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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

SPRUCE S1 Bog Production of Newly-Grown Fine Roots Assessed Using Root Ingrowth Cores in 2013

This dataset contains data from root ingrowth cores from the Spruce and Peatland Responses Under Changing Environments (SPRUCE) experimental site within the Marcell Experimental Forest in northern Minnesota, USA. Root ingrowth cores were used to capture newly-grown fine roots during the 2013 growing season (June - October) at 20 locations at the southern end of the S1-Bog. Rates of fine root production were measured with root ingrowth cores and fine root traits were measured on roots that grew into ingrowth cores. Traits are reported by plant species and root orders and include nitrogen (N) content, root tissue density, specific root length, average diameter. Ingrowth cores were used to compare species-specific depth distribution patterns with minirhizotron analyses and to obtain newly produced roots for chemical analysis and measurement of morphological traits prior to the construction of the SPRUCE experimental plots. This dataset contains two data files in comma-separate values (*.csv) format. Additional metadata are provided: two data dictionaries and a file-level metadata file in comma-separate values (.csv) format and a user guide in PDF (*.pdf) format.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1999

Fast Quantum Algorithm for Predicting Descriptive Statistics of Stochastic Processes

Stochastic processes are used as a modeling tool in several sub-fields of physics, biology, and finance. Analytic understanding of the long term behavior of such processes is only tractable for very simple types of stochastic processes such as Markovian processes. However, in real world applications more complex stochastic processes often arise. In physics, the complicating factor might be nonlinearities; in biology it might be memory effects; and in finance is might be the non-random intentional behavior of participants in a market. In the absence of analytic insight, one is forced to understand these more complex stochastic processes via numerical simulation techniques. In this paper we present a quantum algorithm for performing such simulations. In particular, we show how a quantum algorithm can predict arbitrary descriptive statistics (moments) of N-step stochastic processes in just O(square root of N) time. That is, the quantum complexity is the square root of the classical complexity for performing such simulations. This is a significant speedup in comparison to the current state of the art.

Williams Colin P.↗
NASA NTRS1978

An experiment to verify that the weak interactions satisfy the strong equivalence principle

The construction of a clock based on the beta decay process is proposed to test for any violations by the weak interaction of the strong equivalence principle bu determining whether the weak interaction coupling constant beta is spatially constant or whether it is a function of gravitational potential (U). The clock can be constructed by simply counting the beta disintegrations of some suitable source. The total number of counts are to be taken a measure of elapsed time. The accuracy of the clock is limited by the statistical fluctuations in the number of counts, N, which is equal to the square root of N. Increasing N gives a corresponding increase in accuracy. A source based on the electron capture process can be used so as to avoid low energy electron discrimination problems. Solid state and gaseous detectors are being considered. While the accuracy of this type of beta decay clock is much less than clocks based on the electromagnetic interaction, there is a corresponding lack of knowledge of the behavior of beta as a function of gravitational potential. No predictions from nonmetric theories as to variations in beta are available as yet, but they may occur at the U/sg C level.

Eby, P. B.↗
NASA NTRS1995

Multiparameter linear least-squares fitting to Poisson data one count at a time

A standard problem in gamma-ray astronomy data analysis is the decomposition of a set of observed counts, described by Poisson statistics, according to a given multicomponent linear model, with underlying physical count rates or fluxes which are to be estimated from the data. Despite its conceptual simplicity, the linear least-squares (LLSQ) method for solving this problem has generally been limited to situations in which the number n(sub i) of counts in each bin i is not too small, conventionally more than 5-30. It seems to be widely believed that the failure of the LLSQ method for small counts is due to the failure of the Poisson distribution to be even approximately normal for small numbers. The cause is more accurately the strong anticorrelation between the data and the wieghts w(sub i) in the weighted LLSQ method when square root of n(sub i) instead of square root of bar-n(sub i) is used to approximate the uncertainties, sigma(sub i), in the data, where bar-n(sub i) = E(n(sub i)), the expected value of N(sub i). We show in an appendix that, avoiding this approximation, the correct equations for the Poisson LLSQ (PLLSQ) problems are actually identical to those for the maximum likelihood estimate using the exact Poisson distribution. We apply the method to solve a problem in high-resolution gamma-ray spectroscopy for the JPL High-Resolution Gamma-Ray Spectrometer flown on HEAO 3. Systematic error in subtracting the strong, highly variable background encountered in the low-energy gamma-ray region can be significantly reduced by closely pairing source and background data in short segments. Significant results can be built up by weighted averaging of the net fluxes obtained from the subtraction of many individual source/background pairs. Extension of the approach to complex situations, with multiple cosmic sources and realistic background parameterizations, requires a means of efficiently fitting to data from single scans in the narrow (approximately = 1.2 keV, HEAO 3) energy channels of a Ge spectrometer, where the expected number of counts obtained per scan may be very low. Such an analysis system is discussed and compared to the method previously used.

