivotal in identifying the double helical nature of DNA and in analysis of other molecules, including viruses, and the modified back-projection algorithm universally used in computerized tomography imaging, etc. (From Segen, The Dictionary of Modern Medicine, 1992)
Optical Phenomena
Technetium
The first artificially produced element and a radioactive fission product of URANIUM. Technetium has the atomic symbol Tc, atomic number 43, and atomic weight 98.91. All technetium isotopes are radioactive. Technetium 99m (m=metastable) which is the decay product of Molybdenum 99, has a half-life of about 6 hours and is used diagnostically as a radioactive imaging agent. Technetium 99 which is a decay product of technetium 99m, has a half-life of 210,000 years.
Gated Blood-Pool Imaging
Radionuclide ventriculography where scintigraphic data is acquired during repeated cardiac cycles at specific times in the cycle, using an electrocardiographic synchronizer or gating device. Analysis of right ventricular function is difficult with this technique; that is best evaluated by first-pass ventriculography (VENTRICULOGRAPHY, FIRST-PASS).
Signal Processing, Computer-Assisted
Spectroscopy, Fourier Transform Infrared
Mathematics
Algorithms
Models, Biological
A new filtering algorithm for medical magnetic resonance and computer tomography images. (1/2483)
Inner views of tubular structures based on computer tomography (CT) and magnetic resonance (MR) data sets may be created by virtual endoscopy. After a preliminary segmentation procedure for selecting the organ to be represented, the virtual endoscopy is a new postprocessing technique using surface or volume rendering of the data sets. In the case of surface rendering, the segmentation is based on a grey level thresholding technique. To avoid artifacts owing to the noise created in the imaging process, and to restore spurious resolution degradations, a robust Wiener filter was applied. This filter working in Fourier space approximates the noise spectrum by a simple function that is proportional to the square root of the signal amplitude. Thus, only points with tiny amplitudes consisting mostly of noise are suppressed. Further artifacts are avoided by the correct selection of the threshold range. Afterwards, the lumen and the inner walls of the tubular structures are well represented and allow one to distinguish between harmless fluctuations and medically significant structures. (+info)Wavelet transform to quantify heart rate variability and to assess its instantaneous changes. (2/2483)
Heart rate variability is a recognized parameter for assessing autonomous nervous system activity. Fourier transform, the most commonly used method to analyze variability, does not offer an easy assessment of its dynamics because of limitations inherent in its stationary hypothesis. Conversely, wavelet transform allows analysis of nonstationary signals. We compared the respective yields of Fourier and wavelet transforms in analyzing heart rate variability during dynamic changes in autonomous nervous system balance induced by atropine and propranolol. Fourier and wavelet transforms were applied to sequences of heart rate intervals in six subjects receiving increasing doses of atropine and propranolol. At the lowest doses of atropine administered, heart rate variability increased, followed by a progressive decrease with higher doses. With the first dose of propranolol, there was a significant increase in heart rate variability, which progressively disappeared after the last dose. Wavelet transform gave significantly better quantitative analysis of heart rate variability than did Fourier transform during autonomous nervous system adaptations induced by both agents and provided novel temporally localized information. (+info)A new tool for measuring the suckling stimulus during breastfeeding in humans: the orokinetogram and the Fourier series. (3/2483)
The Fourier series was used to analyse the oral movements recorded by the orokinetogram during breastfeeding in human babies. This is a new method that allows recording of oral movements without introducing any extrinsic element between the nipple and the mouth of the baby. The advantage of displaying suckling activity after fast Fourier transform (FFT) is that this algorithm allows storage, quantification and frequency analysis of the oral movements throughout a suckling bout, which enables the total oral activity to be measured. Two types of oral movements are found: slow high amplitude (SHA) and fast low amplitude (FLA). FLA movements may be derived from peristaltic movements of the tongue that result in tickling stimuli to the mechanoreceptors of the nipple and milk expression. The frequency bandwidth of oral movements is wider (0-8 Hz) than has been described previously (0-3 Hz) and this is due to the presence of the FLA oral movements. An indirect measurement of the energy of oral movements during suckling is obtained by the pattern of energy distribution used in each individual frequency band by oral movements. This pattern changes in relation to the periods of continuous and intermittent suckling activity. SHA and FLA oral movements are more intense during continuous suckling. Statistical analysis showed a correlation between the energy of SHA and FLA waves throughout the suckling bout, and also that the highest level of energy during suckling activity is displayed during the first 2 min. The novel tools described in this paper allow investigation of the role of suckling stimulus in reflex hormone release and other mother-infant interactions. (+info)Effects of nicorandil on aortic input impedance: a comparative study with nitroglycerin. (4/2483)
