Phase errors, correction first-order

The XY problem gives rise to a constant phase error across the spectrum, the delay problem gives a linear phase error. To correct for this, we have two phase adjustment parameters at our disposal zero and first order. 

A phase difference between the carrier frequency and the pulse leads to a phase shift which is almost the same for all resonance frequencies (u)). This effect is compensated for by the so-called zero-order phase correction, which produces a linear combination of the real and imaginary parts in the above equation with p = po- The finite length of the excitation pulse and the unavoidable delay before the start of the acquisition (dead time delay) leads to a phase error varying linearly with frequency. This effect can be compensated for by the frequency-dependent, first-order phase correction p = Po + Pi((o - (Oo), where the factor p is frequency dependent. Electronic filters may also lead to phase errors which are also almost linearly frequency-dependent. 

Figure 2-12 spectra showing zero- and first-order phase errors, (a) The spectrum with frequency-independent (zero-order) phase errors, (b) The spectrum with frequency-dependent (first-order) phase errors, (c) The correctly phased spectrum. 

Here rjji = exp(—27T l j/N). Since the f s and tit s obey bosonic commutation relations up to corrections 0(1/N), one sees from (28c) that Z = 0 + 0(1 /N), i.e., within the bosonic quasi-particle approximation, the action of a phase flip Zi cannot be calculated. However one can draw the conclusion that a single-atom phase error only contributes in first order of 1/N. From the other equations one recognizes an important property if we assume that the initial state Wo is an ideal storage state, i. e., without bright polariton excitations, we find that after tracing out the bright polariton states only decoherence contributions of order 0(1/IV) survive, e.g.,... 

It has already been mentioned in Section 3.2 that the phase of a spectrum needs correcting following Fourier transformation because the receiver reference phase does not exactly match the initial phase of the magnetisation vectors. This error is constant for all vectors and since it is independent of resonance frequencies it is referred to as the zero-order phase correction (Fig. 3.38). Practical limitations also impose the need for a frequency-dependent or first-order phase correction. Consider events immediately after the... 

Typically both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye and the first-order correction is then... 

In real experiments after Fourier transformed the lineshapes are mixtures of absorptive and dispersive signals and are related to the delayed FID acquisition (first-order phase error). The delayed acquisition is a consequence of the minimum time required to change the spectrometer from transmit to receive mode, during this delay the magnetization vectors process according to their chemical shift frequencies. The zero-order phase error arises because of the phase difference between the magnetization vectors and the receiver. In NMR-SIM the delayed acquisition is not necessary because the ideal spectrometer approach does not require any switching time and the first order phase correction is normally zero if no other sources of phase deviations are present. 

Typically, both forms of error occur in a spectrum directly after the FT. The procedure for phase correction is essentially the same on all spectrometers. The zero-order correction is used to adjust the phase of one signal in the spectrum to pure absorption mode, as judged by eye , and the first-order correction is then used to adjust the phase of a signal far away from the first in a similar manner. Ideally, the two chosen resonances should be as far apart in the spectrum as possible to maximise the frequency-dependent effect. Experimentally, this process of phase correction involves mixing of the real and imaginary parts of the spectra produced by the FT process such that the final displayed real spectrum is in pure absorption mode whereas the usually unseen imaginary spectrum is pure dispersion. 

First-order phase correction has one insidious effect, baseline distortion. A frequency-dependent phase shift cannot make up for the data that were lost during d the baseline error is just the DFT of the missing data. However, provided d is small compared to A, this baseline curvature can easily be corrected during postprocessing of the spectrum. In 2D... 

The main consequences are twice. First, it results in contrast degradations as a function of the differential dispersion. This feature can be calibrated in order to correct this bias. The only limit concerns the degradation of the signal to noise ratio associated with the fringe modulation decay. The second drawback is an error on the phase closure acquisition. It results from the superposition of the phasor corresponding to the spectral channels. The wrapping and the nonlinearity of this process lead to a phase shift that is not compensated in the phase closure process. This effect depends on the three differential dispersions and on the spectral distribution. These effects have been demonstrated for the first time in the ISTROG experiment (Huss et al., 2001) at IRCOM as shown in Fig. 14. 

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