Fourier transform time domains

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). 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

Figure 5.17 Cartoon diagram to represent general structure of 4D correlation experiments. This is the same as for 3D correlation experiments (Fig. 5.14) except that an extra resonant population of heteroatom nuclei are involved in generation of transverse magnetisation (in time ts) and magnetisation transfer (during M3). Final pulse sequence generates transverse magnetisation in the Destination Nuclei S that is observed, acquired and digitised in time t/,. Fourier series transformation is used to transform time domain signal information Sfid (ti, ta, ts, 4) into frequency domain (spectral intensity) information, /NMR(fi, F2,...

FFT sizing for fast convolution is illustrated in Fig. 17.13. Since a FFT implements a discrete Fourier transform, frequency-domain multiplication equates to circular convolution in the time domain. The desired... 

The primary result of a pulsed ELDOR measurement is a distribution of dipolar cou-phngs d. This information is contained either in the primary time-domain data (variation of echo intensity as a function of dipolar evolution time t) or in the dipolar spectrum obtained from these data by Fourier transformation. Time- and frequency-domain data contain exactly the same information, since Fourier transformation is a linear operation. However, some features are easier to recognize in time domain (e.g., quahty of least-squares fitting of the data) and others in frequency domain (e.g., orientation selection by missing parts of the Pake pattern). 

While the data are collected in the time domain by scaiming a delay line, they are most easily interpreted in the frequency domain. It is straightforward to coimect the time and frequency domains tln-ough a Fourier transform... 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

Other types of mass spectrometer may use point, array, or both types of collector. The time-of-flight (TOF) instrument uses a special multichannel plate collector an ion trap can record ion arrivals either sequentially in time or all at once a Fourier-transform ion cyclotron resonance (FTICR) instrument can record ion arrivals in either time or frequency domains which are interconvertible (by the Fourier-transform technique). 

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... 

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. 

A computer digitizes the time domain spectmm f(t) and carries out the Fourier transformation to give a digitized F(v). Then digital-to-analogue conversion gives the frequency domain spectmm 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 specttum 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 specttum 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 spectmm the instmment used is usually called a Fourier transform infrared (FTIR) spectrometer although it can be modified to operate in the visible and ultraviolet regions. 

Figure 9.45(b) shows fhe resulf of Fourier transformation (see Section 3.3.3.2) of the signal in Figure 9.45(a) from the time to the frequency domain. This transformation shows clearly that two vibrations, with frequencies of about 3.3 THz (= 3.3 x lo ... 

The Fourier analyzer is a digital deviee based on the eonversion of time-domain data to a frequeney domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minieomputer to solve a set of simultaneous equations by matrix methods. 

Time domains and frequeney domains are related through Fourier series and Fourier transforms. By Fourier analysis, a variable expressed as a funetion of time may be deeomposed into a series of oseillatory funetions (eaeh with a eharaeteristie frequeney), whieh when superpositioned or summed at eaeh time, will equal the original expression of the variable. This... 

The funetion G uj) is the exponential Fourier transform of F t) and is a funetion of the eireular frequeney uj. In praetiee the funetion F t) is not given over the entire time domain but is known from time zero to some finite time T, as shown in Figure 16-2. The time span T may be divided into K equal inerements of At eaeh. For eomputational reasons, let K = 2 where p is an integer. Also, let the eireular frequeney span lu be divided into N parts where N = 2 . (In praetiee, N is often set equal to K.) By setting / = K/NT, the frequeney interval Alu beeomes... 

Spin-spin relaxation is the steady decay of transverse magnetisation (phase coherence of nuclear spins) produced by the NMR excitation where there is perfect homogeneity of the magnetic field. It is evident in the shape of the FID (/fee induction decay), as the exponential decay to zero of the transverse magnetisation produced in the pulsed NMR experiment. The Fourier transformation of the FID signal (time domain) gives the FT NMR spectrum (frequency domain, Fig. 1.7). 

FID Free induction decay, decay of the induction (transverse magnetisation) back to equilibrium (transverse magnetisation zero) due to spin-spin relaxation, following excitation of a nuclear spin by a radio frequency pulse, in a way which is free from the influence of the radiofrequency field this signal (time-domain) is Fourier-transformed to the FT NMR spectrum (frequency domain)... 

