Fourier time domain

Filtering a time series, using Fourier time domain filters, however, involves multiplying die entire time series by a single function, so diat... 

The most common form of deconvolution is Fourier self-deconvolution, which, as the name implies, takes place in the Fourier (time) domain. Deconvolution in the frequency domain is extremely arduous and lengthy. In the time domain, however, deconvolution equates to multiplication—and is therefore much quicker, even after taking into account the extra processing required for the FFT and IFFT. The procedure requires a knowledge of the band shape and full width, half-maximum resolution of the peaks involved. It is also common to enter a resolution enhancement factor—i.e. how much we want to improve the resolution by. As deconvolution is tremendously sensitive to noise, do not attempt it if the signal-to-noise ratio is less than a hundred. Also, enhancement factors tend not to be much greater than 2.5, owing to the noise amplification that occurs. 

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... 

A microwave pulse from a tunable oscillator is injected into the cavity by an anteima, and creates a coherent superposition of rotational states. In the absence of collisions, this superposition emits a free-mduction decay signal, which is detected with an anteima-coupled microwave mixer similar to those used in molecular astrophysics. The data are collected in the time domain and Fourier transfomied to yield the spectrum whose bandwidth is detemimed by the quality factor of the cavity. Hence, such instruments are called Fourier transfomi microwave (FTMW) spectrometers (or Flygare-Balle spectrometers, after the inventors). FTMW instruments are extraordinarily sensitive, and can be used to examine a wide range of stable molecules as well as highly transient or reactive species such as hydrogen-bonded or refractory clusters [29, 30]. 

This is the description of NMR chemical exchange in the time domain. Note that this equation and equation (B2.4.11)) are Fourier transfomis of each other. The time-domain and frequency-domain pictures are always related in this way. 

Once the basic work has been done, the observed spectrum can be calculated in several different ways. If the problem is solved in tlie time domain, then the solution provides a list of transitions. Each transition is defined by four quantities the mtegrated intensity, the frequency at which it appears, the linewidth (or decay rate in the time domain) and the phase. From this list of parameters, either a spectrum or a time-domain FID can be calculated easily. The spectrum has the advantage that it can be directly compared to the experimental result. An FID can be subjected to some sort of apodization before Fourier transfomiation to the spectrum this allows additional line broadening to be added to the spectrum independent of the sumilation. 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

An alternative approach to obtaining microwave spectroscopy is Fourier transfonn microwave (FTMW) spectroscopy in a molecular beam [10], This may be considered as the microwave analogue of Fourier transfonn NMR spectroscopy. The molecular beam passes into a Fabry-Perot cavity, where it is subjected to a short microwave pulse (of a few milliseconds duration). This creates a macroscopic polarization of the molecules. After the microwave pulse, the time-domain signal due to coherent emission by the polarized molecules is detected and Fourier transfonned to obtain the microwave spectmm. 

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-... 

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. 

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,... 

The French physicist and mathematician Jean Fourier determined that non-harmonic data functions such as the time-domain vibration profile are the mathematical sum of simple harmonic functions. The dashed-line curves in Figure 43.4 represent discrete harmonic components of the total, or summed, non-harmonic curve represented by the solid line. 

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. 

Figure 1.4 (a) Free induction decay (FID) in the time domain, (b) Fourier transfor-... 

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-... 

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. 

Frequency spectrum A plot of signal amplitude versus frequency, produced by the Fourier transformation of a time-domain signal. 

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