A study of aortic input impedance was performed to evaluate the effects of nicorandil on the systemic circulation, and the effects were compared with those of nitroglycerin. Sixteen patients with coronary artery disease were divided into 2 age-matched groups. Aortic input impedance was obtained from Fourier analysis of aortic pressure and flow signals at baseline conditions, after intravenous administration of either 4 mg (Group 1) or 8 mg (Group 2) nicorandil, and 20 min after 0.3 mg sublingual nitroglycerin. In Group 1, the first harmonic impedance modulus (Z1, 304+/-140 dyne x s x cm(-5)) and the average of the first to third harmonics (Z1-3, 207+/-99 dyne x s x cm(-5)), indices of wave reflection, significantly decreased (24.4% (p<0.05) and 24.7% (p<0.01), respectively) after nicorandil, and 41.3% (p<0.01) and 33.9% (p<0.01) after nitroglycerin. The effects between the 2 vasodilators were not significantly different. In Group 2, Z1 and Z1-3 (275+/-138 and 196+/-93 dyne x s x cm(-5), respectively) also decreased after administration of nicorandil (28.4% (p<0.01) and 35.9% (p<0.01), respectively), and after administration of nitroglycerin (23.9% (p<0.01) and 28.7% (p<0.01), respectively), without any significant difference between the 2 drugs. Characteristic impedance and total peripheral resistance (R) in both groups remained unchanged except for R after 8 mg nicorandil (from 1830+/-415 to 1433+/-428 dyne x s x cm(-5); p<0.01). Like nitroglycerin, both doses of nicorandil reduced wave reflection. The reduction in R after 8 mg nicorandil is related to decreased tone in the resistance arteries, probably due to potassium channel opener effects. (+info)Oxytocin-induced Ca2+ responses in human myometrial cells. (5/2483)
Complex spatiotemporal changes in intracellular Ca2+ were monitored in an immortalized human myometrial cell line (PHM1-41) and first-passage human myometrial cells after oxytocin stimulation (1. 0-1000 nM). Laser cytometry revealed intracellular Ca2+ oscillations in both culture systems starting at 1.0 nM, which were followed by repetitive Ca2+ transients by 10-15 min that lasted for at least 90 min. The amplitude of the initial Ca2+ spike was dose dependent, while the frequency of Ca2+ oscillations identified by Fast Fourier Transform (FFT) tended to increase with dose. Removal of oxytocin resulted in termination of oscillations. Analysis of the sources of the Ca2+ involved in oscillations indicated that the major contribution to oscillation frequencies of +info)Automated MAD and MIR structure solution. (6/2483)
Obtaining an electron-density map from X-ray diffraction data can be difficult and time-consuming even after the data have been collected, largely because MIR and MAD structure determinations currently require many subjective evaluations of the qualities of trial heavy-atom partial structures before a correct heavy-atom solution is obtained. A set of criteria for evaluating the quality of heavy-atom partial solutions in macromolecular crystallography have been developed. These have allowed the conversion of the crystal structure-solution process into an optimization problem and have allowed its automation. The SOLVE software has been used to solve MAD data sets with as many as 52 selenium sites in the asymmetric unit. The automated structure-solution process developed is a major step towards the fully automated structure-determination, model-building and refinement procedure which is needed for genomic scale structure determinations. (+info)Discrimination of solvent from protein regions in native Fouriers as a means of evaluating heavy-atom solutions in the MIR and MAD methods. (7/2483)
An automated examination of the native Fourier is tested as a means of evaluation of a heavy-atom solution in MAD and MIR methods for macromolecular crystallography. It is found that the presence of distinct regions of high and low density variation in electron-density maps is a good indicator of the correctness of a heavy-atom solution in the MIR and MAD methods. The method can be used to evaluate heavy-atom solutions during MAD and MIR structure solutions and to determine the handedness of the structure if anomalous data have been measured. (+info)Evidence of noncovalent dimerization of calmodulin. (8/2483)
Calcium-binding proteins, such as S-100, dimerize readily, and this phenomenon plays an important role in their regulation of target enzymes [Krebs, J., Quadroni, M. & Van Eldik, L.J. (1995) Nat. Struct. Biol. 2, 711-714; Kilby, P.M., Van Eldik, L.J. & Roberts, G. C. (1996) Structure 4, 1041-1052]. We have investigated by Fourier-transform ion cyclotron resonance (FTICR) MS the conformational states of the calcium-binding protein calmodulin, and present clear evidence for a calmodulin dimer formed as a result of noncovalent interactions between folded monomers. Ultra-high-resolution electrospray ionization (ESI) mass spectra for calmodulin, obtained with a 9.4 T FTICR mass spectrometer, are presented. With the use of denaturing solutions (1 : 1 acetonitrile/water + 1% formic acid), relatively high charge states (20 < z < 10) of monomeric calmodulin ions were detected, whereas when calmodulin was electrosprayed from buffer, monomers ions with only 5-10 charges were detected. CD measurements for calmodulin in buffered solution revealed that its alpha-helical content was significantly higher than that for calmodulin in acetonitrile/water solutions, consistent with a proposition that changes in charge state distributions observed in the MS experiments reflect differing states of calmodulin folding. Under buffered conditions, noncovalently bound calmodulin dimers were observed by ESI FTICR MS. Analytical ultracentrifugation experiments carried out in the same solution conditions as those used in the MS experiments were consistent with the proposed calmodulin dimer-monomer equilibrium. The ultra-high mass resolution achieved with the 9.4 T FTICR mass spectrometer allowed unequivocal identification of the noncovalent, as opposed to covalent, character of the calmodulin dimer. (+info)Fourier analysis is a mathematical technique used in signal processing and physics that decomposes a function or a signal into its constituent sinusoidal components, revealing the frequency spectrum and phase relationships of the original function or signal.
I'm sorry for any confusion, but Fourier Analysis is not a medical term. It is a mathematical concept used in various scientific fields, including physics, engineering, and signal processing.
Fourier Analysis is a method to decompose functions into sinusoidal components (sines and cosines) of different frequencies. This allows for the representation of a function or a signal as a sum of these frequency components. It's particularly useful in analyzing periodic functions, understanding signals, and solving partial differential equations.
If you have any medical terms you would like me to define, please let me know!
"Optical phenomena are perceptible experiences or appearances that result from the interaction of light with the environment, structures, or biological systems, often manifesting as unusual visual effects due to specific conditions or properties."
