Fourier transformation continuous spectra

The maximum entropy method (MEM) is developed to obtain the maximum spectrum information from the limited number of data. It enables us to estimate the power spectrum without an EFT (fast Fourier transform) using a distinct Fourier transform (DFT). The main problems in the FFT method are the so-called spectrum leaks from other frequencies, i.e., in addition to the true range of frequencies, the power spectrum also contains components at other unwanted frequencies, which leads to errors in spectral analysis. To demonstrate the spectrum leaks associated with FFT, suppose that an original continuous signal is the one shown in Fig. 37a. Its Fourier transform power spectrum is shown in Fig. 37b and has one sharp peak. From the limited number of data (c) the FFT is obtained as (d), which is still similar. However, from another set of limited data but half a period longer than the data (c), we obtain a spectrum with a few small peaks. This is the spectrum leak. To cope with this, a windowed Fourier transform shown in (g) with a lenslike window has to be applied to improve the spectrum to (h). 

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

Fourier transform spectroscopy technology is widely used in infrared spectroscopy. A spectrum that formerly required 15 min to obtain on a continuous wave instrument can be obtained in a few seconds on an FT-IR. This greatly increases research and analytical productivity. In addition to increased productivity, the FT-IR instrument can use a concept called Fleggetts Advantage where the entire spectrum is determined in the same time it takes a continuous wave (CW) device to measure a small fraction of the spectrum. Therefore many spectra can be obtained in the same time as one CW spectrum. If these spectra are summed, the signal-to-noise ratio, S/N can be greatly increased. Finally, because of the inherent computer-based nature of the FT-IR system, databases of infrared spectra are easily searched for matching or similar compounds. 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

Fig. 7a- . Plot of Log[Xj(77 K)/ X,(300K)] as a function of k. Points are the experimental values of the logarithm calculated at the extrema of the inverse Fourier transform of the first neighbour peak, a One monolayer of cobalt on copper (111), normal incidence b one monolayer of cobalt on copper (111), grazing incidence c one monolayer of cobalt on copper (110). Continuous lines are the linear regressions corresponding to each spectrum. Related values of = fj2(300 K) - cP(77 K)] are indicated...

Figure 23-3 Infrared absorbance spectra of the amide regions of proteins. (A) Spectra of insulin fibrils illustrating dichroism. Solid line, electric vector parallel to fibril axis broken line, electric vector perpendicular to fibril axis. From Burke and Rougvie.24 Courtesy of Malcolm Rougvie. See also Box 29-E. (B) Fourier transform infrared (FTIR) spectra of two soluble proteins in aqueous solution obtained after subtraction of the background H20 absorption. The spectrum of myoglobin, a predominantly a-helical protein, is shown as a continuous line. That of concanavalin A, a predominantly (3-sheet containing protein, is shown as a broken line. From Haris and Chapman.14 Courtesy of Dennis Chapman.

In practice ultrasound is usually propagated through materials in the form of pulses rather than continuous sinusoidal waves. Pulses contain a spectrum of frequencies, and so if they are used to test materials that have frequency dependent properties the measured velocity and attenuation coefficient will be average values. This problem can be overcome by using Fourier Transform analysis of pulses to determine the frequency dependence of the ultrasonic properties. 

For data in which the variables are expressed as a continuous physical property (e.g. spectroscopy data, where the property is wavelength or wavenumber), the Fourier transform can provide a compressed representation of the data. The Fourier compression method can be applied to one analyzer profile (i.e. spectrum) at a time. 

Continuous-flow 19F LC-NMR spectra were acquired for 16 transients using 60° pulses into 8192 data points over a spectral width of 11 364 Hz, giving an acquisition time of 0.36 s. A relaxation delay of 0.64 s was added to give a total acquisition time for each spectrum of 16 s. The data were multiplied by a line-broadening function of 3 Hz to improve the signal-to-noise ratio and zero-filled by a factor of two before Fourier transformation. The results are presented as a contour plot with 19F NMR chemical shift on the horizontal axis and chromatographic retention time on the vertical axis. 

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