Apodization Function Research Articles

Reduced spatial side lobes in chemical-shift imaging.

Density-weighted k-space sampling with spiral trajectories is used to reduce spatial side lobes in chemical-shift imaging (CSI). In this method, more time is spent collecting data at the center of k space and less time at the edges of k space in order to make the sampling density proportional to a given apodization function, subject to constraints imposed by gradient performance and Nyquist sampling. The efficient k-space coverage of spiral-based trajectories enables good control over the sampling density within practical in vivo scan times. The density-weighted acquisition is compared to a conventional, nonweighted spiral sampling without the application of a window function. For a fixed voxel size and imaging time, the noise variance is observed to be the same for both cases, while spatial side lobes are greatly reduced with the variable-density sampling. This method is demonstrated on a normal volunteer by imaging of brain metabolites at 1.5 T with both single slice CSI and volumetric CSI. Magn Reson Med 42:314-323, 1999.

Fiber grating filter for WDM systems: an improved design

We report on a novel technique for the design of fiber gratings whose reflection characteristic closely approximates the ideal rectangular linear phase filter. The technique is based on the design of a suitable grating period variation, following the apodization function, in order to ensure the Bragg condition to remain constant along the grating length.

A new calculation procedure for spatial impulse responses in ultrasound

A new procedure for the calculation of spatial impulse responses for linear sound fields is introduced. This calculation procedure uses the well known technique of calculating the spatial impulse response from the intersection of a circle emanating from the projected spherical wave with the boundary of the emitting aperture. This general result holds for all aperture boundaries for a flat transducer surface, and is used in the procedure to yield the response for all types of flat transducers. An arbitrary apodization function over the aperture can be incorporated through a simple one-dimensional integration. The case of a soft baffle mounting of the aperture is also included. Specific solutions for transducer boundaries made from lines are given, so that any polygon transducer can be handled. Specific solutions for circles are also given. Finally, a solution for a general boundary is stated, and all these boundary elements can be combined to, e.g., handle annular arrays or semi-circle transducers. Results from an implementation of the approach are given and compared to previously developed solutions for a simple aperture, a complex aperture, and a Gaussian apodized circular transducer.

Phase correction for collision model analysis and enhanced resolving power of fourier transform ion cyclotron resonance mass spectra.

Phase correction of FT-ICR data yields an absorption spectrum that offers a gain by up to a factor of 2 in mass resolving power (at half-maximum peak height), compared to conventional magnitude-mode display. That improvement is equivalent to doubling the applied magnetic field strength, without loss in signal-to-noise (S/N) ratio, provided that the time-domain data are padded with an equal number of zeroes before FFT. Our simple, visual, user-interactive algorithm quickly corrects for zero-order and first-order variation of phase with frequency. We find that the theoretical mass resolving power enhancement for pressure-limited absorption-mode over magnitude-mode line shape depends on the collision mechanism: factor of 1.40 for hard sphere vs 3(1/2) for Langevin (ion: induced dipole). Thus, the experimental enhancement in mass resolving power (factor of 1.43 +/- 0.09) for isotopically resolved peaks in the FT-ICR mass spectra of electrosprayed bovine carbonic anhydrase (approximately 29 kDa) directly supports the hard-sphere collision model. Optimal implementation of phasing requires the following: (a) a delay between excitation and detection of less than half of one sampling interval to avoid baseline "roll" and Gibb's oscillations; (b) accurate analog-to-digital conversion; (c) a sufficiently long acquisition period to yield several data points per absorption-mode peak width at half-maximum peak height; and (d) avoidance of FT-ICR apodization functions (e.g., Hamming and Hanning) that suppress the initial time-domain data. Pulsed single-frequency excitation (duration much less than the reciprocal of the Nyquist bandwidth) can eliminate higher than first-order variation of phase with frequency. Phased FT-ICR spectra should prove especially desirable for analysis of complex mixtures, for resolving isotopic distributions in electrosprayed multiply charged macromolecules and for characterizing ion collisions (and thus ion size and shape).

