# NMR Data Oversampling and NUTS

It is the current trend for NMR spectrometers today to “oversample” the data during acquisition. Oversampling is the act of acquiring data at a faster rate than necessary for the desired spectral window. Since the rate the data is collected is inversely proportional to the observed spectral width after Fourier Transform, then oversampling is when data is acquired for a wider spectral width than necessary for desired spectral region. To make this description easier to follow, let’s define a few shorthand terms:

- SW – the spectral region desired for observation.
- DW – the dwell time at which to acquire data for a spectral width of SW.
- ODW – oversampling dwell time which is greater than DW.
- OSW – oversampled sweep width before “decimation” which is greater than SW.

If the same number of data points were used, data collected at the ODW rate would have a lower digital resolution than would be obtained if the spectrum was collected at the DW rate.

One reason to oversample the data is that audio filters can have their cutoff frequencies set in the spectral region between SW and OSW. This leads to 1) less noise being folded back from frequencies greater than the Nyquist frequency, 2) less non-linear phase changes and 3) less amplitude distortion from filters in SW frequency range. Item 1 translates to increased signal to noise performance. Item 2 leads to spectra where all peaks can be phased with only a zero and first order phase correction. Item 3 leads to NMR spectra where integrals are less distorted.

Another more technical reason to oversample the NMR data is to “spread out” the digitizer quantization noise. When the digitizer samples a signal, it quantitates the signal in discreet steps (digitizer bits). The digitizer does this with some error often referred to as digitizer noise. In 2x oversampled data, two adjacent data points can be averaged to produce a new data point. Averaging two points to get a new data point is the same as if the digitizer has sampled at a slower rate, but also has the effect of averaging the digitizer noise (a factor of 1.4 improvement in signal to noise) and is another bit of digitizer resolution thereby reducing the step quantization error by one bit.

With the availability of faster digitizers and with cheaper memory prices, the advantages for oversampling are cost effective and desirable. How oversampling is implemented by each NMR instrument manufacturer varies, but the key steps are to include some or all of the following to some degree:

- Oversample data acquisition while applying sharp filtering at the Nyquist frequency.
- Apply Digital Quadrature detection.
- Apply digital filter.
- Decimate the data.
- Prepare the data for user consumption.

__Step 1__

Oversampling, as discussed above, is acquiring data at a faster rate than necessary for the desired spectral window. This can be done in quadrature and then Step 2 does not apply, or it can be done in the real mode (single channel) with the desired spectral window offset by some frequency, OF1. Then step 2 does apply.

__Step 2__

If the data is offset by some frequency OF1, then have the computer calculate to quadrature reference frequencies (sin OF1 and cos OF1) and “mix” them with the collected signal and low pass filter the resulting sum and difference frequencies to get the resulting “quadrature detected FID”. This requires a much faster digitizer (which usually means fewer bits per sample) and is computationally intensive. Some manufacturers do this in a dedicated DSP so that the calculation speed issue is greatly reduced.

Step 3

Calculate and apply a lowpass filter such that when the data size is reduced (decimated) there will be no foldback of spectral noise.

__Step 4__

Reduce the data by the degree of oversampling. A decimation of 2 is done by averaging each two adjacent points and generating a new point. This doubles the dwell time and halves the sweep width.

__Step 5__

Some DSP outputs are the correlation between a square wave filter and the NMR data, which is a method of doing the low pass filter. This data starts a zero and builds to a start of an FID at around 50 to 100 points. Other DSP outputs have further calculations to present a more standard FID to the user.

Last updated: 01/22/2003