Nyquist limit

Answer The spectral width is too narrow to allow the Nyquist limit ( 1.3.1) to be satisfied for all the frequencies in the spectrum and the methyl signals are folded into the window. On spectrometers that use a different version of the Fourier transformation, the aliased data may appear at the other end of the frequency spectrum but will still be out of phase with the rest of the signals. Clearly, the spectral width needs to be increased. 

Figure 11 Data sampling. The solid line represents a wave that is sampled at a rate sufficient to identify its frequency. The dashed line represents a wave with a frequency that is above the Nyquist limit. The frequency of this wave cannot be properly identified and will be aliased as having a frequency equal to that of the solid line.

Vice versa, the Nyquist limit frequency, Vnyq, up to which a process can just be correctly detected is given as... 

Another phenomenon that is worth remembering here is that partials falling beyond the Nyquist limit (see Chapter 1) fold over and reflect into the lower portion of the spectrum. 

To make the phase angle plot, we simply use the definition of ZGp(joo). As for the polar (Nyquist) plot, we do a frequency parametric calculation of Gp(jco) and ZGp(joo), or we can simply plot the real part versus the imaginary part of Gptjco).1 To check that a computer program is working properly, we only need to use the high and low frequency asymptotes—the same if we had to do the sketch by hand as in the old days. In the limit of low frequencies,... 

Consider first the corrosion of low alloy steel in HC1 per se, i.e. before the addition of organic inhibitors. As shown in Figures 1 and 2 for N80 steel in 15% and 28% HC1 at 65 C, Nyquist plots for steel in concentrated HC1 typically have only one distinct feature a single capacitance loop (a loop above the Z axis) with a hint of a second capacitance loop at lower frequencies. The low-frequency loop is more fully developed in 28% HC1 than in 15% HC1. Mass transport limitations are not evident except under extreme conditions, e.g. above 28% HC1 and 65 C. 

Harper CL (1996) Evidence for Nb in the early solar system and evaluation of a new p-process cosmochronometer from Nb/ Mo. Astrophys J 466 437-456 Harper CL, Wiesmann H, Nyquist LE, Hartmann D, Meyer B, Howard WM (1991) Interpretation of the Ti- Zr anomaly correlation in CAI NNSE Zr production limits and S/ R/ P decomposition of the bulk solar system abundances. Lunar Planet Sci XXII 517-518... 

There is some critical value of gain at which the G, B plot goes right through the (—1, 0) point. This is the limit of closedloop stability. See Fig. 13.3e. The value of K, at this limit should be the ultimate gain that we have dealt with before in making root locus plots of this system. We found in Chap. 10 that = 64 and Let us see if the frequency-domain Nyquist stability... 

To better understand the diffusion-limited school of thought mentioned above, it is worth digressing momentarily on another noble -metal electrode system silver on YSZ. Kleitz and co-workers conducted a series of studies of silver point-contact microelectrodes, made by solidifying small (200—2000 //m) silver droplets onto polished YSZ surfaces. Following in-situ fabrication, the impedance of these silver microelectrodes was measured as a function of T (600-800 °C), P02 (0.01-1.0 atm), and droplet radius. As an example. Figure 9a shows a Nyquist plot of the impedance under one set of conditions, which the authors resolve into two primary components, the largest (most resistive) occurring at very low frequency (0.01—0.1 Hz) and the second smaller component at moderately low frequency ( 10 Hz). 

Fig. 1 Representative a Nyquist and b Bode plots from electrochemical impedance spectroscopy measurements (HubrechtJ (1998) Metals as Biomaterials, Helsen J, Breme H (eds) John WUey Sons Limited. Reproduced with permission)...

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

The Fourier frequency bandpass of the spectrometer is determined by the diffraction limit. In view of this fact and the Nyquist criterion, the data in the aforementioned application were oversampled. Although the Nyquist sampling rate is sufficient to represent all information in the data, it is not sufficient to represent the estimates o(k) because of the bandwidth extension that results from information implicit in the physical-realizability constraints. Although it was not shown in the original publication, it is clear from the quality of the restoration, and by analogy with other similarly bounded methods, that Fourier bandwidth extrapolation does indeed occur. This is sometimes called superresolution. The source of the extrapolation should be apparent from the Fourier transform of Eq. (13) with r(x) specified by Eq. (14). 

Therefore, the Nyquist plot must dip down into the third quadrant. If is does, there are two values of gain that represent the limits of closedloop stability. The maximum... 

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