Extending the Range of Beer's Law in FT-IR Spectrometry. Part I: Theoretical Study of Norton—Beer Apodization Functions

Beer's law is obeyed when the measured value of absorbance varies linearly with concentration in the absence of chemical effects. The absorbance range over which bands measured on any type of spectrometer obey Beer's law depends on the ratio of the resolution to the full width at half-height of the analytical band, often called the resolution parameter, rho, and the instrument line shape (ILS) function of the spectrometer. Apodization functions are used widely in Fourier transform infrared (FT-IR) spectrometry to change the instrument line shape function. For Lorentzian bands, it is well known that when rho is about 1 and the spectrum is computed with a triangular apodization function, there is a negative deviation from Beer's law when the true peak absorbance is high, whereas the use of boxcar truncation results in a positive deviation. In this paper, the possibility of extending the linear range of Beer's law by applying optimized apodization functions of the type proposed by Norton and Beer has been investigated. This theoretical study shows that the linear range of Beer's law plots for spectra measured at moderate resolution can be extended to peak absorbances greater than 2 with the use of any of the Norton-Beer apodization functions. Linear plots of the apparent absorbance of a band against analyte concentration can be obtained even for bands of absorbance substantially greater than 3, although a nonzero intercept on the absorbance axis may be observed.

Extending the Range of Beer's Law in FT-IR Spectrometry. Part II: Theoretical Study of Continuous Apodization Functions

Beer's law is obeyed when the measured value of absorbance varies linearly with concentration in the absence of chemical effects. The absorbance range over which bands measured on any type of spectrometer obey Beer's law depends on the ratio of the resolution to the full width at half-height of the analytical band, often called the resolution parameter, rho, and the instrument line shape (ILS) function of the spectrometer. Apodization functions are used widely in Fourier transform infrared (FT-IR) spectrometry to change the instrument line shape function. For Lorentzian bands, it is well known that when rho is about 1 and the spectrum is computed with a triangular apodization function, there is a negative deviation from Beer's law when the true peak absorbance is high, whereas the use of boxcar truncation results in a positive deviation. In this paper, the possibility of extending the linear range of Beer's law by applying optimized apodization functions of the type proposed by Norton and Beer has been investigated. This theoretical study shows that the linear range of Beer's law plots for spectra measured at moderate resolution can be extended to peak absorbances greater than 2 with the use of any of the Norton-Beer apodization functions. Linear plots of the apparent absorbance of a band against analyte concentration can be obtained even for bands of absorbance substantially greater than 3, although a nonzero intercept on the absorbance axis may be observed.

Quantitation of ion abundances in fourier transform ion cyclotron resonance mass spectrometry

To improve the analytical usefulness of Fourier transform ion cyclotron resonance mass spectrometry (FTICR-MS), an extensive survey of various methods for quantitation of peak magnitudes has been undertaken using a series of simulated transient response signals with varying signal-to-noise ratio. Both peak height (five methods) and peak area (four methods) were explored for a range of conditions to determine the optimum methodology for quantitation. Variables included dataset size, apodization function, damping constant, and zero filling. Based on the results obtained, recommended procedures for optimal quantitation include: apodization using a function appropriate for the peak height ratios observed in the spectrum (i.e., Hanning for ratios of about 1:10, three-term Blackman-Harris for ratios of ∼1:100, or Kaiser-Bessel for ratios of ∼1:1000); zero filling until the peaks of interest are represented by 10–15 points (generally obtained with one order of zero filling); and use of the polynomial y = ( ax 2 + bx + c) n and the three data points of highest intensity of the peak to locate the peak maximum, Y max = (− b 2/4 a + c) n . In this peak fitting procedure, which we have termed the “Comisarow method,” n is 5.5, 9.5, and 12.5 for the Hanning, three-term Blackman-Harris, and Kaiser-Bessel apodization functions, respectively. Accuracy of quantitation using an optimal peak height determination is about equal to that for peak area measurements. These recommendations were found to be valid when tested with real FTICR-MS spectra of xenon isotopes.

New Window Function for Very Short Acquisition Times

An apodization function, the sinc-TRAF function (S-TRAF), has been developed for application to exponentially decaying data sets where the acquisition times ( AT) is less than or equal to one time constant, T2. For heavily truncated time-domain signals, S-TRAF is able to remove sinc ripple with only minor losses in linewidth (LW) and signal-to-noise ratio (S/N). Ripple removal achieved with the application of S-TRAF rivals that observed from use of the “ultimate ripple-free resolution enhancement” function (URFRE) for AT as low as 0.3 T2*. S-TRAF maintains LW and S/N better than URFRE for all acquisition times examined.

