Fourier pairs

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. 

Fourier pairs not only exist in time-/frequency domain but also in any other domain combined by a quantity q and the belonging dimension-inverted quantity l/q. 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

NMR data i,s usually processed using one or more processing functions, some of which are applied in the time domain, others in the frequency domain. Each processing function in the time domain f(t) also has its counterpart F(f) in the frequency domain and forms a Fourier pair. In principle the same effect in the final spectrum S"(f) may be obtained with a given processing function, applied either in the time or the frequency domain as long as a few important rules are followed when performing the... 

With s(t) and f(t) beeing the FID of an NMR signal and a processing function or a second FID respectively and with S(f) and F(f) beeing the corresponding frequency domain counterparts, the following rules govern the manipulation of these Fourier pairs ... 

Figure 11 Some well characterized Fourier pairs. The white spectrum and the impulse function (a), the boxcar and sine functions (b), the triangular and sintf functions (c), and the Gaussian pair (d)...

Factor analysis, 79 0-mode, 84 / -mode, 85 target transform, 91 Feature extraction, 54 Feature selection, 54 Filtering, 41 Flicker noise, 31 Fourier integral 41 Fourier pairs 42 Fourier transform 28 Furthest neighbours clustering, 103, 107 Fuzzy clustering, 104, 115... 

Note that the above equation and Eq. 6.116 form a Fourier pair. Equations 6.116 and 6.119 are completely equivalent but the solution via the characteristic function is strikingly simpler than the time-domain solution. 

Fig. 4.1.2 The rectangular pulse function (a) and the sine function (b) form a Fourier pair, transformation,...

The spectra resulting from two-dimensional Fourier transformation of phase and amplitude modulated data sets can be determined by using the following Fourier pair... 

The FID s(t) and spectrum S(v) form a Fourier pair which can be transfen ed into one another by (forward) ... 

A time domain function can be expressed as a Fourier series, an infinite series of sines and cosines. However in practise integrals related to the FOURIER series, rather than the series themselves are used to perform the Fourier transformation. Linear response theory shows that in addition to NMR time domain data and frequency domain data, pulse shape and its associated excitation profile are also a FOURIER pair. Although a more detailed study [3.5] has indicated that this is only a first order approximation, this approach can form the basis of an introductory discussion. 

Hence, s(t) and S(f) are equivalent representations of a signal and are often referred to as a Fourier pair. For the simplest infinite oscillatory signal of fi equency V the Fourier pair is... 

Conventional discrete sampling influences a spectrum in quite a straightforward way because of the simplicity of the corresponding Fourier pair ... 

The two functions Es t, co) and Es(t,r) form a Fourier pair, related to each other by... 

The Fourier pair relationship between the time (f) and frequency (co) is expressed in the form... 

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