Some Fourier Transform Applications

As will be discussed in Chapter 6, one of the main uses of the Fomier transform is for analyzing sounds. We can pick peaks in the frequency domain and determine whether they behave like sinusoids. Those peaks can be used to calibrate an additive or modal synthesizer model like we did with the coffee mug in Chapter 4. We can look at gross spectral envelope features, and use those to control a filter-based model. We can also use frequency domain convolution to implement filters, multiplying the spectnun of an input signal by a shaping function spectrum to yield the final filtered output spectrum. 

Phase corruptions, such as those caused by echoes in rooms, can change the shape of a waveform quite radically, but the pitch of the source is still quite evident to human listeners. The frequency domain provides us with much more robust methods of pitch estimation. 

The frequencies of the five strongest detected peaks of the sss consonant are 5970, 7816, 8236, 8279, and 8435 Hz. The HNR was computed to be -7 dB, reflecting the fact that the soimd is really noise. The frequencies of the five strongest detected peaks of the ahh vowel are 1325, 2450, 3412, 3980, and 4730 Hz. These have a common divisor of about 670 Hz (it s a baby saying ahh, thus the high pitch). The HNR was computed to be 8.7 dB. 

Another method of estimating the pitch of a signal uses a special form of convolution called autocorrelation defined as  

This is a time (lag) domain function that expresses the similarity of a signal to lagged versions of itself. It can also be viewed as the convolution of a signal with the time-reversed version of itself. Pure periodic signals exhibit periodic peaks in the autocorrelation fimction. Autocorrelations of white noise sequences exhibit only one clear peak at the zero lag position, and are small (zero for infinite length sequences) for all other lags. 

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In the following, some examples of applications of Fourier transform infrared (FTIR) Spectroscopy and of solid-state nuclear magnetic resonance (NMR) to the study of polymorphism in polymers are described. 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

Recent work in our laboratory has shown that Fourier Transform Infrared Reflection Absorption Spectroscopy (FT-IRRAS) can be used routinely to measure vibrational spectra of a monolayer on a low area metal surface. To achieve sensitivity and resolution, a pseudo-double beam, polarization modulation technique was integrated into the FT-IR experiment. We have shown applicability of FT-IRRAS to spectral measurements of surface adsorbates in the presence of a surrounding infrared absorbing gas or liquid as well as measurements in the UHV. We now show progress toward situ measurement of thermal and hydration induced conformational changes of adsorbate structure. The design of the cell and some preliminary measurements will be discussed. 

Conventional use has been made of the radioisotope C, and details need hardly be given here. Illustrative examples include the elucidation of pathways for the anaerobic degradation of amino acids (Chapter 7, Part 1) and purines (Chapter 10, Part 1). Some applications have used C with high-resolution Fourier transform NMR in whole-cell suspensions, and this is equally applicable to molecules containing the natural or the synthetic P nuclei. As noted later, major advances in NMR have made it possible to use natural levels of C. 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

In the mid-IR, routine infrared spectroscopy nowadays almost exclusively uses Fourier-transform (FT) spectrometers. This principle is a standard method in modem analytical chemistry45. Although some efforts have been made to design ultra-compact FT-IR spectrometers for use under real-world conditions, standard systems are still too bulky for many applications. A new approach is the use of micro-fabrication techniques. As an example for this technology, a miniature single-pass Fourier transform spectrometer integrated on a 10 x 5 cm optical bench has been demonstrated to be feasible. Based upon a classical Michelson interferometer design, all... 

The most obvious drawback of Fourier space approaches is the computational cost of the Fourier transformation itself. However, this can be circumvented in some virtual screening applications. Gaussian functions are frequently used to approximate electron densities. Interestingly, the Fourier transform of a Gaussian function is again a Gaussian function and hence amenable to analytic transformation. 

A special case is the application domain of discrete functions (e.g., measurements on some spatial grid). The Fourier transform of a discrete function can be computed quite efficiently by a special algorithm (Fast Fourier Transform) at discrete points in Fourier space [132]. 

To introduce the application of ultrashort laser sources in microscopy, we want to review some properties of femtosecond pulses first for a comprehensive introduction the reader may refer to one of the established textbooks on femtosecond lasers (Diels and Rudolph 2006). The most important notion is the Fourier transform relation between the temporal shape of a pulse and the spectrum necessary to create it. This leads to the well-known time-bandwidth product for the pulse temporal width (measured as full width at half maximum, FWHM) At and the pulse spectral width Av. 

Fourier-transform infrared (FTIR) spectrometers encode infrared wavenumbers by moving a mirror in a Michelson interferometer which results in a unique, path-dependent pattern of interference for each light wavelength in the IR beam. FTIRs have come to totally dominate the IR market and are the means by which most of the work described in this review was accomplished. Only for some special applications (modulation spectra and time-dependence studies) are dispersive-based (scanning monochromator or tuned laser) spectrometers still used. The advantages of the FTIR approach are that the entire spectral region of interest can... 

FTIR Fourier transform infrared spectroscopy has been used for many years to measure atmospheric gases. Because FTIR has become such a common analytical method, we do not describe the technique itself here but rather refer the reader to several excellent books and articles on the subject (e.g., see Griffiths and de Haseth, 1986 Wayne, 1987). For reviews of some atmospheric applications, see Tuazon etal. (1978,1980), Marshall et al. (1994), and Hanst and Hanst (1994). 

The molecular specificity of Fourier transform infrared (FTIR) lends itself quite well to applications in pharmaceutical development labs, as pointed out in a review article with some historical perspective.10 One of the more common applications of mid-IR in development is a real-time assessment of reaction completion when used in conjunction with standard multivariate statistical tools, such as partial least squares (PLS) and principal component analysis (PCA).18,19 Another clever use of FTIR is illustrated in Figure 9.1, where the real-time response of a probe-based spectroscopic analyzer afforded critical control in the charge of an activating agent (trifluoroacetic anhydride) to activate lactol. Due to stability and reactivity concerns, the in situ spectroscopic approach was... 

X-ray scattering from coal was the subject of several early studies which led to the postulation that coal contains aromatic layers about 20 to 30 A in diameter, aligned parallel to near-neighbors at distances of about 3.5 A (Hirsch, 1954). Small-angle x-ray scattering, which permits characterization of the open and closed porosity of coal, has shown a wide size distribution and the radius of gyration appears to be insufficient to describe the pore size. Application of the Fourier transform technique indicated that some coals have a mesoporosity with a mean radius of 80 to 100 A (Guet, 1990). 

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