The effect of apodisation and finite resolution on Fourier transform infrared and Raman spectra

The interferograms measured on Fourier transform infrared (FTIR) and Raman (FT-Raman) spectrometers are routinely apodised before the Fourier transform is carried out. The effects of finite resolution and different apodisation functions on synthetic bands of known Lorentz-Gauss character have been investigated. (The apodisation functions chosen were those commonly available on commercial FTIR and FT-Raman spectrometers). The effects are dramatic, the ratio of resolution/(true bandwidth) required to obtain less than 10% distortion (as judged by the peak height and width) in the band varied from 1 for boxcar to less than 0.2 for Norton-Beer (strong). Increasing Gaussian character of the band results in initially (i.e. at high resolution) the band being less altered by resolution and apodisation than a pure Lorentzian band. As the resolution is degraded, the band is more strongly affected. These results are explained in terms of the effect of truncation and distortion of the Fourier transform of the bandshape. A comparison of calculated and experimental results gave good agreement for the 1216 cm −1 band of chloroform in deuterochloroform. The experimental results also showed an increasing degree of Gaussian character as the resolution became comparable with the full width at half height of the band.

An apodized spherical transducer: Analytical solution

There is a current interest in spherical transducers with apodization. We present an analytic expression for the field due to such a focused transducer when the apodization function is Gaussian.

Contrast response analysis for medical ultrasound imaging

Traditional measures of spatial resolution are helpful but incomplete for characterizing the imaging performance of medical ultrasound systems because the challenge is not so much resolving strong point reflectors as limiting the impact of stronger scattering regions on weaker regions. This paper discusses the use of an alternative measure which is based on the ability of a system to accurately image an anechoic region surrounded by a uniform scattering medium. This measure, called contrast response, includes effects of both mainlobe width and sidelobe level. Since the impulse response of a linear system to a target is the convolution of the system spatial impulse response (beam pattern) and the target scattering function, the system output can be described by a three dimensional integral. The contrast response is defined as the ratio of this system output when the anechoic target is not present to the output when it is present, evaluated over a range of target sizes. Contrast response analysis is useful for both system design and performance comparison. Examples are presented which compare the performance of systems with different apodization functions and aperture sizes. This analysis approach suggests why the strong window functions popular in signal processing have not been used for apodization in medical ultrasound and why 256 channel systems have not demonstrated a dramatic performance improvement over 128 channel systems.

Regularized Method of Spectral Curve Deconvolution

A regularized method of spectral curve deconvolution is proposed. This method is based on three fundamental principles: the regularized method of solving the convolution equation; the use, instead of the apodization function, of the digital low-pass filter, which permits exact knowledge of its characteristics; and the use of the Fourier transform modulus of the spectrum being treated for obtaining a priori information about the frequency characteristics of the solution and noise, required for determination of the optimum parameters of the regularizing operator. The regularized method of de-convolution permits the acquisition of an approximately stable solution for the deconvolution problem of spectral curves, which moves toward an exact solution with the decrease of the experimental spectrum error. Examples are given of the application of the regularized method of deconvolution to simulated and experimental IR spectra. A conclusion about the expediency of using the given method for resolution enhancement in complex spectra is made.

On the use of tapered linearly chirped gratings as dispersion-induced distortion equalizers in SCM systems

We report on the use of tapered linearly chirped fiber gratings to equalize dispersion-induced distortion in long distance/broad-band subcarrier multiplexed (SCM) systems. Different apodization functions for the coupling factor of the fiber grating have been considered and their equalizing capability of fiber-induced distortion in a subcarrier system have been compared. The results of the theoretical analysis show that both harmonic and intermodulation distortions are significantly reduced when the equalizer parameters have been properly designed. Higher reduction of the nonlinear distortion is obtained when the length of the filter increases, even though a saturation effect on this reduction figure appears when the equalizer length is about four or five times its minimum value to achieve equalization, which is derived theoretically based on the SCM optical system parameters. The performance of a Tanh-type tapered linearly chirped grating equalizer on a realistic broadband SCM system is also considered. The results reveal that the dispersion-induced distortion can be neglected when the equalizer is employed.

Gifa V. 4: A complete package for NMR data set processing

The Gifa program is designed for processing, displaying and analysing 1D, 2D and 3D NMR data sets. It has been constructed in a modular fashion, based on three independent modules: a set of commands that perform all the basic processing operations such as apodisation functions, a complete set of Fourier Transforms, phasing and baseline correction, peak-picking and line fitting, linear prediction and maximum entropy processing; a set of command language primitives that permit the execution of complex macro commands; and a set of graphic commands that permit to build a complete graphic user interface, allowing the user to interact easily with the program. We have tried to create a versatile program that can be easily extended according to the user's requirements and that is adapted to a novice as well as an experienced user. The program runs on any UNIX computer, with or without graphic display, in interactive or batch mode.