Wheaton, Wm. A.↗
NASA NTRS1999

Dynamics of Single Chains of Suspended Ferrofluid Particles

We present an experimental study of the dynamics of isolated chains made of super-paramagnetic particles under the influence of a magnetic field. The motivation of this work is to understand if the chain fluctuations exist and, if it does, how does the fluctuation affect chain aggregation. We find that single chains strongly fluctuate and that the characteristic frequency of their fluctuations is inversely proportional to the magnetic field strength. The higher the field the lower the characteristic frequency of the chain fluctuations. In the high magnetic field limit, chains behave like rigid rods without any internal motions. In this work, we used ferrofluid particles suspended in water. These particles do not have any intrinsic magnetization. Once a magnetic field is applied, a dipole moment is induced in each particle, proportional to the magnetic field. A dipolar magnetic interaction then occurs between particles. If dipole-dipole magnetic energy is higher than the thermal energy, the result is a structure change inside the dipolar fluid. The ratio of these two energies is expressed by a coupling constant lambda as: lambda = (pi(a(exp 3))(chi(exp 2))(mu(sub 0))(H(sub 0))(exp 2))/18kT Where a is the particle radius, mu(sub 0) is the vacuum magnetic permeability, H(sub 0) the applied magnetic field, k the Boltzmann constant and T the absolute temperature. If lambda > 1, magnetic particles form chains along the field direction. The lateral coalescence of several chains may form bigger aggregates especially if the particle volume fraction is high. While many studies and applications deal with the rheological properties and the structural changes of these dipolar fluids, this work focuses on the understanding of the chain dynamics. In order to probe the chain dynamics, we used dynamic light scattering (DLS) in self-beating mode as our experimental technique. The experimental geometry is such that the scattering plane is perpendicular to the magnetic field. Therefore, only motions in this plane are probed. A very dilute sample of a ferrofluid emulsion with a particle volume fraction of 10(exp -5) is used in this experiment. We chose such a low volume fraction to avoid multiple light scattering as well as lateral chain-chain aggregation. DLS measures the dynamic structure factor S(q,t) of the sample (q is the scattering wave vector, t is the time). In the absence of the magnetic field, identical particles of ferrofluid droplets are randomly distributed and S(q,t) reduces to exp(-q(exp 2)2D(sub 0)t). D(sub 0)=(kT/(6(pi)(eta)(a)) is the diffusion coefficient of Brownian particles (where Xi = (6(pi)(eta)(a)) is the Stokes frictional coefficient of a spherical particle in a fluid of viscosity eta). If interactions or polydispersity can not be ignored, an effective diffusion coefficient is introduced. Formally, D(sub eff) is defined as: D(sub eff) = - q(exp -2) partial derivative of (ln(S(q,t)) with respect to time, as t goes to 0. D(sub eff) reduces to D(sub 0) if no interactions and only a few particles size are present. Therefore, we can use DLS to measure particle size. The particle radius was found to be a=0.23 mu m with 7% of polydispersity. In this case, if we vary the scattering angle theta (and so q) we do not have any change in the measured diffusion coefficient: it is q-independent. When a magnetic field is applied, particles aggregate into chains if lambda > 1. We first studied the kinetics of the chain formation when lambda = 406. At a fixed scattering angle, we measured diffusion coefficient D(sub eff) as a function of time. Experimentally, we find that D(sub eff) decreases monotonously with time. Physically, this means that chains are becoming longer and longer. Since we are only sensitive to motions in the scattering plane and since chains have their main axis perpendicular to this plane, the measured diffusion coefficient is the trans-verse diffusion coefficient. We can relate D(sub eff) to the mean number of particles per chain N(t) at a given time and to the diffusion coefficient of an isolated particle D(sub 0) as D(sub eff)=f(N(t))D(sub 0). Since f(N) is known from other recent work, N can be expressed as a function of the time. We found a square root dependency: N(t) proportional to the square root of t. As expected for very low volume fraction, this behavior is characteristic of a diffusion-limited aggregation as suggested by several authors and by our previous work. In this study, we focus on the dependence of the effective diffusion coefficient on the scattering angle and the magnetic field strength. After the magnetic field is applied (lambda = 406) for a long time, typically 6 hours, kinetics of chain formation becomes very slow. Chain size does not vary much over the next hour period. Thus, we can perform different interesting experiments. First, at a fixed magnetic field, we measure the effective diffusion coefficient as a function of the scattering angle (from 5 to 130 deg). Our results show that the measured diffusion coefficient increases linearly with the scattering angle: D(sub eff) proportional to q. If we do the same experiment for different lambda values, D(sub eff) depends on lambda as D(sub eff) proportional to lambda(exp -1/2). We also find for different lambda values that the same asymptotic D(sub eff) value is obtained when q approaches zero. The angle dependency of D(sub eff) suggests that an additional motion exists besides chain drifting. Chain size is constant during experiment, which was verified by measuring the same diffusion coefficient at the beginning and at the end of the angle switching. If chains are rigid, D(sub eff) is independent of q. Therefore, we found that D(sub eff) not only measures the motion of the entire chain but also its internal fluctuations. These internal motions are the fluctuations of the particles in the chain. To understand the q dependency of D(sub eff), let us look at the probing length used. In our study, the characteristic length scale probed is l=2pi/q which is in the range of 0.9<l/a<20. When l is much larger than the particle radius (l/a going to infinity) we are mainly sensitive to the center of mass diffusion of the chain. D(sub eff) is then the diffusion of the entire chain and depends only on N but not on q and lambda. The value D(sub eff) allows us to obtain the number of particles per chain N. In the opposite limit (i.e. l/a < 1), we are sensitive to motions on the size of individual particles. The main contributions to the measured diffusion coefficient come from internal motions of the chains (i.e. particles' fluctuations inside the chain). We investigated also the effect of the chain size on D(sub eff). For the same value of the magnetic interaction lambda and different chain sizes, we found that the slope of D(sub eff) versus q decreases when N increases as a power law D(sub eff) proportional to N(exp -0.7). This is an important behavior because this means that the longer the chain the slower its internal fluctuations besides the whole chain motion.

