Real 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.

The spectra are phase sensitive, so apply real Fourier Transformation with TPPl. 

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. 

The spectra of Figure 15 were obtained by real Fourier transformation, so only absolute values of the frequencies are obtained ... 

Equation 11 indicates that E(S)ngt is the cosine (or real) Fourier transform of B(F). Unfortunately, this is rarely the case due to the practical constraints of the apparatus. A frequency-dependent phase error, 9v> is often present and affects the phase angle 2 p8. The source of this phase error can be misalignment of the interferometer, dispersion by the beamsplitter, errors in data acquisition, or caused by electronic filters in the detector amplifier. If the phase error is considered Equation 11 becomes ... 

In the TPPI method a single data set with 512 increments is collected. In each successive increment the phase of the 90° pulse at the end of the period is incremented by 90° with respect to the phase of the corresponding pulse in the previous increment. (An equivalent experiment can be performed in which the phases of the pulses before the ti period are shifted by 90°). This is equivalent to changing the reference frame in so that the transmitter in the dimension appears to be shifted to one edge of the spectrum. After performing a real Fourier transformation, all peaks will appear to be shifted to one side of the transmitter in /. The main disadvantage of this technique is that phase distortions can appear for resonances in strongly coupled spin systems. 

Analysis of the ESEEM signal can be performed either directly in the time domain, by measuring the amplitude of echo modulation, or in the frequency domain, after numerical Fourier transformation of V T). Because theoretical descriptions of ESEEM " predict that the stimulated echo is modulated according to a cosine function, with t=x + T sls time variable, the real Fourier transform must be used to get the frequency spectra ... 

If X, y and h are functions with Fourier transforms X, Y and H (real problem), we can write equation (9) in the frequency domain ... 

The 2-D / -space data set is Fourier transformed, and the magnitude image generated from the real and imaginary outputs of the Fourier transform. 

One of the major advantages of SEXAFS over other surface structutal techniques is that, provided that single scattering applies (see below), one can go direcdy from the experimental spectrum, via Fourier transformation, to a value for bond length. The Fourier transform gives a real space distribudon with peaks in at dis-... 

The real space pair distributions gy(rj is the inverse Fourier transform of (Sy(Q)-l), that is ... 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

Given a real-valued sequence (Jq, Ci,. ..,(T v-i, the the correlation function R t) and Fourier Transformation may be defined in both a continuous and discrete form ... 

We now apply these results to compute 1 v(2>) the Fourier transform of Kuv(x), in terms of its imaginary part Im OL p). Causality asserts that J uv(p) is an analytic function of p0 in Imp0 > 0, and hence that there exists a dispersion relation relating the real and imaginary parts of... 

Often one of the diatomic bond distances r or r2 can be used as s. Insertion of Eq. (41) into Eq. (40), coupled with arguments such as those in Section IIC to connect < >/( ) to RWP iterates, then leads to an expression for Eq. (40) within the RWP framework [13]. The relevant reaction probability expression, Eq. (18) of Ref. [13], which need not be detailed here, involves Fourier transformation of ls=so ( ) / ls=so ( ) requires the real wave packet and its derivative... 

The first Fourier transformation of the FID yields a complex function of frequency with real (cosine) and imaginary (sine) coefficients. Each FID therefore has a real half and an imaginary half, and when subjected to the first Fourier transformation the resulting spectrum will also have real and imaginary data points. When these real and imaginary data points are arranged behind one another, vertical columns result. This transposed data... 

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. 

Figure 3.6 The first set of Fourier transformations across <2 yields signals in V2, with absorption and dispersion compronents corresponding to real and imaginary parts. The second FT across /, yields signals in V, with absorption (i.e., real) and dispersion (i.e., imaginary) components quadrants (a), (b), (c), and (d) represent four different combinations of real and imaginary components and four different line shapes. These line shaptes normally are visible in phase-sensitive 2D plots.

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-... 

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). 

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

Real and imaginary parts Two equal blocks of frequency that result from Fourier transformation of the FIDs. 

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