Quadrature channel

As described in the Section 3.3, two supposedly identical detectors are arranged to sample the signal simultaneously along x and y. However, the two detectors are usually not quite identical, and the reference signals to the detectors may not differ by precisely 90°. Also, the sample-and-hold circuits that follow the detectors may have slightly different characteristics. Three types of artifacts result (1) a DC offset between quadrature channels, (2) a gain difference between channels, and (3) a phase difference between channels. In terms of Eq. 3.5, artifact (1) means that there are constant terms of different magnitude added to the sine and cosine terms, while (2) and (3) imply different values of C and 4 rf respectively, for the sine and cosine terms. 

The signal is then digitised for storage in the computer memory. There are two ways in which the data can be captured when the two quadrature channels are being used. A data point for each channel can be captured at the same time (called simultaneous acquisition) or, alternatively, one data point is taken at each dwell time but routed to the different channels of the phase-sensitive detector (called sequential acquisition). The relationship of the spectral width and DW is slightly different for the two approaches ... 

Figure 3.23. Data sampling schemes for the two common quadrature detection methods, (a) Simultaneous sampling the two quadrature channels (representing x and y magnetisation) are sampled at the same point in time, (b) Sequential sampling the two channels are sampled alternately at twice the rate of method (a), and the phase inverted for alternate pairs of data points (see text).

B) In-phase spectrum after phase rotation of solution ferricyanide and ferrocyanide bands into the quadrature channel. Reprinted from Ref. [30] with permission. 

The magnitude of the signal represents the total optical response of the system to the modulation frequency d>, while the phase represents the phase delay. Provided the phase delay across the solution band is constant (the species is in a homogeneous environment), the value of is measured for a solution species, and then the entire complex spectrum is rotated into the quadrature channel, as shown in Fig. 3.50. As a result, only the absorption due to surface species with different phase delays remains in the in-phase channel. This technique was used for disentangling strong bands due to the ferro/ferricyanide Faradaic species that were superimposed on weak bands of these species in the adsorbed state on a Pt electrode [263]. 

Figure 3.50. Projection of solution and surface iDands into in-phase and quadrature channels...

Figure 21.24. (a) Magnitude (dashed line) and phase (solid line) spectra of ferricyanide and ferrocya-nide with hexacyanoferrate adsorbed on the electrode surface (b) in-phase spectrum after phase rotation of solution-phase ferricyanide and ferrocyanide bands into the quadrature channel. 10 mM ferrocyanide 1 Hz modulation frequency potential modulation limits, 0.02 to 0.42 V versus SCE. (Reproduced from [20], by permission of the American Chemical Society copyright 1997.)... 

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 ... 

Bruker instruments use quadrature detection, with channels A and B being sampled alternately, so the dwell time is given by ... 

In quadrature detection, the transmitter offset frequency is posidoned at the center of the F domain (i.e., at F2 = 0 in single-channel detection it is positioned at the left edge). Frequencies to the left (or downfield) of the transmitter offset frequency are positive those to the right (or upheld) of it are negative. 

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. 

CYCLOPS A four-step phase cycle that corrects dc imbalance in the two channels of a quadrature detector system. 

Quadrature images Any imbalances between the two channels of a quadrature detection system cause ghost peaks, which appear as symmetrically located artifact peaks on opposite sides of the spectrometer frequency. They can be eliminated by an appropriate phase-cycling procedure, e.g., CYCLOPS. 

Quadrature detection Preferred system of signal detection using two detection channels with reference signals offset by 90°. 

From the practical point of view, dual-channel phase detectors operated in quadrature appear to be the best hardware-detection choice. Unlike other techniques, phase detection is sensitive to RF frequency offset from resonance and to the RF phase which, on the one hand, makes the signals more complex to use (as well as and more sensitive to instrument instabilities) but, on the other hand, leads to a number of important advantages, such as ... 

One can avoid overlap in the remote dimension if the carrier was moved away sufficiently from the spectral region of interest. In this case there will be no need for explicit quadrature detection neither in the direct, nor in the remote dimension, while pure phase character is still retained. Two-channel (e.g., quadrature) detection in the direct acquisition dimension offers, however, a sensitivity advantage of factor of /2 [23] with no extra cost in acquisition time, so it is worthwhile to retain. 

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. 

Both FIDs are acquired with the same t value, and both are encoded with the same frequency 2a in t, but they are 90° out of phase (cosine vs. sine modulation in t ), just as the real and imaginary channels of the receiver (Mx and My) are 90° out of phase. This gives us our quadrature detection in F, allowing us to put zero F audio frequency in the center of the F spectral window. 

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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