Fourier time-domain response

Fourier transform refers to NMR experiments in which the time domain response of the spin system is transformed into a frequency spectrum. 

FID). FID is a time domain response of all nuclei irradiated by r.f. excitation pulse and can be analyzed by Fourier transformation into the frequency domain. 

Multiple-point fluorescent deteclion has been proposed to enhance detection sensitivity. This method is based on the use of a detector function, such as the Shah function. The time-domain signals were first detected, and they were converted into a frequency-domain plot by Fourier transformation. Therefore, this technique was dubbed Shah convolution Fourier transform detection (SCOFT). As a comparison, the single-detection point time-domain response is commonly known as the electropherogram [698,699,701]. 

To provide an example of the two-dimensional response from a system containing well-defined intramolecular vibrations, we will use simulations based on the polarized one-dimensional Raman spectrum of CCI4. Due to the continuous distribution of frequencies in the intermolecular region of the spectrum, there was no obvious advantage to presenting the simulated responses of the previous section in the frequency domain. However, for well-defined intramolecular vibrations the frequency domain tends to provide a clearer presentation of the responses. Therefore, in this section we will present the simulations as Fourier transformations of the time domain responses. Figure 4 shows the Fourier transformed one-dimensional Raman spectrum of CCI4. The spectrum contains three intramolecular vibrational modes — v2 at 218 cm, v4 at 314 cm, and vi at 460 cm 1 — and a broad contribution from intermolecular motions peaked around 40 cm-1. We have simulated these modes with three underdamped and one overdamped Brownian oscillators, and the simulation is shown over the data in Fig. 4. 

Once the diffusive reorientation contribution has been subtracted from the deconvolved time-domain response, a final Fourier transform yields the intermolecular spectrum (often referred to as the reduced spectral density). 

To summarize, all of the information of the system is available from either means of exciting the resonance, driving it and sweeping the frequency, or hitting it with an impulse for its time response. The first experiment is performed in the frequency domain and the second in the time domain. The mathematical transformation of one representation into the other is the Fourier transform. The time domain response and the frequency domain response are called Fourier transform... 

We will now investigate the relationship between frequency domain and time domain responses. In most cases considered in this section a transient electromagnetic field is excited by a step function current in the source. Moreover the theory of the transient induction logging described in this monograph will be developed for this type of excitation. For this reason the relationship between frequency response and transient response corresponding to this single type of excitation will be our principal concern. The information we need is obtained through use of the Fourier transform which takes the well... 

In the previous section, we established a correspondence between the transient time-domain response (exponentially damped cosine wave) to a sudden "impulse" excitation and the steady-state frequency-domain response (Lorentzian absorption and dispersion spectra) to a continuous excitation. The Fourier transform may be thought of as the mathematical recipe for going from the time-domain to the frequency-domain. In this section, we shall introduce the mathematical forms of the transforms, along with pictorial examples of several of the most important signal shapes. 

Figure 9. Apodizations of chi experimental FT-NMR signal. (A) Fourier transform of the original unweighted free induction decay (F.I.D.) time-domain response to a 90 -pulse excitation. (B) Signal-to-noise enhancement F.I.D. weighted by the factor, exp(- r>LB-t), with LB = 3.0 Hz, before F.T.

Deconvolution of response to frequency-sweep excitation, (a) Cosine Fourier transform of linearly increasing frequency sweep time-domain waveform, (b) Magnitude spectrum of excitation, (c) Cosine Fourier transform of time-domain response to excitation, (d) Magnitude spectrum of response, (e) = (c)/(a). (f) Magnitude spectrum... 

The steady-state ESR absorption spectrum, A((u), is equivalent to the Fourier transform of the time-domain response, f(t), to an impulse excitation, if the magnetic resonance system is linear (i.e., absence of saturation). Since f(t) is causal (i.e., f(t) = 0 for t < 0), there is a simple mathematical relation between the absorption spectrum, A(w), and its corresponding dispersion spectrum, D(w) ... 

Figure 2. Overall measurement concept for EIS based on binary signal stimulation and Fast Fourier Transform (FFT) transformation of the time domain response.