Optical phenomena refer to the various observable patterns and effects that occur due to the interaction of light with the environment or with structures in our eye. These can include natural phenomena such as rainbows, mirages, and halos around the sun or moon, as well as visual artifacts created by the eye itself, such as afterimages, floaters, and flashes of light. Some optical phenomena are caused by the refraction, reflection, or interference of light waves, while others may result from abnormalities in the eye's structure or function. Understanding these phenomena can provide insight into the properties of light and the functioning of the visual system.
Technetium is a synthetic radioactive metallic element (symbol Tc, atomic number 43) used in medical imaging as a radioisotope for diagnostic purposes, such as technetium-99m, which decays quickly and emits gamma rays that can be detected by gamma cameras to visualize organ function and blood flow.
Technetium is not a medical term itself, but it is a chemical element with the symbol Tc and atomic number 43. However, in the field of nuclear medicine, which is a branch of medicine that uses small amounts of radioactive material to diagnose or treat diseases, Technetium-99m (a radioisotope of technetium) is commonly used for various diagnostic procedures.
Technetium-99m is a metastable nuclear isomer of technetium-99, and it emits gamma rays that can be detected outside the body to create images of internal organs or tissues. It has a short half-life of about 6 hours, which makes it ideal for diagnostic imaging since it decays quickly and reduces the patient's exposure to radiation.
Technetium-99m is used in a variety of medical procedures, such as bone scans, lung scans, heart scans, liver-spleen scans, brain scans, and kidney scans, among others. It can be attached to different pharmaceuticals or molecules that target specific organs or tissues, allowing healthcare professionals to assess their function or identify any abnormalities.
'Gated Blood-Pool Imaging' is a nuclear medicine technique that uses the injection of radioactive tracers into the bloodstream, which are then captured by the heart and used to create images of the heart's chambers during different phases of the cardiac cycle, allowing for the evaluation of cardiac function and structure.
Gated Blood-Pool Imaging (GBPI) is a type of nuclear medicine test that uses radioactive material and a specialized camera to create detailed images of the heart and its function. In this procedure, a small amount of radioactive tracer is injected into the patient's bloodstream, which then accumulates in the heart muscle and the blood pool within the heart chambers.
The term "gated" refers to the use of an electrocardiogram (ECG) signal to synchronize the image acquisition with the heart's contractions. This allows for the visualization of the heart's motion during different phases of the cardiac cycle, providing valuable information about the size, shape, and contraction of the heart chambers, as well as the movement of the walls of the heart.
GBPI is often used to assess patients with known or suspected heart disease, such as valvular abnormalities, cardiomyopathies, or congenital heart defects. It can help diagnose and evaluate the severity of these conditions, guide treatment decisions, and monitor the effectiveness of therapy.
"Computer-assisted signal processing refers to the use of computers and specialized software to analyze, interpret, and manipulate electronic signals, such as biological data, to extract relevant information, improve signal quality, and facilitate clinical decision-making."
Computer-assisted signal processing is a medical term that refers to the use of computer algorithms and software to analyze, interpret, and extract meaningful information from biological signals. These signals can include physiological data such as electrocardiogram (ECG) waves, electromyography (EMG) signals, electroencephalography (EEG) readings, or medical images.
The goal of computer-assisted signal processing is to automate the analysis of these complex signals and extract relevant features that can be used for diagnostic, monitoring, or therapeutic purposes. This process typically involves several steps, including:
1. Signal acquisition: Collecting raw data from sensors or medical devices.
2. Preprocessing: Cleaning and filtering the data to remove noise and artifacts.
3. Feature extraction: Identifying and quantifying relevant features in the signal, such as peaks, troughs, or patterns.
4. Analysis: Applying statistical or machine learning algorithms to interpret the extracted features and make predictions about the underlying physiological state.
5. Visualization: Presenting the results in a clear and intuitive way for clinicians to review and use.
Computer-assisted signal processing has numerous applications in healthcare, including:
* Diagnosing and monitoring cardiac arrhythmias or other heart conditions using ECG signals.
* Assessing muscle activity and function using EMG signals.
* Monitoring brain activity and diagnosing neurological disorders using EEG readings.
* Analyzing medical images to detect abnormalities, such as tumors or fractures.
Overall, computer-assisted signal processing is a powerful tool for improving the accuracy and efficiency of medical diagnosis and monitoring, enabling clinicians to make more informed decisions about patient care.
Fourier Transform Infrared (FTIR) Spectroscopy is a technique that measures the absorption or transmission of infrared radiation by a sample, transformed through Fourier analysis to produce a spectrum revealing the molecular vibrations and chemical bonds present in the sample.
Fourier Transform Infrared (FTIR) spectroscopy is a type of infrared spectroscopy that uses the Fourier transform mathematical technique to convert the raw data obtained from an interferometer into a more interpretable spectrum. This technique allows for the simultaneous collection of a wide range of wavelengths, resulting in increased sensitivity and speed compared to traditional dispersive infrared spectroscopy.
FTIR spectroscopy measures the absorption or transmission of infrared radiation by a sample as a function of frequency, providing information about the vibrational modes of the molecules present in the sample. This can be used for identification and quantification of chemical compounds, analysis of molecular structure, and investigation of chemical interactions and reactions.
In summary, FTIR spectroscopy is a powerful analytical technique that uses infrared radiation to study the vibrational properties of molecules, with increased sensitivity and speed due to the use of Fourier transform mathematical techniques and an interferometer.
"Mathematics is not typically defined in the field of medicine, as it is a distinct subject that involves the study of numbers, quantities, structure, space, and change, and while it may not be a medical term, it is widely used in various medical fields for research, analysis, and understanding of data."
