Frequency-dependent phase

Figure 6.10 Frequency-dependent phase shifts and demodulations (bottom panel) for IR-I44 (open symbols) and DOTCI (solid symbols), together with the best single exponential lit to each dataset. The residuals for the 1R-144 data set are depicted in the upper panel. Reproduced from Ref. 25 with permission.

Because the initial emphasis of this study was on extending ACC to liquid-fueled combustors, a simple closed-loop controller, which had been well tested in the previous studies involving gaseous fuel, was utilized. Such a controller, however, may not be effective in a combustor where the oscillation frequencies drift significantly with the control. The main problem was the frequency-dependent phase shift associated with the frequency filter. For such a case, it would be more useful to employ an adaptive controller that can rapidly modify the phase setting depending on the shift in the dominant oscillation frequencies. 

Coherent light sources are characterized by a spectral intensity distribution E(oj) and a frequency-dependent phase 0(w). According to first-order perturbation theory, linear absorption probabilities are given by the overlap between the spectrum of the light source E(w) and the optical transition, and are independent of the phase function In non-linear processes (2nd... 

If the penetration depth of the light is much smaller than the film thickness, illumination of the nanocrystalline electrode from the electrolyte side is expected to give a characteristic spiral in the high frequency IMPS response (Fig. 8.29). This spiral is typical for a constant time lag (i.e., frequency dependent phase shift), and it arises simply from the transit time required for carriers to move from the front face to the rear contact. By contrast, if the electrode is illuminated through the substrate, electrons are generated close to the contact and the transit time is much smaller (Fig. 8.29). This is reflected in the high value of o)min. 

In general, any sequence which introduces a delay between the pulse(s) and the beginning of data collection will produce this type of phase error. Inevitably, there is always a short delay between the end of the pulse and the beginning of acquisition, but if the delay is only a few microseconds then the frequency-dependent phase error is less than 360° and is easily removed. However, in the spectrum above, the delay is 14 ms so that frequency differences of several kHz will introduce phase errors of thousands of degrees these are virtually impossible to remove with currently available phasing algorithms. 

The frequency-dependent phase errors seen above are the reason why virtually all modem pulse sequences contain refocussing Tt-pulses in the middle of lengthy evolution periods. 

For those spins further from resonance, the angle 0 becomes greater and the net rotation toward the x-y plane diminishes until, in the limit, 0 becomes 90 . In this case the bulk magnetisation vector simply remains along the -f-z-axis and thus experiences no excitation at all. In other words, the nuclei resonate outside the excitation bandwidth of the pulse. Since an off-resonance vector is driven away from the y-axis during the pulse it also acquires a (frequency dependent) phase difference relative to the on-resonance vector (Fig. 3.6). This is usually small and an approximately linear function of frequency so can be corrected by phase adjustment of the final spectrum (Section 3.2.8). 

Figure 3.39. First-order (frequency dependent) phase errors arise from a dephasing of magnetisation vectors during the pre-acquisition delay which follows the excitation pulse. When data collection begins, vectors with different frequencies have developed a significant phase difference which varies across the spectrum.

Fig. 4.8 Illustration of the appearance of a frequency dependent phase error in the spectrum. In (a) the line which is on resonance (at zero frequency) is in pure absorption, but as the offset increases the phase error increases. Such an frequency dependent phase error would result from the use of a pulse whose RF field strength was not much larger than the range of offsets. The spectrum can be returned to the absorption mode, (c), by applying a phase correction which varies with the offset in a linear manner, as shown in (b). Of course, to obtain a correctly phased spectrum we have to choose the correct slope of the graph of phase against offset.

The usual convention is to express the frequency dependent phase correction as the value that the phase takes at the extreme edges of the spectrum. So, for example, such a correction by 100° means that the phase correction is zero in the middle (at zero offset) and rises linearly to +100° at on edge and falls linearly to —100° at the opposite edge. 

A phase correction has to be applied for two reasons The zero-order phase correction PHCO arises because of the phase difference between the receiver and the detection pulse. Additional frequency dependent phase deviations arising from chemical shift evolution in the short delay between the last pulse and the signal detection can be compensated by a first-order phase correction PHCl . 

It is important to realize that this approximately linear dependence of the phase angle on the offset is an intrinsic property of the experiment and has nothing to do with other mechanisms which can give rise to phase shifts such as instrumental delays in filters or inappropriate choice of the time origin in the Fourier transform process. In fact, since all these effects are approximately linear in x, they are usually lumped together and treated as one effect. Since the choice of the time origin is arbitrary and is entirely under the experimenter s control, all contributions to such frequency dependent phase shifts can be cancelled by an appropriate choice of the time origin and this is done quite often in FT of very wide lines. 

In FT NMR, it is not trivial to set the detector phase so that it will be set to the absorption mode for all lines in the spectrum. In addition to a phase offset error due to improper setting of the detector for the absorption mode, there will be frequency dependent phase changes across the spectrum. Lines at the start of the spectrum might appear absorption-like... 

The receiver system may also introduce phase shifts. The most basic cause for frequency dependent phase shifts, though, is that a finite rotates magnetizations at different offsets by different amounts (see II.A.2.). 

A symptom of the above problem appears when we try to phase correct the spectrum after the Fourier transformation. In the absence of this problem, the phase correction required is independent of the interval between pulses. However, with this problem the projection of the magnetizaton on the x -y plane might lead or lag the rotating y axis depending on its frequency in the spectrum and the time between pulses. So we experience an additional frequency dependent phase shift which depends on the interval between pulses in the range of intervals comparable to or less than T. ... 

Phase shifters range from simple delay lines to mechanically or electronically controlled delay lines to those based on phase locked loops or other active circuits. Quadrature and 180° hybrids as well as DBM s can be used as fixed value broadband phase shifters. This last point is important since most continuously adjustable phase shifters are frequency dependent. Phase shifters in NMR spectrometers are required to set the reference phase in a phase sensitive detector and also to set the relative phases of rf pulses as required for various pulse sequences. 

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