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. 

Pulse fluorometry uses a short exciting pulse of light and gives the d-pulse response of the sample, convoluted by the instrument response. Phase-modulation fluorometry uses modulated light at variable frequency and gives the harmonic response of the sample, which is the Fourier transform of the d-pulse response. The first technique works in the time domain, and the second in the frequency domain. Pulse fluorometry and phase-modulation fluorometry are theoretically equivalent, but the principles of the instruments are different. Each technique will now be presented and then compared. 

Similarly to non-selective experiments, the first operation needed to perform experiments involving selective pulses is the transformation of longitudinal order (Zeeman polarization 1 ) into transverse magnetization or ly). This can be achieved by a selective excitation pulse. The first successful shaped pulse described in the literature is the Gaussian 90° pulse [1]. This analytical function has been chosen because its Fourier transform is also a Gaussian. In a first order approximation, the Fourier transform of a time-domain envelope can be considered to describe the frequency response of the shaped pulse. This amounts to say that the response of the spin system to a radio-frequency (rf) pulse is linear. An exact description of the... 

Fig. 2. The behavior of the magnetization vector (i) is shown in response to the application of a single 7i/2 r.f. pulse along V, (ii). The decay of the magnetization vector in the x -y plane yields the received time-domain signal, called the FID, shown in (iii). The result of a Fourier transformation of the FID is the spectrum shown in (iv). For a liquid-like sample, the full-width at half-maximum-height of the spectral signal is l/itV) (Section II.A.2).

As used here, a DC model is characterized entirely in terms of dielectric constants (e) of the pure solvent (i.e., in the absence of the solute and its cavity) and the structure of the molecular cavity (size and shape) enclosing the solute [3], We confine ourselves to dipolar medium response, due either to the polarizability of the solvent molecules or their orientational polarization1 [15,16]. Within this framework, in its most general space and time-resolved form, one is dealing with the dielectric function s(k, >), where k refers to Fourier components of the spatial response of the medium, and oj. to the corresponding Fourier components of the time domain [17]. In the limit of spatially local response (the primary focus of the present contribution), in which the induced medium polarization (P) at a point r in the medium is specified entirely by the electric field (E) at the same point, only the Tong wavelength component of s is required (i.e., k = 0) [18,19]. 

It is possible that a skin, which is moist and cool gives exactly the same electrical response to measurements made at a single frequency as a skin, which is dry and warm. To separate and specify potentially confounding influences such as water content, temperature change, and sweat gland activity, it is necessary to use some form of electrical spectroscopic technique, that is, stimulation at three or more different frequencies, or a time-domain approach followed by Fourier transformation.44-46... 

Whatever the excitation, the transformation of the response from the frequency to the time domain (Fig. 11.21) is done with the inverse Fourier transform, normally as the FFT (fast Fourier transform) algorithm, just as for spectra of electromagnetic radiation. Remembering that the Fourier transform is a special case of the Laplace transform with... 

Fourier transform voltammetry — Analysis of any AC or transient response using (fast) Fourier transformation (FFT) and inverse (fast) Fourier transformation (IFFT) to convert time domain data to the frequency domain data and then (often) back to time domain data but separated into DC and individual frequency components [i-ii]. See also - Fourier transformation, AC voltamme-... 

In order to actually cover 19 decades in frequency, dielectric spectroscopy makes use of different measurement techniques each working at its optimum in a particular frequency range. The techniques most commonly applied include time-domain spectroscopy, frequency response analysis, coaxial reflection and transmission methods, and at the highest frequencies quasi-optical and Fourier transform infrared spectroscopy (cf. Fig. 2). A detailed review of these techniques can be found in Kremer and Schonhals [37] and in Lunkenheimer [45], so that in the present context only a few aspects will be summarized. 

The analysis of chromatographic data is commonly carried out by determining the first and second moments of the response peak. This method is simple and convenient although somewhat more accurate results may be obtained by more sophisticated methods such as Fourier transform or direct matching of the response curves in the time domain(10-14). 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

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