I'm sorry for any confusion, but there seems to be a misunderstanding. Mathematics is not a medical term; it is a branch of science dedicated to the study of numbers, shapes, and structures. However, mathematics does have many applications in medicine, such as in modeling disease spread, analyzing medical images, or designing clinical trials. If you have any questions related to mathematics in a medical context, I'd be happy to help clarify those for you!
In the context of medicine, an algorithm refers to a step-by-step procedure or set of rules that healthcare professionals use to diagnose and manage specific medical conditions, often presented in a flowchart format for easy understanding and implementation.
An algorithm is not a medical term, but rather a concept from computer science and mathematics. In the context of medicine, algorithms are often used to describe step-by-step procedures for diagnosing or managing medical conditions. These procedures typically involve a series of rules or decision points that help healthcare professionals make informed decisions about patient care.
For example, an algorithm for diagnosing a particular type of heart disease might involve taking a patient's medical history, performing a physical exam, ordering certain diagnostic tests, and interpreting the results in a specific way. By following this algorithm, healthcare professionals can ensure that they are using a consistent and evidence-based approach to making a diagnosis.
Algorithms can also be used to guide treatment decisions. For instance, an algorithm for managing diabetes might involve setting target blood sugar levels, recommending certain medications or lifestyle changes based on the patient's individual needs, and monitoring the patient's response to treatment over time.
Overall, algorithms are valuable tools in medicine because they help standardize clinical decision-making and ensure that patients receive high-quality care based on the latest scientific evidence.
Biological models are simplified representations or systems that mimic specific biological processes, structures, or organisms, used to understand, predict, or test various aspects of biological phenomena, such as disease progression or drug response, through experimental manipulation and observation.
Biological models, also known as physiological models or organismal models, are simplified representations of biological systems, processes, or mechanisms that are used to understand and explain the underlying principles and relationships. These models can be theoretical (conceptual or mathematical) or physical (such as anatomical models, cell cultures, or animal models). They are widely used in biomedical research to study various phenomena, including disease pathophysiology, drug action, and therapeutic interventions.
Examples of biological models include:
1. Mathematical models: These use mathematical equations and formulas to describe complex biological systems or processes, such as population dynamics, metabolic pathways, or gene regulation networks. They can help predict the behavior of these systems under different conditions and test hypotheses about their underlying mechanisms.
2. Cell cultures: These are collections of cells grown in a controlled environment, typically in a laboratory dish or flask. They can be used to study cellular processes, such as signal transduction, gene expression, or metabolism, and to test the effects of drugs or other treatments on these processes.
3. Animal models: These are living organisms, usually vertebrates like mice, rats, or non-human primates, that are used to study various aspects of human biology and disease. They can provide valuable insights into the pathophysiology of diseases, the mechanisms of drug action, and the safety and efficacy of new therapies.
4. Anatomical models: These are physical representations of biological structures or systems, such as plastic models of organs or tissues, that can be used for educational purposes or to plan surgical procedures. They can also serve as a basis for developing more sophisticated models, such as computer simulations or 3D-printed replicas.
Overall, biological models play a crucial role in advancing our understanding of biology and medicine, helping to identify new targets for therapeutic intervention, develop novel drugs and treatments, and improve human health.
Fourier analysis
... can be done on cosets, even discrete cosets. Conjugate Fourier series Generalized Fourier series Fourier- ... Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a ... similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise ... and Fourier Analysis)". Sixty Symbols. Brady Haran for the University of Nottingham. Introduction to Fourier analysis of time ...
Multiplier (Fourier analysis)
The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, ... Fourier Analysis on Groups, Interscience Torchinsky, Alberto (2004), Real-Variable Methods in Harmonic Analysis, Dover, ISBN 0- ... In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on ... and the Fourier and inverse Fourier transforms are given by the formulae f ^ ( ξ ) := ∫ R n f ( x ) e − 2 π i x ⋅ ξ d x f ( x ...
List of Fourier analysis topics
Fourier analysis) Fourier shell correlation Pinsky phenomenon Generalized Fourier series Regressive discrete Fourier series ... This is a list of Fourier analysis topics. See also the list of Fourier-related transforms, and the list of harmonic analysis ... analysis Convergence of Fourier series List of Fourier-related transforms Fourier transform on finite groups Fractional Fourier ... Fast Fourier transform Cyclotomic fast Fourier transform Sparse Fourier transform Fourier amplitude sensitivity testing Fourier ...
Indirect Fourier transformation
B. Vestergaard and S. Hansen (2006). "Application of Bayesian analysis to indirect Fourier transformation in small-angle ... In a Fourier transformation (FT), the Fourier transformed function f ^ ( s ) {\displaystyle {\hat {f}}(s)} is obtained from f ... In indirect Fourier transformation, a guess on the largest distance in the particle D m a x {\displaystyle D_{max}} is given, ... An indirect Fourier transform (IFT) is a solution to this problem. In small-angle scattering on single molecules, an intensity ...
Discrete Fourier series
Fourier analysis). ... ISBN 0-13-754920-2. samples of the Fourier transform of an ... A specific example is the inverse discrete Fourier transform (inverse DFT). The general form of a DFS is: Discrete Fourier ... In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising ... The Fourier series coefficients can be interpreted as a sequence of finite length for k=0,...,(N-1), and zero otherwise, or as ...
