NMR Processing: Convolution: Trapezoidal Filter

Function:
 

 
The figure shows the continuous example FID after tm with TM1 = 0.2, TM2 = 0.8, corresponding to dataset fid/110 in the download package.
The parameters TM1 and TM2 are given as fractions of the acquisition time, with TM1 < TM2, and determine at which point the linear initial ascent reaches the plateau value (TM1), or at which point the final descent starts (TM2).
The equation above the FID is my construction, and, as my math is horrible, may contain errors.

Examples/Exercises:
 

Click on the figure to see how tm with TM1 = 0.1 and TM2 = 0.9 applied to the synthetic dataset synth/114 in the download package affects the spectrum. The initial ascent serves to increase resolution, which is rather minor in the present case; the descending part of the filter works for apodization. Note the negative "trough" under the peaks.

Properties/Purpose:

• the resolution enhancement is only modest
• can be used for apodization
• may result in negative wings of the peaks -> integration is hampered
• The trapezoidal function is a rather old filter function and mentioned in many textbooks and reviews. In my opinion, its merits are the easy implementation on old computers and therefore has historical importance. However, given the Gaussian or sine-bell multiplication, I don't see any advantages in the trapezoidal function and didn't find any advantages in the more recent literature, either.

Literature:

• Hoffman, Levy, Progr. NMR Spectrosc. 1991, 23, 211; doi:10.1016/0079-6565(91)80005-M
• Lindon, Ferrige, Progr. NMR Spectrosc. 1980, 14, 27; doi:10.1016/0079-6565(80)80002-1
• Gassner, Jardetzky, Conover, J. Magn. Reson. 1978, 30, 141; doi:10.1016/0022-2364(78)90234-2