Quadrature spectrum

Figure 1.34 (a) Reduced S/N ratio resulting from noise folding. If the Rf carrier frequency is placed outside the spectral width, then the noise lying beyond the carrier frequency can fold over, (b) Better S/N ratio is achieved by quadrature detection. The Rf carrier frequency in quadrature detection is placed in the center of the spectrum. Due to the reduced spectral width, noise cannot fold back on to the spectrum. 

A serious problem associated with quadrature detection is that we rely on the cancellation of unwanted components from two signals that have been detected through different parts of the hardware. This cancellation works properly only if the signals from the two channels are exactly equal and their phases differ from each other by exactly 90°. Since this is practically impossible with absolute efficiency, some so-called image peaks occasionally appear in the center of the spectrum. How can you differentiate between genuine signals and image peaks that arise as artifacts of quadrature detection ... 

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. 

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). 

Figure 14. Contour plot of the 360 MHz H-NMR correlation spectrum of dl-camphor. A 64 x256 data set was accumulated with quadrature phase detection in both dimensions and the data set was zero filled once in the dimension and symmetrized. T was 5 sec and t was incremented by 1.63 msec. Total accumulation time was 24 minutes and data workup and plotting took 15 min.

A pre-requisite for the successful extraction of key NMR parameters from an experimental spectrum is the way it is processed after acquisition. The success criteria are low noise levels, good resolution and flat baseline. Clearly, there are also experimental expedients that can further these aims, but these are not the subject of this review per se. In choosing window functions prior to FT, the criteria of low noise levels and good resolution run counter to one another and the optimum is just that. Zero filling the free induction decay (FID) to the sum of the number acquired in both the u and v spectra (in quadrature detection) allow the most information to be extracted. 

DC-Correction is applied to compensate for a DC-offset of the FID, i.e. a vertical shift of the FID with respect to the zero-line, which occurs in quadrature detection mode if the two channels are not matched to each other. The effect is most pronounced for very weak samples and manifests itself, after Fourier transformation, as a spike in the centre of the spectrum at the center or carrier frequency. 

A complete description of the translational motion-internal motion coupling and its effects is, in principle, contained in eqs. (11-29), (11-30), and (11-31). Unfortunately, even for the simplified form of hypothetical molecular spectrum studied by Rice, McLaughlin, and Jortner, it has not yet been possible to perform the indicated quadratures. Even without actual calculation, our previous analysis of the theory of radiationless processes suffices to define the following general properties of the photo-dissociative act ... 

In summary, we have shown how the absorption of a photon leads to the formation of a resonant scattering state. Explicit formulas involving quadrature over the system energy spectrum have been presented but not evaluated. When the resonant scattering state may be approximated in terms of a set of quasistationary bound states, an explicit relationship is obtained for the rate of dissociation in terms of the matrix elements coupling zero-order states and the corresponding densities of states. In principle this permits the use of experimental rate data to evaluate the matrix elements vx and v2, if px and p2 can be estimated. 

The AB and AX systems of all 13C —13C bonds appear in one spectrum when the INADEQUATE pulse sequence (Fig. 2.48) is applied. Complete interpretation usually becomes difficult in practice due to signal overlapping, isotope shifts and AB effects (Section 2.9.4). A separation of the individual 13C— 13C two-spin systems by means of a second dimension would be desirable. It is the frequency of the double quantum transfer (d e) in Fig. 2.48 which introduces a second dimension to the INADEQUATE experiment. This double quantum frequency vDQ characterizes each 13CA — I3CX bond, as it depends on the sum of the individual carbon shieldings vA and vx in addition to the frequency v0 of the transmitter pulse located in the center of the spectrum if quadrature detection is applied [69c, 71] ... 

Figure 5.3 (a) Static IR absorbance spectrum of unoriented neat LDPE, (b) In-phase and quadrature step-scan dynamic spectra of neat LDPE in CH2 bending region, (c) 2-dimensional FT-IR synchronous correlation plot of neat LDPE in the CH2 bending, (d) 2-dimensional FT-IR asynchronous correlation plot of neat LDPE in CH2 bending region... 

Fig. 6. Top 2D MAT sequence for correlating isotopic chemical shift and CSA with two separate experiments P+ and P . All pulses following CP are 90°. A four-step phase cycling is used with 6 = —y, x, —y, x. and 62 = —y, x, x, -y. The receiver phases are x, -x, — y, -y for the P+ pulse sequence and x, —x,y, y for the P pulse sequence. (The sign of receiver phases with an asterisk depends on the relation between the pulse phase and the receiver phase of the particular spectrometer in use. These receiver phases must be changed in sign when the quadrature phase cycle (x,y, —x, -y) of the excitation pulse and the receiver phase in a single-pulse test experiment result in a null signal.) Phase alternation of the first H 90° pulse and quadrature phase cycling of the last 13C 90° pulse can be added to the above phase cycle. The time period T can be any multiple of a rotor period except for multiples of 3. Bottom 2D isotropic chemical shift versus CSA spectrum of calcium formate powder with a three-fold MAT echo extension. (Taken from Gan and Ernst178 with permission.)...

Answer The spectrum was produced by the misapplication of a trick commonly used in solid-state NMR spectroscopy ( 8.4 and 9.6). If the early data points in the FID are distorted by the recovery time of the probe and/or receiver, one can often improve the appearance of the baseline by left-shifting the data file and adding a zero to the righthand end for each left-shift. If quadrature detection is in use, it is essential that an even number of left-shifts be performed. In this example, a single left-shift was applied and the frequency sense of the spectrum has been reversed the signals have been flipped from left to right in the frequency domain because we have, in effect, reversed the x and y axes in the experiment ( 1.3.2). 

The two copies of the COSY spectrum and the fi = 0 responses can all be separated without phase cycling if one is prepared to sacrifice digital resolution by increasing the fi-spectral width. The following spectmm was collected without phase cycling, with quadrature detection OFF,... 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

The large spectral widths required by some of the applications also put more severe demands on pulse power, if uniform excitation is to be achieved across the full width of the spectrum. If both components of the complex magnetization are detected (quadrature phase detection) the carrier can be placed at the centre of the spectrum without any rf carrier folding occurring as in single-channel detection better uniformity of excitation is thus achieved at a given transmitter power. 

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