Nyquist frequency

It was shown in Eqs. 2.12 and 2.13 that to compute the complete spectrum from 0 to oo cm, the inteiferogram would have to be sampled at infinitesimally small increments of retardation. That is, of course, impossible, as an infinite number of data points must be collected and computer storage space would be exhausted. Even if these data could be collected, the Fourier transform would take forever to be computed. Obviously, interferograms must be sampled discretely. Just how often the interferogram should be sampled is a problem that has been solved mathematically. 

Fourier Transform Infrared Spectrometry, Second Edition, by Peter R. Griffiths and James A. de Haseth Copyright 2007 John Wiley Sons, Inc. 

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Fig. 40.8. (a) Sine function sampled at the Nyquist frequency (2 points per period), (b) An under-sampled sine function. 

Fig. 40.11. Aliasing or folding, (a) Sine of 8 Hz sampled at 16 Hz (Nyquist frequency), (b) Sine of 11 Hz sampled at 16 Hz (under-sampled), (c) A sine of 5 Hz fitted through the data points of signal (b).

Bode-Nyquist frequency domain design and Root locus... 

The filter parameters are based on the reverberation time at zero frequency and the Nyquist frequency, notated T, (0) and 7 ,(7C), respectively ... 

Let us consider the accuracy required for the sampling clock in a DSP system, van de Plassche has the following analysis If the input is full range and near the Nyquist frequency, then we have the greatest slope. Let us use a simple sinusoid as the input, i.e., V = A sin (at). The variation in the output of the converter that depends on the... 

Figure 8.7 Digital Sine function - the frequency response for a zero order hold interpolator sample rate converter with L = 4, which puts the original Nyquist frequency at 0.25 7t. We can see rolloff in the passband of about -3.9 dB and very poor rejection of images outside of the passband, which result in artifacts perceived as pitch shifting distortion.

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. 

Nyquist Frequency Highest frequency which can be accurately... 

The Nyquist frequency is the maximum frequency that can be present in the continuous sequence h(t), if h(t) is to be perfectly represented by the sampled... 

The Nyquist frequency is not only important in instrumental analysis. Consider sampling a geological core where depth relates to time, to determine whether the change in concentrations of a compound, or isotopic ratios, display cyclicity. A finite amount of core is needed to obtain adequate quality samples, meaning that there is a limitation in samples per unit length of core. This, in turn, limits the maximum frequency that can be... 

As we saw in Section 3.4, quadrature phase detection discriminates between frequencies higher and lower than the pulse frequency, but it does not prevent foldover from frequencies higher than the Nyquist frequency. For a desired spectral width FT, there are two common methods for carrying out quadrature phase detection, as was indicated in Section 3.4. One method uses two detectors and samples each detector at FT points per second, thus acquiring 2 FT data in the form of FT complex numbers. The other (commonly called the Redfield method ) requires only a single detector and samples at 2 FT points per second while incrementing the phase of the receiver by 90° after each measurement. (In two-dimensional NMR studies, a variant of this method is usually called the rime-proportional phase incrementation, or TPPI, method.) Because these methods result in quite different treatment of folded resonances, we now consider these approaches in more detail. 

FIGURE 3.6 (a) Depiction of magnetization M precessing in the rotating frame at the frequencies indicated. W/2 is the Nyquist frequency, and sampling of both phase-sensitive detectors is illustrated at t- /W and 2/ H7to obtain projections along both x and y. ... 

In this example W/2 becomes the Nyquist frequency.) Although the projections of M along the x axis at the sampling times are identical for (W/2) + 8 and (W/2) 4- 8, the projections along y are different, and these frequencies can thus be distinguished from each other. However, measurements for (W/2) + 8 and... 

FIGURE 3.6 (Continued) (b) Example of foldover of frequencies above the Nyquist frequency with two phase-sensitive detectors. The lower spectrum (spectral width 5000 Hz) faithfully reproduces the true spectrum, but the upper spectrum (spectral width 3600 Hz) shows that peaks near + 2000 Hz now display an aliased frequency near — 1800 Hz and appear near the right-hand end of the spectrum. Sample D-glucorono-6,3-lactone-l,2-acetonide in DMSO-<4 at 500 MHz. Spectra courtesy of Joseph J. Barchi (National Institutes of Health). 

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