Cutillas, S.↗
DOE OSTI2022

Spatial mapping of plant N-glycosylation cellular heterogeneity inside soybean root nodules provided insights into legume-rhizobia symbiosis

Although ubiquitously present, information on the function of complex N-glycan posttranslational modification in plants is very limited and it is often neglected. In this work, we adopted an enzyme-assisted matrix-assisted laser desorption/ionization mass spectrometry imaging strategy to visualize the distribution and identity of N-glycans in soybean root nodules at a cellular resolution. We additionally performed proteomics analysis to probe the potential correlation to proteome changes during symbiotic rhizobia-legume interactions. Our ion images reveal that intense N-glycosylation occurs in the sclerenchyma layer, and inside the infected cells within the infection zone, while morphological structures such as the cortex, uninfected cells, and cells that form the attachment with the root are less N-glycosylated. Notably, we observed different N-glycan profiles between soybean root nodules infected with wild type rhizobia and those infected with mutant rhizobia incapable of efficiently fixing atmospheric nitrogen. The majority of complex N-glycan structures, particularly those with characteristic Lewis-a epitopes, are overexpressed in the mutant nodules. Our proteomic results revealed that these glycans likely originated from proteins that maintain the redox balance crucial for proper nitrogen fixation, but also from enzymes involved in N-glycan and phenylpropanoid biosynthesis. These findings indicate possible involvement of Lewis-a glycans in these critical pathways during legume-rhizobia symbiosis.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS1995

