Special Topic Oversampling and Digital Filtering

For a typical spectral width of 6250 Hz (12.5 ppm for a H spectrum on a 500 MHz spectrometer), the sampling rate is 12,500 samples per second. For a 13C spectrum the spectral width is larger, so that a 13C spectrum with a 250 ppm spectral width on a 600 MHz spectrometer would require a sampling rate of 

A wider filter function gives a narrower frequency response. Consider a digital filter that is four points wide, with all four values given equal weight. A frequency of 4 x SW is 

The filter is then moved to the next position and the weighted average is again calculated (Fig. 3.24, bottom). The value for point 8 is 

This process of moving the filter function through the raw data and calculating weighted averages is called convolution, and the digitally filtered data d, d2, 3,. .. are called the 

The convolution theorem states that the Fourier transform of the convolution (d) is simply the product of the Fourier transforms of the two functions (c and r) that are combined by convolution to make d. Thus convolution in time domain is equivalent to simple multiplication in frequency domain  

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