The frequency and time domains are related by the Fourier transform,... 

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... 

Most of the early vibration analysis was carried out using analog equipment, which necessitated the use of time-domain data. The reason for this is that it was difficult to convert time-domain data to frequency-domain data. Therefore, frequency-domain capability was not available until microprocessor-based analyzers incorporated a straightforward method (i.e.. Fast Fourier Transform, FFT) of transforming the time-domain spectmm into its frequency components. 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

Most predictive-maintenance programs rely almost exclusively on frequency-domain vibration data. The microprocessor-based analyzers gather time-domain data and automatically convert it using Fast Fourier Transform (FFT) to frequency-domain data. A frequency-domain signature shows the machine s individual frequency components, or peaks. 

The evolution period tl is systematically incremented in a 2D-experiment and the signals are recorded in the form of a time domain data matrix S(tl,t2). Typically, this matrix in our experiments has the dimensions of 512 points in tl and 1024 in t2. The frequency domain spectrum F(o l, o 2) is derived from this data by successive Fourier transformation with respect to t2 and tl. 

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Fourier transformation of Rf pulses (which are in the time domain) produces frequency-domain components. If the pulse is long, then the Fourier components will appear over a narrow frequency range (Fig. 1.24) but if the pulse is narrow, the Fourier components will be spread over a wide range (Fig. 1.25). The time-domain signals and the corresponding frequency-domain partners constitute Fourier pairs. 

In pulse NMR we measure in the time domain i.e., the variation of signal amplitude with time (FID) is recorded. These time-domain data are then subjected to Fourier transformation to convert them into the frequency domain. 

Two-dimensional NMR spectroscopy may be defined as a spectral method in which the data are collected in two different time domains acquisition of the FID tz), and a successively incremented delay (tj). The resulting FID (data matrix) is accordingly subjected to two successive sets of Fourier transformations to furnish a two-dimensional NMR spectrum in the two frequency axes. The time sequence of a typical 2D NMR experiment is given in Fig. 3.1. The major difference between one- and two-dimensional NMR methods is therefore the insertion of an evolution time, t, that is systematically incremented within a sequence of pulse cycles. Many experiments are generally performed with variable /], which is incremented by a constant Atj. The resulting signals (FIDs) from this experiment depend... 

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

Another resolution-enhancement procedure used is convolution difference (Campbell et ai, 1973). This suppresses the ridges from the cross-peaks and weakens the peaks on the diagonal. Alternatively, we can use a shaping function that involves production of pseudoechoes. This makes the envelope of the time-domain signal symmetrical about its midpoint, so the dispersionmode contributions in both halves are equal and opposite in sign (Bax et ai, 1979,1981). Fourier transformation of the pseudoecho produces signals... 

The next step after apodization of the t time-domain data is Fourier transformation and phase correction. As a result of the Fourier transformations of the t2 time domain, a number of different spectra are generated. Each spectrum corresponds to the behavior of the nuclear spins during the corresponding evolution period, with one spectrum resulting from each t value. A set of spectra is thus obtained, with the rows of the matrix now containing Areal and A imaginary data points. These real and imagi-... 

There are actually two independent time periods involved, t and t. The time period ti after the application of the first pulse is incremented systematically, and separate FIDs are obtained at each value of t. The second time period, represents the detection period and it is kept constant. The first set of Fourier transformations (of rows) yields frequency-domain spectra, as in the ID experiment. When these frequency-domain spectra are stacked together (data transposition), a new data matrix, or pseudo-FID, is obtained, S(absorption-mode signals are modulated in amplitude as a function of t. It is therefore necessary to carry out second Fourier transformation to convert this pseudo FID to frequency domain spectra. The second set of Fourier transformations (across columns) on S (/j, F. produces a two-dimensional spectrum S F, F ). This represents a general procedure for obtaining 2D spectra. 

A Fourier Transform Relationship between Time-Domain and Frequency-Domain Excitation Functions. 

Fourier transformation A mathematical operation by which the FIDs are converted from time-domain data to the equivalent frequency-domain spectrum. 

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