Non-uniform discrete Fourier transform
Non-Uniform Fourier Transform: A Tutorial. NFFT 3.0 - Tutorial NUFFT software library (Fourier analysis, Transforms, Digital ... signals Discrete Fourier transform Fast Fourier transform Least-squares spectral analysis Lomb-Scargle periodogram Spectral ... of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in ... Plonka, Gerlind; Potts, Daniel; Steidl, Gabriele; Tasche, Manfred (2019). Numerical Fourier Analysis. Birkhäuser. doi:10.1007/ ...
Fourier optics
... just as in ordinary Fourier analysis and Fourier transforms. Let's consider an imaging system where the z-axis is the optical ... This issue brings up perhaps the predominant difficulty with Fourier analysis, namely that the input-plane function, defined ... These mathematical simplifications and calculations are the realm of Fourier analysis and synthesis - together, they can ... Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is ...
Regressive discrete Fourier series
A R package for RDFS at Github Discrete Fourier transform Fourier series Arruda, J.R.F., 1992a: Analysis of non-equally spaced ... In applied mathematics, the regressive discrete Fourier series (RDFS) is a generalization of the discrete Fourier transform ... Arruda, J.R.F., 1993: Spatial domain modal analysis of lightly-damped structures using laser velocimeters. Journal of Vibration ... Arruda, J.R.F., 1992b: Surface smoothing and partial spatial derivatives using a regressive discrete Fourier series. Mechanical ...
Fourier amplitude sensitivity testing
... (FAST) is a variance-based global sensitivity analysis method. The sensitivity value is ... McRae, G.J., J.W. Tilden and J.H. Seinfeld (1982). Global sensitivity analysis-a computational implementation of the Fourier ... is the maximum order of the calculated Fourier coefficients. After calculating the estimated Fourier coefficients, the first ... The Fourier coefficients can be approximately calculated as A m j ≈ 1 2 π ∫ − π π f ( X 1 ( s ) , X 2 ( s ) , … , X n ( s ) ) ...
Finite Fourier transform
Fourier and Wavelet Analysis (Springer, 2004), p. 264 Morelli, E., "High accuracy evaluation of the finite Fourier transform ... In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a ... or another name for the Fourier series coefficients. or another name for one snapshot of a short-time Fourier transform. ... 52-53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as " ...
Fourier inversion theorem
... the inverse Fourier transform and the identity transform are all examples of operators. Folland, G. B. (1992). Fourier Analysis ... More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an operator (see Fourier transform ... The Fourier inversion theorem is analogous to the convergence of Fourier series. In the Fourier transform case we have f : R n ... Tables of Fourier transforms may easily be used for the inverse Fourier transform by composing the looked-up function with the ...
Fractional Fourier transform
Fourier analysis, Time-frequency analysis, Integral transforms, Articles containing video clips). ... Fractional Fourier transform can be used in time frequency analysis and DSP. It is useful to filter noise, but with the ... In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations ... Wiener, N. (April 1929). "Hermitian Polynomials and Fourier Analysis". Journal of Mathematics and Physics. 8 (1-4): 70-73. doi: ...
Fast Fourier transform
... is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis ... Fast Fourier Transform for Polynomial Multiplication - fast Fourier algorithm Fast Fourier Transforms, Connexions online book ... Welch, Peter D. (1969). "A fixed-point fast Fourier transform error analysis". IEEE Transactions on Audio and Electroacoustics ... Danielson, Gordon C.; Lanczos, Cornelius (1942). "Some improvements in practical Fourier analysis and their application to x- ...
Fourier operator
Least-squares spectral analysis Advances in machine vision: strategies and applications, Colin Archibald and Emil Petriu, ed., ... The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform, ... Diagonal slices through the Fourier operator give rise to chirps. Thus rotation of the Fourier operator gives rise to the ... The Fourier operator defines a continuous two-dimensional function that extends along time and frequency axes, outwards to ...
Fourier series
Discrete Fourier transform Fast Fourier transform Fejér's theorem Fourier analysis Fourier sine and cosine series Fourier ... The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that ... Weisstein, Eric W. "Fourier Series". MathWorld. Joseph Fourier - A site on Fourier's life which was used for the historical ... Also see Fourier transform § Negative frequency). The Fourier series in amplitude-phase form is: Fourier series, amplitude- ...
Sparse Fourier transform
CS1 errors: missing periodical, Fourier analysis, Big data). ... By Fourier series, xn can be written as x n = ( F ∗ X ) n = ∑ k ... The sparse Fourier transform (SFT) is a kind of discrete Fourier transform (DFT) for handling big data signals. Specifically, ... We can use Fourier transform to estimate their coefficients. X k ′ = 1 L ∑ l = 1 L x n ′ e − j 2 π N n ′ ℓ {\displaystyle X_{k ... Recently, the sparse Fourier transform (SFT) has gained a considerable amount of attention, for it performs well on analyzing ...
Short-time Fourier transform
Freeware for short-time Fourier transforms and time-frequency analysis National Taiwan University, Time-Frequency Analysis and ... Fourier analysis, Time-frequency analysis, Transforms, Signal processing). ... 74-80 (March 2003). Jont B. Allen (June 1977). "Short Time Spectral Analysis, Synthesis, and Modification by Discrete Fourier ... The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase ...
Fourier-transform infrared spectroscopy
doi:10.1016/0020-0891(69)90023-2. Singh, Iqbal (2008-09-01). "Renal geology (quantitative renal stone analysis) by 'Fourier ... Thus, a discrete Fourier transform is needed. The fast Fourier transform (FFT) algorithm is used. The first FTIR spectrometers ... The term Fourier-transform infrared spectroscopy originates from the fact that a Fourier transform (a mathematical process) is ... Discrete Fourier transform - Type of Fourier transform in discrete mathematics − for computing periodicity in evenly spaced ...
