Complex Fourier transform

Figure 1.33 The underlying principle of the Redfield technique. Complex Fourier transformation and single-channel detection gives spectrum (a), which contains both positive and negative frequencies. These are shown separately in (b), corresponding to the positive and negative single-quantum coherences. The overlap disappears when the receiver rotates at a frequency that corresponds to half the sweep width (SW) in the rotating frame, as shown in (c). After a real Fourier transformation (involving folding about n = 0), the spectrum (d) obtained contains only the positive frequencies.

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. 

Fourier transformation in (Fti), spectra are obtained with real (R) and imaginary (/) data points. For detection in the quadrature mode with simultaneous sampling, a complex Fourier transformation is performed, with a phase correction being applied in F. (c) A normal phase-sensitive transform P— RR and I- RI. (d) Complex FT is applied to pairs of columns, which produces four quadrants, of which only the RR quadrant is plotted. 

The matrix obtained after the F Fourier transformation and rearrangement of the data set contains a number of spectra. If we look down the columns of these spectra parallel to h, we can see the variation of signal intensities with different evolution periods. Subdivision of the data matrix parallel to gives columns of data containing both the real and the imaginary parts of each spectrum. An equal number of zeros is now added and the data sets subjected to Fourier transformation along I,. This Fourier transformation may be either a Redfield transform, if the h data are acquired alternately (as on the Bruker instruments), or a complex Fourier transform, if the <2 data are collected as simultaneous A and B quadrature pairs (as on the Varian instruments). Window multiplication for may be with the same function as that employed for (e.g., in COSY), or it may be with a different function (e.g., in 2D /-resolved or heteronuclear-shift-correlation experiments). 

However, for illustration, only one side of the interferogram and its spectrum will be shown, usually the function of the positive spatial and spectral variable. In other operating modes of the interferometer, asymmetric interferograms are produced that have a complex Fourier transform. Asymmetric interferograms will not be treated in this work. For a more complete discussion of Fourier transform spectroscopy, the reader should consult Bell (1972), Vanasse and Strong (1958), Vanasse and Sakai (1967), Steel (1967), Mertz (1965), the Aspen International Conference on Fourier Spectroscopy (Vanasse et al., 1971), and the two volumes of Spectrometric Techniques (Vanasse, 1977, 1981). A review of early work, which includes several major contributions of his own, is given by Connes (1969). Another interesting paper on the earlier historical development of Fourier transform spectroscopy is that by Loewenstein (1966). 

F), is complex and yields after a complex Fourier transformation a real and an imaginary... 

Consider now the complex Fourier transform of the memory function,... 

In the simulation, the value of the F(t) function is calculated at discrete points in a finite time interval then discrete complex Fourier transform is performed on this array to obtain the simulated spectrum.101... 

Now we have exactly the same kind of data we have with States mode acquisition cosine modulation in t for the real FID and sine modulation in ti for the imaginary FID. In I2 we can equate Mx with the real part and My with the imaginary part, so that in each case we have Mx = —sin(f2bD) and My = cost f2b/2). This represents a vector starting on the y axis at t2 = 0 and rotating counter clockwise (positive offset 2) at the rate of f2b rad s-1. Once these rearrangements have been made in the computer, the data is processed just like States data, with a complex Fourier transform in t. ... 

Using I(t), V(t) or their complex Fourier transforms /(o ) and v(oi), one can deduce the relations describing the dielectric characteristics of a sample being tested either in the frequency or the time domain. The final form of these relations depends on the geometric configuration of the sample cell and its equivalent representation [79-86],... 

If the phase-sensitive detectors are adjusted to give a phase angle (Eq. 3.8) ( — 4>r ) = 0, the real part of the FT spectrum corresponds to pure absorption at the pulse frequency, but off-resonance lines display phase angles proportional to their off-resonance frequency as a consequence of limited rf power and nonzero pulse width (Eq. 2.55). However, acquisition of data as complex numbers from the two phase-sensitive detectors and subsequent processing with a complex Fourier transform permit us to obtain a spectrum that represents a pure absorption mode. 

After the application of weighting functions (primarily in NMR), the next step in data processing is to zero fill the data to at least a factor of two (called one level of zero filling). The reason for this step is that the complex Fourier transform of np data points consists of a real part (from the cosine part of the FT) and an imaginary part (from the sine part of the FT), each containing np/2 points in the frequency domain. Therefore, the actual spectrum displayed is described by only half of the original number of points. The technique of zero... 

The result of Eq. (48) is obtained by applying a complex Fourier transform in t2 followed by real Fourier transform in tp For either phase or amplitude modulation, 2D NMR signals have to be acquired in such a manner that both pure absorption spectra and sign discrimination will be obtained. 

Two further aspects of Fourier transformation with respect to NMR data must be mentioned. With quadrature detection a complex Fourier transformation must be performed, there is a 90° phase shift between the two detectors and the sine and cosine dependence of the sequential or simultaneous detected data points are different. In addition because the FID is a finite number of data points, the integral of the continuous Fourier transform pair must be replaced by a summation. 

Since the complex signal s-H(t) is proportional M+(t) the complex Fourier transform pair s+(t), detector 2... 

After complex Fourier transformation the spectrum signal S(co) is described by ... 

In complete analogy to NMR, FT EPR has been extended into two dimensions. Two-dimensional correlation spectroscopy (COSY) is essentially subject to the same restrictions regarding excitation bandwidth and detection deadtime as was described for one-dimensional FT EPR. In 2D-COSY EPR a second time dimension is added to the FID collection time by a preparatory pulse in front of the FID detection pulse and by variation of the evolution time between them (see figure B1.15.10(B)). The FID is recorded during the detection period of duration t, which begins with the second 7r/2-pulse. For each the FID is collected, then the phase of the first pulse is advanced by 90°, and a second set of FIDs is collected. The two sets of FIDs, whose amplitudes oscillate as functions of t, then undergo a two-dimensional complex Fourier transformation, generating a spectrum over the two frequency variables co and co,. 

The two signals (Eqs. (42) and (43)) are then considered as real and imaginary parts of a complex function, and a complex Fourier transform is performed in the computer. A further technical detail of the NMR spectrometer sketched in Fig. 13 is the so-called field-frequency lock. This implies that the rf-frequency is controlled in relation to the applied static magnetic field by monitoring the NMR signal of or F nuclei. 

Fourier Transform ftSR Complex Fourier Transforms... 

Complex Fourier transform information. Muons in Si at 5 K, 40 G.Top Power spectrum fron data taken with two orthogonal pairs of positron telescopes. Middle Real part of the same spectrum. Bottom Imaginary part of the same spectrum. 

That the two radical precessions have opposite senses can be demonstrated by complex Fourier transformation of orthogonal time spectra. An example is given in Table 2. The precession observed at 25.8 MHz in the real transform appears at 357.3 MHz in the complex transform. Given the periodic nature of the discrete Fourier transformation, the larger frequency is the "wraparound" of a negative precession frequency of 25.8 MHz (i.e., i max " I "D Thus, it can be seen that the true difference between the two frequencies A I = (135.4 + 25.8)MHz = 161.2 MHz, and that the two frequencies are indeed close to being equidistant from the diamagnetic precession. (See Equation 37). 

Equation 16 states that the spectrum B (5) is the complex Fourier transform of the product of E( >)()et od U(8). By recalling that the product of two functions in one domain (distance, 8) is the convolution of the functions in the other domain (spectral, u) then... 

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