Ulysses-ARTEMIS radio observation of energetic flare electron

Type 3 radio bursts allow us to follow energetic electrons ejected by solar flares into the interplanetary medium, even when the observer is far away from the electrons. The emission frequency f(sub p) is related to the ambient density n(sub e) by f(sub p) varies as the square root of n(sub e), and as a function of the distance r to the sun we have approximately n(sub e) varies as r(exp -2); as a consequence, on a 1/f - t dynamic spectrum type 3 bursts appear as nearly straight traces, whose slope gives an estimation of the source speed. We used the data of the URAP radio receiver on Ulysses (1-1000 kHz), observing sources in the solar wind, and the ground data of the ARTEMIS spectrograph (100-500 MHz), observing sources of the corona, over the years 1991-1994. We found a surprisingly large number of excellent high-frequency - low-frequency associations. A type 3 burst group on ARTEMIS (10 to 100 bursts over 1 to 10 minutes) typically gives rise to one isolated burst on Ulysses. As bursts often start in high frequencies during the maximum phase of flares, this demonstrates in a very convincing manner that some of the flare electrons themselves make it all the way to the interplanetary medium. We discuss decorrelation cases in the context of geometrical configuration between the active region and the two observing sites. We also study how apparent electron speeds vary with the distance to the sun.

Hoang, S.↗
NASA NTRS2006

Microthruster Propulsion for the Space Technology 7 (ST7) Technology Demonstration Mission

For future applications to precision formation flying missions, NASA's New Millennium Program is scheduled to test colloid micro-Newton thrusters (CMNTs) on the ST7 technology demonstration mission. These CMNTs are part of a disturbance reduction system (DRS) on the ESA SMART-2 Spacecraft or LISA Pathfinder. The goal of the ST7 DRS is to demonstrate technologies necessary to meet the nanometer precision positioning control requirements of the LISA mission. In order to achieve these goals, the CMNTs are required to demonstrate a thrust resolution of less than 0.1 micro-N and a thrust noise of less than 0.1 micro-N/[square root]Hz for thrust levels between 5 and 30 micro-N. Developed by Busek Co. with support from JPL in testing an design, the CMNT has been developed over the last four years into a flight-ready microthrust system. The development, validation testing, and flight unit production of the CMNTs are described. Development tests and analysis include preliminary wear tests, propellant loading process verification, flow testing, and performance verification. Validation and flight unit verification includes thermal and structural analysis, life testing, thermal and dynamic load testing, and performance verification. Final delivery of the units is planned in 2007 with and planned launch and flight demonstration 2009.

microthruster↗
DOE OSTI2025

Phytoremediation potential of Nerium oleander and Salix alba for heavy metal removal in rock-amended soils: a natural and cost-effective approach

Enhanced weathering (EW) through the application of ground rock is a competitive carbon removal strategy. Adoption of this technology at a meaningful scale requires a systematic assessment of its long-term feasibility, especially with regard to soil quality from the application of rock amendments that contain varying levels of heavy metal (loid)s (HM) such as Cu, Ni, Cr, Co, and Pb. The potential accumulation of these metal (loid)s could be an unintended consequence of repeated large-scale EW applications, necessitating careful evaluation for use in croplands. This study explores the idea of using phytoremediation as a natural, low-cost means of remediating rock-amended soils. Specifically, we examined the ability of Nerium oleander and Salix alba species to remove HM from rock-amended soils in their tissues (i.e., leaves, stems, and roots). In this study, the relative abundance of HM accumulation in hyperaccumulator plants followed the order: Si > Rb > Cu > Sn > Cr > Cd > Pb > Ni > Mo > Co > As > Sb > Se > Cs. Our results indicate that HM accumulation in soils treated with rock were significantly below permissible limits set by the Environmental Protection Agency (EPA). Further, in reasonable amounts, some of these HM serve as essential micronutrients required by plants. In fact, we found increased growth and higher biomass for both plants under rock application than without. We further found a significant uptake of most HM in rock-amended soils planted with S. alba. Comparatively, uptake of certain HM like Ni, Mo, Cs, Pb and Cu was relatively higher in the roots of N. oleander. In contrast to N. oleander, S. alba accumulated higher levels of HM in its stems and leaves than in its roots. Interestingly, 31–35% weathering of metabasalt (applied rock) was observed across plant types over a 5-month period. Overall, we conclude that S. alba has a greater potential for phytoremediation in rock-amended soils, although both plants may be useful in remediating soils with varying levels and types of contamination.

