Pole-zero coincidences

The only difference between Froissart doublets for the noise-free c and noise-corrupted cn + rn time signals from Figures 4.10 and 4.11, respectively, is that the latter are more irregularly distributed than the former. This is expected due to the presence of the random perturbation rn in the noise-corrupted time signal. Flowever, this difference is irrelevant since the only concern to SNS is that noise-like or noisy information is readily identifiable by pole-zero coincidences. Note that the full auxiliary lines on each subplot in Figures 4.10 and 4.11 are drawn merely to transparently delineate the areas with Froissart doublets. 

Thus, the distances between poles z q and zeros z p are proportional to the amplitudes df. Hence, df = 0 for the exact pole-zero coincidences in the Froissart doublets (19). It is vital to have full control over the locations of all... 

Reconstructed resonances were assessed as to whether they were genuine via the concept of Froissart doublets. This was done throughout the entire Nyquist range, with special attention to the range between 1.3 and 3.3 ppm, that is, the range of interest. Of a total of 750 resonances, 741 were found to be spurious, that is, with zero amplitudes and the pole-zero coincidences. For each of the three time signals, there were nine true resonances. 

Coincidence and Dead-time Losses in y-Spectrometry. The influence of electronic effects at high-count rates on the performance of Ge(Li) detectors is considerable. The resolution of a detector can be degraded by effects within the amplification system, but these can be minimized by (i) the use of pole-zero cancellation, to prevent the pulse-height error caused by the tail of a preceding pulse and (n) baseline (or D.C.) restoration facilities to prevent similar errors caused by shifts in the apparent pulse baselines. The latter are a result of capacitative effects between the various stages of the overall amplifier, biased amplifier, and multi-channel analyser system. These effects can degrade the resolution of the detector but should not change the y-ray peak area. 

Back-reflection pinhole camera. If the incident beam is normal to the sheet specimen and therefore parallel to the fiber axis, and a projection like that shown in Fig. 9-10(b) is made (projection plane parallel to sheet), then both the incident beam and the fiber axis coincide with the center of the projection. If the texture is ideally sharp, the (hkl) pole figure will consist of one or more concentric circles centered on the center of the projection, and the chance that one of these pole circles will coincide with the concentric reflection circle is essentially zero no reflection will occur. But if the texture has enough scatter, one of the pole circles will broaden into a band wide enough to touch the reflection circle at all points a Debye ring of uniform intensity will be formed. See Prob. 9-7. Thus a uniform Debye ring is not always evidence for randomly oriented grains. 

The FPT provides the exact separation of genuine (physical) from spurious (unphysical, noise and /or noise-like) information encountered either in theory or measurements involving time signals. This is accomplished by means of Froissart doublets [16] that are coincident pairs of poles z = 2 q and zeros zfp in the response functions or complex-valued spectra... 

Note that engineers use various tricks to improve the response further. For example, they may spread the two zeros symmetrically around the LC double pole (rather than coinciding with it). One reason to put a zero (or two) slightly before the LC pole location is that the LC pole can produce a very dramatic 180° phase shift, and this can sometimes lead to conditional stability. So the zero absorbs some of that abruptness in a sense. 

There is a practical difficulty involved in using the full-blown transconductance op-amp compensation scheme discussed above — because the pole and zero from HI are not independent. They will even tend to coincide if say Rf2 is much smaller than Rfl (i.e. if the desired output voltage is almost identical to the reference voltage). In that case, the pole and zero coming from HI will cancel each other out completely. Therefore, we can t proceed anymore, because we were counting on the zero from HI to change the open-loop gain from —2, to —1, just in time before it crossed over. 

The z-domain transfer function is shown to be the ratio of two Mth-order polynomials in z, namely, JV(z) and D z). The values of z for which N z) = 0 are termed the zeros of the filter, whereas those for which D z) = 0 are the poles. The poles of such an FIR filter are at the origin, that is, z = 0, in the z plane. The positions of the zeros are determined by the weighting coefficients, that is, foji, fc = 0,1,..., M. The poles and zeros in the z plane for the simple moving average filter are shown in Fig. 8.96. The zeros, marked with a circle, are coincident with the unit circle, that is, the contour in the z plane for which... 

The main difference compared to the non-feedback case is that the adjusting procedure gives a much more accurate response of a seismometer within the passband than for the non-feedback sensor. Generally the high- and low-cutoff frequencies are extended by 2-3 octaves. In parallel with the feedback adjustment, a fine-timing of the output filter is done. In case the closed-loop response is actually flat over the passband, the device characteristics coincide with the output filter characteristics. This decreases the complexity of further device response presentation in terms of poles and zeroes. 

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