Fourier Self-Deconvolution: Parameter Determination and Analytical Band Shapes

On the basis of general theories of Fourier self-deconvolution and curve fitting, an objective procedure to obtain the deconvolution parameters halfwidth, σ, and breaking point, L, is described. In the curve fitting routine, the analytical accessible line shape functions, derived for different apodization functions, are used to simulate the deconvoluted spectra. The fully automated deconvolution program was applied to the IR spectra of the amide I band of dry protein films to determine the secondary structure content.

Analysis and optimization of wedge transducers for Lamb wave excitation.

Wedge transducers are commonly used to excite and detect Lamb waves in solid plates for NDE purposes. In this paper, the analysis and design of wedge transducers for efficient, single mode Lamb wave excitation in general multilayered anisotropic plates is presented. The incident acoustic field distribution on the surface of the plate due to a wedge transducer with arbitrary apodization is calculated using the angular spectrum method. Then, the particle velocity and stress field distributions in the plate in response to this source is obtained by the surface impedance approach. This field distribution can be expressed as a sum of Lamb wave modes in the structure, since the Lamb waves form an orthogonal mode basis. Once the mode amplitudes are found, the excitation efficiency for the desired Lamb wave mode is calculated. The results indicate a trade‐off between efficiency and multimode excitation, as expected. To achieve both high efficiency and single mode excitation, an optimization scheme is used where the parameters involve the frequency of operation, dimension of the wedge, and the apodization function. The results of this analysis are presented for various practical examples.

Design of apodized linearly chirped fiber gratings for dispersion compensation

We investigate and compare the equalization performance of various apodization functions in linearly chirped fiber gratings, with a view of determining an optimum profile for the design and fabrication of chromatic dispersion compensators. A discussion concerning the relation of these apodization functions with the group delay characteristic of the equalizer is presented yielding important results connected with the grating design. The analysis has been extended to consider nonideal effects such as nonconstant averaged index and a realistic optical source.

Gradient‐based receiver structures for detecting sound directions

The use of a pressure gradient microphone for detecting the source direction of an incident plane wave is well known. Furthermore, any receiving aperture that applies a specified apodization can be regarded as extracting a point‐sample of a spatially filtered version of the incident wave. Our result is that when the gradient method is applied to the spatially filtered wave (which is also a plane wave), the answer can be expressed in terms modified apodizations, formed from the originally specified apodization function and its vector gradient, integrated over the original aperture and/or its boundary, together with simple temporal filters. Fortunately, this seemingly complicated structure collapses to a much simpler one in many cases of interest, and it is even possible to extend the method to the higher order case in which two or more wave directions are detected simultaneously. Alternatively, one wave direction can be measured while steering a broadband null in another specified direction, and the steered‐null direction can be altered dynamically even after beamforming. Examples are given. Our results are especially applicable to transient sounds of arbitrary bandwidth. [Work supported by ARL:UT IR&D.]

Curve Fitting and Deconvolution of Instrumental Broadening: A Simulated Annealing Approach

A curve-fitting procedure based on the simulated annealing algorithm has been developed for the analysis of spectral Raman data. By the inclusion of a priori information about the instrumental broadening in the definition of the cost function that is minimized, effects of the finite instrumental resolution are eliminated from the resulting fit. The ability of the method to reproduce original band shapes is tested on synthesized spectra and FT-Raman spectra of diamond recorded at different resolutions with different apodization functions. The procedure yields the global optimum of the fitted parameters and is easily implemented on a personal computer.

The human lacrimal gland synthesizes apolipoprotein D mRNA in addition to tear prealbumin mRNA, both species encoding members of the lipocalin superfamily

Apolipoprotein D (apoD), a glycoprotein originally characterized as a component of the high density lipoprotein fraction of human plasma and known to be a member of the lipocalin protein superfamily, has been found in human tear fluid by Western blot analysis. Unlike serum it seems that in the tear fluid apoD exists mainly as a disulphide linked homodimer which is not associated with lecithin/cholesterol acyltransferase (LCAT) or apolipoprotein A-I (apo A-I). By reverse-transcription-PCR (RT-PCR) of mRNA extracted from a human lacrimal gland and use of specific primers we could demonstrate expression of the apoD gene in this tissue. The amplified cDNA was cloned and a subsequent sequence analysis confirmed the identity of apoD mRNA in the human lacrimal gland. These investigations indicate that the lacrimal gland is the site of synthesis of the tear fluid apoD. Although the physiological function of apoD is unknown, it has the ability to bind phospholipids, cholesterol and other small hydrophobic molecules. Therefore, this protein might interact with meibomian lipids present in human tear fluid and probably contribute to the surface spreading of these lipids or it may function as a clearance factor, protecting the cornea from harmful lipophilic molecules.