Data sets, imaginary

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

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). 

With 2D experiments the situation is a little more complicated as the size of the overall digitised matrix depends on the number of time increments in tl as well as parameters specific to the 2D acquisition mode. Nevertheless, a digitised matrix of TD(2) X TD(1) complex data points is acquired and stored. Similar to ID the effective number o measured data points used for calculation TD(used) and the total number of data points SI to be transformed in t2 and tl may be defined prior to Fourier transformation. These parameters may be inspected and defined in the General parameter setup dialog box accessible via the Process pull-down menu. With 2D WIN-NMR the definitions for TD(2) and TD(1) are the same as for TD with ID WIN-NMR. However, unlike ID WIN-NMR, with 2D WIN-NMR SI(2) and SI(1) define the number of pairs of complex data points, instead of the sum of the number of real and imaginary data points. Therefore the 2D FT command (see below) transforms the acquired data of the current data set into a spectrum consisting of SI data points in both the real and the imaginary part. 

Generally, the impedance spectrum of an electrochemical system can be presented in Nyquist and Bode plots, which are representations of the impedance as a function of frequency. A Nyquist plot is displayed for the experimental data set Z(Zrei,Zim.,mi), (/ = 1,2,. ..,n) of n points measured at different frequencies, with each point representing the real and imaginary parts of the impedance (Zrei Zim4) at a particular frequency . 

For a validated data set, the measured real part and transformed imaginary part, Equation C.l, will match. Similarly, the measured imaginary dispersion and the transformed real part, Equation C.2, will match. In contrast, for a corrupted data set neither the measured real part and transformed imaginary part, nor the measured imaginary dispersion and the transformed real part will match. For systems that are not completely stable, significant deviations between the experimental and transformed data are usually apparent at low frequencies due to the longer acquisition time. 

Examine the imaginary residual errors to determine whether they fall within the error structure. Should a few points lie outside the error structure at intermittent frequency values, do not be concerned. Assess prediction of the real part of the impedance by examining real residual plots with confidence intervals displayed. Real residual data points that are outside the confidence interval are considered to be inconsistent with the Kramers-Kronig relations and should be removed from the data set. 

As stated in Section 7-4b, digital resolution in the v domain is a function of the number of increments (ni) and the spectral width (swi). Spectral data describing the v dimension can be acquired in the either the phase-sensitive or the absolute-value mode. Real and imaginary ui-domain data sets exist for both types of acquisition, but are treated differently. The imaginary data are discarded in phase-sensitive acquisition, just as with the V2 dimension data previously described. By contrast, with absolute-value data, both the absorptive (real) and dispersive (imaginary) components of the v domain are used to describe the spectrum. The important point is that, for both kinds of data, the acquisition of 2M increments yields M points, after Fourier transformation, to characterize spectra in the ui dimension. Therefore, if swi = 2,100 Hz and ni = 512, then DR = swi/(ni/2) = 2,100 Hz/(512/2) = 8.2 Hz/point. If one level of zero filling is carried out, then the effective ni = 1,024 and it follows that DRi = 2,100 Hz/( 1,024/2) = 4.1 Hz/point. 

A 2D NMR experiment can lead to a data set that is either phase modulated or amplitude modulated as a function of fj, depending on the particular experiment and coherence pathways selected. A regular ID spectrum consists of absorption A(p) and dispersion peaks corresponding to the real and imaginary parts of the spectral lines, respectively. In 2D experiments, phase modulation in fj results in twisted 2D real lineshapes as a result of the Fourier transformation of bi-exponential time domain... 

The data set obtained for the electropherogram, which is represented in the time domain, is converted into a data set in the frequency domain using an FT. When the number of the data points is N, FT of the data in the time domain yields N/2 + 1 complex points that are pairs of real and imaginary points in the frequency domain. These complex data are represented in terms of their magnitude as follows ... 

Following the Fourier transform, two data sets are generated representing the real and imaginary spectra (Section 3.3) so the real part with which one usually deals contains half the data points of the original FID (in the absence of further manipulation), and its data size, SI, is therefore TD/2. Digital resolution is then ... 

Figure 5.18. The four quadrants of a phase-sensitive data set. Only the RR quadrant is presented as the 2D spectrum and this is phased to contain absorption-mode lineshapes in both dimensions to provide the highest resolution (R = real, I = imaginary). Positive contours are in black and negative in red.

The cosine and sine data sets are transformed with respect to t2 and the real parts of each are taken. Then a new complex data set is formed using the cosine data for the real part and the sine data for the imaginary part ... 

The cosine data set is transformed with respect to tx and the imaginary part discarded to give... 

A new complex data set is now formed by using the signal from Eq. [9] as the real part and that from Eq. [10] as the imaginary part... 

Add a column of zeros for the imaginary component of the signal, inC13 C524, then Fourier transform the resulting data set. 

The Ruben/State/Haberkorn mode [2.40] This quadrature detection mode leads to phase sensitive 2D spectra and is based on the different symmetry properties of the sine and cosine functions after Fourier transformation. As shown in equation [2-15] the first term of imaginary part of the sine data set Im[s(coi, CO2)] has a negative sign in contrast to the real part of the cosine data set. Simple mathematical addition of the real part and the imaginary part of the sine and cosine modulated data sets after Fourier transformation... 

Complex data point. A data point consisting of both a real and an imaginary component. The real and imaginary components allowan NMR data set to be phase sensitive, insofar as there can occur a partitioning of the data depicted between the two components. An array of complex data points therefore consists of two arrays of ordinates as a function of the abscissa. 

If the data set to be Fourier transformed is expressed as d(j), j = 0,1,2,- ,N-1, in which d(j) consists of real and imaginary parts, then the discrete Fourier transform is given by ... 

The Kramers-Kronig frequency domain transformations enable the calculation of one component of the impedance from another or the determination of the phase angle from the magnitude of the impedance or the polarization resistance Rp from the imaginary part of the impedance. Furthermore, the Kramers-Kronig (KK) transforms allow the validity of an impedance data set to be checked. Precondition for the application of KK transforms is, however, that the impedance must be finite-valued... 

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