Phase errors, correction zero-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. 

The antiphase doublet (Fig. 6.14(c)) is dispersive because /-coupling evolution to the antiphase state moves the vectors by 90°, from the +x axis to the +/ and —/ axes. This dispersive antiphase doublet can be phase corrected by moving the reference axis from the +x axis to the +/ axis (90° zero-order phase correction). Now the C = a peak is positive absorptive and the C = ft peak is negative absorptive (Fig. 6.15) and the central 12CH3l peak is pure dispersive because the vector is on the -hx/ axis and the reference axis is now +y (90° phase error). 

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. 

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. 

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