Fourier sine and cosine series
Fourier series Fourier analysis Least-squares spectral analysis Byerly, William Elwood (1893). "Chapter 2: Development in ... In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two ... Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, ... If f is an odd function with period 2 L {\displaystyle 2L} , then the Fourier Half Range sine series of f is defined to be f ( ...
Graph Fourier transform
Orphaned articles from August 2020, All orphaned articles, Graph theory, Fourier analysis). ... Fourier and the science of today / Fourier et la science d'aujourd'hui. 20 (5): 474-488. Bibcode:2019CRPhy..20..474R. doi: ... Similar to classical noise reduction of signals based on Fourier transform, graph filters based on the graph Fourier transform ... Analogously to the classical Fourier transform, graph Fourier transform provides a way to represent a signal in two different ...
List of Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set ... These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which ... Hartley transform Short-time Fourier transform (or short-term Fourier transform) (STFT) Rectangular mask short-time Fourier ... Integral transform Wavelet transform Fourier-transform spectroscopy Harmonic analysis List of transforms List of mathematic ...
Generalized Fourier series
In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of ... So we can form a generalized Fourier series (known as a Fourier-Legendre series) involving the Legendre polynomials, and f ( x ... Banach space Eigenfunctions Fractional Fourier transform Function space Hilbert space Least-squares spectral analysis ... The generalized Fourier series of a square-integrable function f : [ a , b ] → F {\displaystyle f:[a,b]\to \mathbb {F} } , with ...
Fourier transform on finite groups
... for the purpose of the Fourier transform on finite groups. Fourier transform Least-squares spectral analysis Representation ... Terras, Audrey (1999), Fourier Analysis on Finite Groups and Applications, Cambridge University Press, p. 251, ISBN 978-0-521- ... Fourier analysis, Finite groups, Pages that use a deprecated format of the math tags). ... This generalization of the discrete Fourier transform is used in numerical analysis. A circulant matrix is a matrix where every ...
Discrete-time Fourier transform
... also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is ... Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). ... which is by far the most common method of modern Fourier analysis. Both transforms are invertible. The inverse DTFT is the ... the summations over n are a Fourier series, with coefficients x[n]. The standard formulas for the Fourier coefficients are also ...
Planar Fourier capture array
Wikipedia articles with possible conflicts of interest from July 2013, Fourier analysis, Image sensors). ... A planar Fourier capture array (PFCA) is a tiny camera that requires no mirror, lens, focal length, or moving parts. It is ... By making them all unique, each sensor of the PFCA relates a distinct component of the 2D Fourier transform of the far-away ... PFCAs do not perform an exact Fourier transform since outputs are real-valued and are not perfect sinusoidal transforms of the ...
Joseph Fourier
Fourier analysis Fourier-Deligne transform Heat equation Least-squares spectral analysis List of things named after Joseph ... The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the ... A. Rosenbaum and E. L. Davis, Fourier's Theorem, . A. Rosenbaum and E. L. Davis Fourier 1824a. Fourier 1827a. Weart, S. (2008 ... Fourier made important contributions to dimensional analysis. The other physical contribution was Fourier's proposal of his ...
Fourier transform
Folland, Gerald (1992), Fourier analysis and its applications, Wadsworth & Brooks/Cole. Fourier, J.B. Joseph (1822), Théorie ... DFT matrix Fast Fourier transform Fourier integral operator Fourier inversion theorem Fourier multiplier Fourier series Fourier ... Fourier 1878, p. 408. (Jordan 1883) proves on pp. 216-226 the Fourier integral theorem before studying Fourier series. ... These can be generalizations of the Fourier transform, such as the short-time Fourier transform or fractional Fourier transform ...
Conjugate Fourier series
In the mathematical field of Fourier analysis, the conjugate Fourier series arises by realizing the Fourier series formally as ... Fourier analysis, Fourier series, All stub articles, Mathematical analysis stubs). ... Classical Fourier analysis, Graduate Texts in Mathematics, vol. 249 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/ ...
Discrete Fourier transform
Fourier analysis, Digital signal processing, Numerical analysis, Discrete transforms, Unitary operators). ... The Discrete Fourier Transform Discrete Fourier Transform Indexing and shifting of Discrete Fourier Transform Discrete Fourier ... The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. In digital ... matrix Fast Fourier transform FFTPACK FFTW Generalizations of Pauli matrices Least-squares spectral analysis List of Fourier- ...
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential ... Microlocal analysis Fourier transform Pseudodifferential operator Oscillatory integral operator Symplectic category Hörmander, ... A Fourier integral operator T {\displaystyle T} is given by: ( T f ) ( x ) = ∫ R n e 2 π i Φ ( x , ξ ) a ( x , ξ ) f ^ ( ξ ) d ... In fact, if we allow the sound speed c in the wave equation to vary with position we can still find a Fourier integral operator ...
Fourier analysis - Wikipedia
Fourier analysis can be done on cosets, even discrete cosets. Conjugate Fourier series Generalized Fourier series Fourier- ... Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a ... similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise ... and Fourier Analysis)". Sixty Symbols. Brady Haran for the University of Nottingham. Introduction to Fourier analysis of time ...
Mid-infrared supercontinuum-based Fourier transform spectroscopy for plasma analysis | Scientific Reports
Product analysis of CO2/CH4 plasmas. A bigger challenge for the spectroscopic analysis of a complex mixture of reaction ... Mid-infrared supercontinuum-based Fourier transform spectroscopy for plasma analysis. *R. Krebbers1, ... Data analysis. For real-time analysis of the detected interferograms, a LabVIEW-based data processing system is developed. A ... To demonstrate the possibilities of the system for plasma analysis and study, a quantitative analysis of the products formed in ...
