Time-domain spectra

The process of going from the time domain spectrum f t) to the frequency domain spectrum F v) is known as Fourier transformation. In this case the frequency of the line, say too MFtz, in Figure 3.7(b) is simply the value of v which appears in the equation... 

Figure 3.7 (a) The time domain spectrum and (b) the corresponding frequency domain spectrum... 

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

The newer instruments (Figure 2.4c) utilize a radiofrequency pulse in place of the scan. The pulse brings all of the cycloidal frequencies into resonance simultaneously to yield a signal as an interferogram (a time-domain spectrum). This is converted by Fourier Transform to a frequency-domain spectrum, which then yields the conventional m/z spectrum. Pulsed Fourier transform spectrometry applied to nuclear magnetic resonance spectrometry is explained in Chapters 4 and 5. 

Fourier transformation (FT) Mathematical operation to convert a time domain spectrum (FID) to a frequency domain spectrum (normal NMR spectrum). 

Figure 10.4 Free induction decay signal, which appears in the receiver coils after a 90° pulse. The FID is a time-domain spectrum, showing RF intensity as a function of time 12- It is a composite of the RF absorption frequencies of all nuclei in the sample. The Fourier transform decomposes an FID into its component frequencies, giving a spectrum like that shown in Fig. 10.1. Figure generously provided by Professor John M. Louis.

Fig. 4.4. (a) Excitation-ionization spectrum of the H atom Balmer series around the ionization limit in a static homogenous magnetic field, (b) Fourier-transformed time domain spectrum of the spectrum shown in (a). The square of the absolute value is plotted. The time scale is given in units of the cyclotron period Tc = 2 k/u c. Reprinted from Main, Holle, Wiebusch, and Welge (1987). 

Figure 4.2b is a presentation of the FID of the decoupled 13C NMR spectrum of cholesterol. Figure 4.2c is an expanded, small section of the FID from Figure 4.2b. The complex FID is the result of a number of overlapping sine-waves and interfering (beat) patterns. A series of repetitive pulses, signal acquisitions, and relaxation delays builds the signal. Fourier transform by the computer converts the accumulated FID (a time domain spectrum) to the decoupled, frequency-domain spectrum of cholesterol (at 150.9 MHz in CDC13). See Figure 4.1b.

Figure 3.9(a) shows a time domain spectrum corresponding to the frequency domain spectrum in Figure 3.9(b) in which there are two lines, at 25 and 100 MHz, with the latter having half the intensity of the former, so that... 

A computer digitizes the time domain spectrum fit) and carries out the Fourier transformation to give a digitized F( v). Then digital-to-analogue conversion gives the frequency domain spectrum F(v) in the analogue form in which we require it. 

For radiofrequency and microwave radiation there are detectors which can respond sufficiently quickly to the low frequencies (<100 GHz) involved and record the time domain spectrum directly. For infrared, visible and ultraviolet radiation the frequencies involved are so high (>600 GHz) that this is no longer possible. Instead, an interferometer is used and the spectrum is recorded in the length domain rather than the frequency domain. Because the technique has been used mostly in the far-, mid- and near-infrared regions of the spectrum the instrument used is usually called a Fourier transform infrared (FTIR) spectrometer although it can be modified to operate in the visible and ultraviolet regions. 

The conversion of an oscillating electric field E(t), the so-called time domain spectrum, into a frequency domain spectrum is known as a Fourier transformation. A simple but neat description of this transformation is given by Hollas [16]. The oscillating electric field arising from a molecular emission line following the radiation pulse is converted into an oscillating voltage f(t) with a frequency v, which we may write... 

Figure 10.17. (a) Time domain spectrum, (b) frequency domain spectrum for radiation of a single frequency. 

We first consider the intermolecular modes of liquid CS2. One of the details that two-dimensional Raman spectroscopy has the potential to reveal is the coupling between intermolecular motions on different time scales. We start with the one-dimensional Raman spectrum. The best linear spectra are based on time domain third-order Raman data, and these spectra demonstrate the existence of three dynamic time scales in the intermolecular response. In Fig. 3 we have modeled the one-dimensional time domain spectrum of CS2 for 3 cases (A) a single mode represented by the sum of three Brownian oscillators, (B) three Brownian oscillators, and (C) a distribution of 20 arbitrary Brownian oscillators. Case (A) represents the fully coupled, or isotropic case where the liquid is completely homogeneous on the time scales of the simulation. Case (B) deconvolutes the linear response into the three time scales that are directly evident in the measured response and is in the limit that the motions associated with each of the three timescales are uncoupled. Case (C) is an example where the liquid is represented by a large distribution of uncoupled motions. 

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