Ibrahim, Naira [Jackson State Univ., Jackson, MS (↗
DOE OSTI2020

Nitrogen fixation and resorption efficiency differences among twelve upland and lowland switchgrass cultivars

In nitrogen (N)-limited terrestrial ecosystems, plants employ various strategies to acquire and conserve N, including translocation of N in perennial tissues and stimulation of N fixation in roots and soils. Switchgrass (Panicum virgatum L.) is a genotypically and phenotypically diverse perennial grass with two distinct ecotypes (lowland and upland) and numerous genotypes. It grows well in low-N soils, likely because of its ability to translocate N and to associate with N-fixing microbes, but little is known about variation in these traits among cultivars or even ecotypes. We measured N translocation, N fixation potential in roots and soils, soil net N mineralization, soil net nitrification, and biomass yields in 12 switchgrass cultivars grown in a replicated block experiment in southwestern Michigan, USA. Lowland cultivars had higher yields, rates of N translocation, soil net N mineralization, and N fixation potentials on washed, non-sterile roots, while upland cultivars exhibited higher N fixation potentials in root-free soil. N resorption efficiencies averaged 53±5 [SE] % for lowland versus 29±3% for upland cultivars. Additionally, there were significant among-cultivar differences for all response variables except mineralization and nitrification, with differences likely explained by cultivar-specific physiologies and microbial communities. Here, the ideal cultivar for biofuels is one that can maintain high yields with minimal fertilizer addition and there appear to be several cultivars that meet these criteria. In addition, results suggest substantial N cycle differences among cultivars that might be exploited by breeders to create new or improved high-yielding, N-conserving switchgrass lines.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2014

Superconducting Quantum Arrays for Wideband Antennas and Low Noise Amplifiers

Superconducting Quantum Iinetference Filters (SQIF) consist of a two-dimensional array of niobium Josephson Junctions formed into N loops of incommensurate area. This structure forms a magnetic field (B) to voltage transducer with an impulse like response at B0. In principle, the signal-to-noise ratio scales as the square root of N and the noise can be made arbitrarily small (i.e. The SQIF chips are expected to exhibit quantum limited noise performance). A gain of about 20 dB was recently demonstrated at 10 GHz.

SQUID Detectors↗
NASA NTRS1980

Incipient Failure Detection (IFD) of antifriction ball bearings

Two methods were investigated for retrieving the ball bearing signals buried by the overall signature of an operating large engine such as the main space shuttle engine. The signal averaging method averages n** signals over a time duration equaling the known period of the ball bearing defect frequency. With this method, the desired ball bearing defect signal will be enhanced over the noise by a factor of the square root of n. A second method is to perform a further frequency analysis on the logarithmic spectrum in order to detect and separate different families of harmonics and sidebands, thereby giving information about modulating frequencies. These two alternate IFD methods were carried out with a Norland 3001 waveform and data system. The obtained results are then compared with the power spectral density (PSD) of the overall machine signals and conclusions are made accordingly.

Wang, D. Y.↗
NASA NTRS1999

Parameter Estimation in Astronomy with Poisson-Distributed Data: The (CHI)2(gamma) Statistic - 1

Applying the standard weighted mean formula, [Sigma (sub i)n(sub i)ssigma(sub i, sup -2)], to determine the weighted mean of data, n(sub i), drawn from a Poisson distribution, will, on average, underestimate the true mean by approx. 1 for all true mean values larger than approx.3 when the common assumption is made that the error of the i th observation is sigma(sub i) = max square root of n(sub i), 1).This small, but statistically significant offset, explains the long-known observation that chi-square minimization techniques which use the modified Neyman'chi(sub 2) statistic, chi(sup 2, sub N) equivalent Sigma(sub i)((n(sub i) - y(sub i)(exp 2)) / max(n(sub i), 1), to compare Poisson - distributed data with model values, y(sub i), will typically predict a total number of counts that underestimates the true total by about 1 count per bin. Based on my finding that weighted mean of data drawn from a Poisson distribution can be determined using the formula [Sigma(sub i)[n(sub i) + min(n(sub i), 1)](n(sub i) + 1)(exp -1)] / [Sigma(sub i)(n(sub i) + 1)(exp -1))], I propose that a new chi(sub 2) statistic, chi(sup 2, sub gamma) equivalent, should always be used to analyze Poisson- distributed data in preference to the modified Neyman's chi(exp 2) statistic. I demonstrated the power and usefulness of,chi(sub gamma, sup 2) minimization by using two statistical fitting techniques and five chi(exp 2) statistics to analyze simulated X-ray power - low 15 - channel spectra with large and small counts per bin. I show that chi(sub gamma, sup 2) minimization with the Levenberg - Marquardt or Powell's method can produce excellent results (mean slope errors approx. less than 3%) with spectra having as few as 25 total counts.

