Sine bell for truncation

The sine bell is a window function commonly used to enhance the appearance of truncated data, such as often recorded for multidimensional NMR spectra. The sine bell ensures the signal decays to zero and fourier transformation will not produce artifacts. Shifting the sine bell also allows some resolution enhancement to be applied.

Window functions are applied to the raw FID data to increase signal-to-noise, improve resolution, or reduce artifacts. A sine bell multiplication is often used for multidimensional NMR data. The Bruker software defines the sine bell apodisation function as follows

SINM(t) = sin((π-π/SSB)(t/AQ) + π/SSB)

where t is a point in the FID, SSB is a parameter which defines the shift of the sine bell, and AQ is the the acquisition time or length of the FID. Also available is a squared sine bell (QSIN) which is simply the sine bell function squared.

QSIN(t) = [sin((π-π/SSB)(t/AQ) + π/SSB)]²

The figure below shows several sine bells with different values of SSB. Note how all the sine bells reach zero at the end of the FID. The red sine bell with SSB=0 is not shifted at all and its first point has zero intensity. This is a very harsh resolution enhancing function and is really only used for COSY spectra to reduce the broad diagonal signals. The blue sine bell with SSB=4 starts with an intensity of ~0.7 (sin(π/4)) and is resolution enhancing, while the green sine bell with SSB=2 starts with an intensity of 1 (sin(π/2)) and is sometimes referred to as a cosine bell. Shown in yellow is a squared sine bell with SSB=2. This apodisation function shows a smooth transition from 1 for the first point of the FID to zero for the last and is the default apodisation for nearly all multidimensional spectra.

Most multidimensional NMR spectra are truncated because it it is too time consuming to record the full signal decay in the indirect dimensions. Fourier transformation of truncated data leads to sinc wiggles which appear as t₁ noise. Apodisation functions that do not scale the end of the FID to zero, like the exponential and gaussian multiplications, will produce poor quality spectra. The sine bell functions, since they scale the final points to near zero, give much better results.

The figure below shows expansions of a NOESY spectrum collected on cylcomarin A in chloroform-d processed with different window functions. All spectra are plotted with the same contour levels. The spectra in the left column show strong vertical streaks, t₁ noise, because the data is truncated. The spectra in the right column were processed with sine bell window functions and have much reduced t₁ noise. Looking closely at the second spectrum in the right column, the one processed with SSB=4, one can observe negative streaks about the diagonal because of the strong resolution enhancement.

Taking slices from the processed data gives another view of the impact of the apodisation functions. In the stackplot below rows at 1.31 ppm were extracted from the 2D spectra above. The lower three slices, where no apodisation, an exponential, and a gaussian multiplication were used, clearly show the sinc wiggles in the baseline. The upper three slices, which used sine bell windows, have much better baselines. The peaks above 4.0 ppm are mostly the result of t₁ noise, and its obvious that in the slices from the sine bell processed spectra this is much reduced.

When processing multidimensional spectra the squared cosine bell (QSIN,SSB=2) is a good first choice. If you find that your spectrum has lots of t₁ noise check that you are using the most appropriate apodisation, and if you need to resolve overlapped peaks reprocessing with a more resolution enhancing function may be worthwhile.