Fourier Theory without Complex Analysis | School of Mathematics and Statistics
Many of the basic proofs of Fourier theory use complex analysis. ... One of the reasons why the Fourier transformation is a useful ... Many of the basic proofs of Fourier theory use complex analysis.. This is a beautiful theory, but unstable under perturbations ... One of the reasons why the Fourier transformation is a useful tool is that it converts differentiation (hard to understand) to ... In recent years, there has been interest in avoiding the use of complex analysis. ...
FFT (Fast Fourier Transform) Waveform Analysis
... is one of the most useful analysis tools available. Learn how it works in laymans terms in this application note. ... Other Fourier Analysis Software Issues. The needs of any Fourier analysis application are best served by a graphics based ... FFT (Fast Fourier Transform) Waveform Analysis. Order a DI-188 WinDaq starter kit for $49 and get WinDaq addon for FREE to ... Obviously, a Fourier analysis software package that offers a choice of several windows is desirable to eliminate spectral ...
Fourier Transform Infrared Absorption Spectroscopy for Quantitative Analysis of Gas Mixtures for Homeland Security Applications...
Fourier Transform Infrared Absorption Spectroscopy for Quantitative Analysis of Gas Mixtures for Homeland Security Applications ... Fourier Transform Infrared Absorption Spectroscopy for Quantitative Analysis of Gas Mixtures for Homeland Security Applications ... www.nist.gov/publications/fourier-transform-infrared-absorption-spectroscopy-quantitative-analysis-gas-mixtures ... test concentration and associated uncertainties in the analyte test concentration to be validated by independent analysis, ...
Fourier Analysis
... Produces the Fourier analysis of a data set by computing the Discrete Fourier Transform (DFT) of an input ... For more information on Fourier analysis, refer to the corresponding Wikipedia article. ... Inverse: When checked, calculates the inverse Discrete Fourier Transform.. Polar: When checked, the results are in polar ... The source data for this example is the same of the FOURIER function page. ...
Surface profile determination by additive-subtractive phase-modulated ESPI with Fourier analysis | (1996) | Wang | ...
Surface profile determination by additive-subtractive phase-modulated ESPI with Fourier analysis Author(s): Lewis S. Wang; ... Laser Interferometry VIII: Techniques and Analysis. Malgorzata Kujawinska; Ryszard J. Pryputniewicz; Mitsuo Takeda, Editor(s) ... SPIE 2860, Laser Interferometry VIII: Techniques and Analysis, (17 July 1996); doi: 10.1117/12.276302. Show Author Affiliations ... Fourier transform approach is employed to extract the phase data. A simplified method for calculating the surface height ( ...
Newest 'fourier-analysis' Questions - Mathematics Stack Exchange
Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, ... Newest fourier-analysis questions feed Subscribe to RSS Newest fourier-analysis questions feed To subscribe to this RSS feed, ... Fourier Transform of $\sin(\sin(x))$ [closed] How can I calculate the Fourier Transformation or Fourier series of $\sin(\sin(x ... Theorem 9.11 in Real and Complex Analysis) In Walter Rudins Real and Complex Analysis, he gives a proof of the Fourier ...
PalaeoMath: Part 24 - The Centre Cannot Hold I: Z-R Fourier Analysis | The Palaeontological Association
... data analysis procedure(s).. Before we move on to demonstrating the Fourier analysis part of Z-R Fourier analysis, I want to ... Fourier analysis as a more generalized approach to the Fourier-based analysis of outlines than the traditional radial Fourier ... As with radial Fourier analysis, we can use the Fourier expansion to define a set of geometrically comparable and mutually ... The Centre Cannot Hold I: Z-R Fourier Analysis. Written by Norm MacLeod - The Natural History Museum, London, UK (email: n. ...
Graduate Thesis Or Dissertation | Fourier Analysis and Equidistribution on the p-adic Integers | ID: cz30q0563 |...
ap.analysis of pdes - Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi...
If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $,k,,N$, then $\,f\,_{L^\infty}\leq \frac CN ... Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\ ... This problem is suggested by the decay of generalized porous medium equation $\partial_t u= \Delta F\circ u$ is the Fourier ... In the Fourier variables we expect the time decay $,\widehat{u(t)},^2\leq c ,\widehat{u(0)},^2$. The latter is true if we have ...
soft question - Why does Fourier analysis of Boolean functions 'work'? - Theoretical Computer Science Stack Exchange
... then standard Fourier analysis is often no longer enough, and it can be helpful to move to higher-order Fourier analysis. $\ ... Over the years I have gotten used to seeing many TCS theorems proved using discrete Fourier analysis. The Walsh-Fourier ( ... the concept of symmetry appears to be paramount in Fourier analysis. the more "symmetric" the function, the more that Fourier ... By the way, Fourier analysis is just a special case of the representation theory of finite groups. This theory considers the ...
real analysis - Understanding Fourier Transform Theorem in Folland: Theorem 8.35 - Mathematics Stack Exchange
While reading Follands Real Analysis textbook, I came across the Fourier Analysis section and became troubled by Theorem 8.35 ... begingroup$ The Fourier transform is linear, so it makes perfect sense to take the Fourier transform of $f_1+f_2$ (which equals ... About the proof of this proposition about Schwartz space in Steins Fourier Analysis ... In particular, I have not seen any mention of the Fourier transform if $f \in L^p$ for $p , 2$ (could we use Schwartz functions ...
