Nyquist interval

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. 

Fig. 5 Object (shaded) used in computer simulations Its diffraction image is the solid curve The data image (dashed curve) is the diffraction image plus 4% amplitude random noise All plotted points are spaced by one-half the Nyquist interval. Hence to resolve the central dip in the object would require superresolution.

The image data were formed by adding 4% (of peak image value) Gaussian noise to the signal image. Sampling was at one-half the Nyquist interval so as to permit superresolution of the central dip in the object. 

The experimental results mainly obtained by EIS supported the mechanism of the build up of an oxide passive layer. They lead to the determination of quantitative parameters related to the change of surface reactivity. The measurements constituted a series of impedance diagrams obtained at successive time intervals. Examples given in Eig. 11 represent [12] the time variation of the Nyquist diagram resulting from the build up of an insulating layer, after reaction ofSClonan initially bare hydrophobic... 

Nyquist, L., Lindstrom, D., Mittlefehldt, D. et al. (2001c) Manganese-chromium formation intervals for chondrules from the Bushunpur and Chainpur meteorites. Meteoritics and Planetary Science, 36, 911-938. 

For a finite sampling interval A<5, more than one superposition of cosine/sine waves can give rise to the recorded interferogram. For the transformed spectrum to be unique, the sampling interval A<5 must be sufficiently small to detect modulations in the interferogram due to the shortest wavelength present in the spectrum, the so-called Nyquist criterion [66] ... 

Figure 4. Three-dimensional representation of the time-resolved data near 8 = 0 as they appear following one interferometric scan. The sampling interval employed was 1.2656 jxm, corresponding to a Nyquist wavenumber of 3950.7 cm 1. Selection of an interferogram at any time delay following the photolysis laser pulse is possible, and is shown here for t = 150/is. Reproduced with permission from Ref. 37.

The sampling time At must be sufficiently short to avoid aliasing from signal intensity at frequencies above the Nyquist frequency (0Ny = n I At, which are folded back into the frequency interval -0)Ny < (0 < 0)Ny As a rule, At < Tm( /10 should be fulfilled, where Tmin is the shortest diffusion time constant. 

A = sampling interval (e.g., time between samples) The Nyquist critical frequency or critical angular frequency is 1 K... 

Normally a FID is acquired using simultaneous quadrature detection, sequential quadrature detection is explained in section 2.3.1. The FID consists of the values of the X- and y-components of the magnetization measured at discrete intervals during the acquisition period T q. The time interval between two successive pairs of data points is called the dwell time For a given (chosen) spectral width SW [Hz], a total number of data points TD and acquisition time T q, the following relation is valid (Nyquist theorem) ... 

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