Mighell, Kenneth J.↗
DOE OSTI2023

Sun's Series via Cyclotomic Multiple Zeta Values

We prove and generalize several recent conjectures of Z.-W. Sun surrounding binomial coefficients and harmonic numbers. We show that Sun's series and their analogs can be represented as cyclotomic multiple zeta values of levels N ϵ {4,8,12,16,24} , namely Goncharov's multiple polylogarithms evaluated at N -th roots of unity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

JWST Noise Floor. I. Random Error Sources in JWST NIRCam Time Series

James Webb Space Telescope (JWST) transmission and emission spectra will provide invaluable glimpses of transiting exoplanet atmospheres, including possible biosignatures. This promising science from JWST, however, will require exquisite precision and understanding of systematic errors that can impact the time series of planets crossing in front of and behind their host stars. Here, we provide estimates of the random noise sources affecting JWST Near-Infrared Camera (NIRCam) time-series data on the integration-to-integration level. We find that 1/f noise can limit the precision of grism time series for two groups (230–1000 ppm depending on the extraction method and extraction parameters), but will average down like the square root of N frames/reads. The current NIRCam grism time-series mode is especially affected by 1/f noise because its GRISMR dispersion direction is parallel to the detector fast-read direction, but could be alleviated in the GRISMC direction. Care should be taken to include as many frames as possible per visit to reduce this 1/f noise source: thus, we recommend the smallest detector subarray sizes one can tolerate, four output channels, and readout modes that minimize the number of skipped frames (RAPID or BRIGHT2). We also describe a covariance-weighting scheme that can significantly lower the contributions from 1/f noise as compared to sum extraction. We evaluate the noise introduced by pre-amplifier offsets, random telegraph noise, and high dark current resistor capacitor (RC) pixels and find that these are correctable below 10 ppm once background subtraction and pixel masking are performed. We explore systematic error sources in a companion paper.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1973

Modulation of the Jovian decametric radio emission by IO

It is argued that the sweeping of trapped energetic MeV protons by Io can result in induced emissions. The instability is caused by the density gradient occurring at the edge of the cavity. It is shown that the azimuthal Larmor drift of the energetic protons which is transverse to both the local magnetic field and the density gradient can excite extraordinary waves with frequencies slightly above the local electron cyclotron frequency. It is found that the emission coefficient is proportional to the square root of (n sub p/n sub t), where n sub p and n sub t denote the density of the energetic protons and the density of the thermal electrons, respectively. The theory also predicts a lower frequency cutoff. This cutoff frequency depends upon the density of thermal electrons in the source region.

Wu, C. S.↗
NASA NTRS1976

Partial correlation properties of pseudonoise /PN/ codes in noncoherent synchronization/detection schemes

This concise paper considers the effect on the autocorrelation function of a pseudonoise (PN) code when the acquisition scheme only integrates coherently over part of the code and then noncoherently combines these results. The peak-to-null ratio of the effective PN autocorrelation function is shown to degrade to the square root of n, where n is the number of PN symbols over which coherent integration takes place.

Cartier, D. E.↗
NASA NTRS1976

Fourier and Hadamard transform spectrometers - A limited comparison

An encoding figure of merit is established for a detector-noise limited Fourier transform spectrometer (FTS) and compared to the comparable figure for a Hadamard transform spectrometer (HTS). The limitation of the Fourier system is partly that it does not truly Fourier analyze the radiation. Instead a cosine squared modulation is imposed on the different spectral frequencies. An additional difficulty is that neither the cosine nor the cosine squared functions form an orthonormal set. This makes the Fellgett's advantage (root-mean-squared figure of merit) for a single detector Michelson interferometer a factor of the square root of (N/8) greater than for a conventional grating instrument - rather than the square root of (N/2). The theoretical limit would be the square root of N.

Tai, M. H.↗