A fractional Fourier transform analysis of a bubble excited by an ultrasonic chirp - Strathprints
Barlow, Euan and Mulholland, Anthony J. (2011) A fractional Fourier transform analysis of a bubble excited by an ultrasonic ... A fractional Fourier transform analysis of a bubble excited by an ultrasonic chirp ... The fractional Fourier transform is proposed here as a model based, signal processing technique for determining the size of a ... The fractional Fourier transform is used to produce a more detailed description of each response, and two-dimensional cross ...
Reconstructability analysis with Fourier transforms
| Emerald Insight
Reconstructability analysis with Fourier transforms - Author: Martin Zwick ... Reconstructability analysis with Fourier transforms. Martin Zwick (Portland State University, Portland, Oregon, USA) ... Fourier methods used in two‐ and three‐dimensional image reconstruction can be used also in reconstructability analysis (RA). ... Fourier methods may thus enhance the power of RA for data analysis and data mining. ...
Analysis of Retinal Nerve Fiber Layer Data Obtained by Optical Coherence Tomograph Using Fourier Based Analysis | IOVS | ARVO...
Purpose: : (1) To evaluate the efficacy of Wavelet-Fourier Analysis (WFA) and Fast Fourier Analysis (FFA) (Essock et al. IOVS ... Analysis of Retinal Nerve Fiber Layer Data Obtained by Optical Coherence Tomograph Using Fourier Based Analysis ... Analysis of Retinal Nerve Fiber Layer Data Obtained by Optical Coherence Tomograph Using Fourier Based Analysis ... Analysis of Retinal Nerve Fiber Layer Data Obtained by Optical Coherence Tomograph Using Fourier Based Analysis . Invest. ...
![2008.10871] Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{ö}dinger operators](data:image/png;base64,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)
2008.10871] Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{ö}dinger operators
Title:Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{ö}dinger operators. Authors: ... Download a PDF of the paper titled Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{\o} ... Download a PDF of the paper titled Analysis of the Feshbach-Schur method for the Fourier Spectral discretizations of Schr{\o} ... Abstract:In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schr ...
Mathematical Analysis of Computer Generated Binary Fourier Transform Holograms - MADOC
Mathematical Analysis of Computer Generated Binary Fourier Transform Holograms. Zhang, Eryi ; Hesser, Jürgen ; Dietrich, ... Fourier-Hologramm , Fourier-Transformation Abstract: A rigorous mathematical treatment of a general binarization process for ... computer generated binary Fourier transform hologram is developed. Further a generalized error diffusion coefficient matrix is ...
MATLAB and Simulink Courseware - MATLAB & Simulink
Fourier Analysis - Aurora Vision
... software and libraries that are easy-to-use and combine reliability with high performance of image processing and analysis. ... You are here: Start » Function Reference » Computer Vision » Fourier Analysis. Fourier Analysis. Select a function from the ... Transforms an image in frequency domain back to spatial domain using inverse Fourier transformation. ...
Did Joseph Fourier discover the greenhouse effect? - Rwmansiononpeachtree.com
What is the main purpose of Fourier analysis?. Fourier analysis is a type of mathematical analysis that attempts to identify ... The Fourier transform and Fouriers law of conduction are also named in his honour. Fourier is also generally credited with the ... What was the crazy idea proposed by Fourier?. Fouriers idea was to model a complicated heat source as a superposition (or ... What did Jean Baptistier Fourier discover?. Discovery of the greenhouse effect Fourier was the first person to study the ...
Real-time implementation of discrete fourier transform phase analysis and fault tolerant control for PMSM in electric vehicles ...
... for elecrical vehicles by using discrete Fourier transform phase (DFT) method. Findings - A novel method is proposed with ... Real-time implementation of discrete fourier transform phase analysis and fault tolerant control for PMSM in electric vehicles ... Real-time implementation of discrete fourier transform phase analysis and fault tolerant control for PMSM in electric vehicles ... for elecrical vehicles by using discrete Fourier transform phase (DFT) method. Findings - A novel method is proposed with ...
Fourier Analysis - Reuben Cohn-Gordon
Extending Fourier analysis to linear functionals. Theres a complex story re. the analysis part of Fourier analysis, namely ... Fourier Analysis. 26 January, 2020. $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\C}{\mathbb{C}}$ $\newcommand{\N}{\mathbb{N ... Less abstract version: the general idea of Fourier analysis is to express (sufficiently nice) functions as sums (or integrals) ... Since particles in quantum mechanics are vectors in a function space, all the machinery of Fourier analysis is perfectly ...
Analysis of structured texture using the multiresolution Fourier transform - WRAP: Warwick Research Archive Portal
1992) Analysis of structured texture using the multiresolution Fourier transform. University of Warwick. Department of Computer ... A multiresolution approach to the analysis of structural texture is presented. The multiresolution Fourier transform (MFT) is ... Analysis and synthesis of both artificial and natural images demonstrate the capability of the algorithm. ...
Fourier Analysis Questions and Answers - Matchmaticians
MATH 382 Fourier Analysis 2023-24 - Catalog
Fourier analysis is the study of waves and frequencies. More precisely, the goal of Fourier analysis is to decompose a ... MATH 382 - T1 (S) TUT Fourier Analysis MATH 382 - T1 (S) TUT Fourier Analysis ... The goal of this course is to cover the basic theory (fourier series, the fourier transform, the fast fourier transform) and ... Williams » Catalog » Courses and Programs 2023-24 » Mathematics » Mathematics Spring 2023-24 Class List » MATH 382 Fourier ...
