Nyquist sampling theorem

For image slicers, contiguous slices of the sky are re-arranged end-to-end to form the pseudo-slit. In that case it is obvious that the sky can be correctly sampled (according to the Nyquist sampling theorem) by the detector pixels on which the slices are projected in the same way as required for direct imaging. 

Equation (2.3.13) states the Nyquist sampling theorem. If At is set too high, the spectral width is too low, and signals at higher frequencies will appear at false positions (Fig. 2.3.9). This phenomenon is called signal aliasing [Deri]. 

Nyquist sampling theorem A law of sampling theory stating that if a continuous bandwidth-limited signal contains no frequency components higher than half the frequency at which it is sampled, then the original signal can be recovered without distortion. 

One critical mathematical preHminary for the processing of discrete-time waveforms, which originate from sampling continous-time waveforms, is the issue of sampUngrate and aliasing. The Nyquist sampling theorem can be summarized as follows ... 

Failure to obey the Nyquist sampling theorem results in aliasing, that is, the effect of frequencies greater than the Nyquist frequency are reflected into the band of frequencies between DC and one-half the sampling rate. This situation is demonstrated in Fig. 20.70, where the samples of the high-frequency sinusoid are indistinguishable from a uniform sampling of the low-frequency sinusoid. 

The continuous-time FS coefficients can also be approximated using the DFT If a continuous-time waveform x(t) of period T is sampled at a rate of Ts samples per second to obtain x(n), in accordance with the Nyquist sampling theorem, then the FS coefficients are approximated by... 

Nyquist rate To obey the Nyquist sampling theorem, a bandhmited waveform should be sampled at a rate which is at least twice the Nyquist frequency. This minimum sampling rate is known as the Nyquist rate. Failure to follow this restriction results in ahasing. 

In a coherent picture then the imaging experiment reduces to a series of measurements as a function of the wavenumber in three-dimensional space and taking the inverse Fourier transform of the NMR signal returns the real space image. At this level of approximation, the field of view and spatial resolution are given directly by the Nyquist sampling theorem. 

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