Double zero-filling

Another feature of FT is to yield better resolved frequency spectra from larger data sets, which not necessarily need to contain real data. Simply adding a row of zeros to the end of the experimental transient is beneficial in that it smoothes the peak shape by increasing the number of data points per m/z interval. This trick is known as zero-filling. The number of attached zeros normally equals the number of data points, sometimes even twice as many are filled in (double zero-filling). 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

Without zero-filling, data points in the transformed spectrum occur only at multiples of 1/T Hz, which correspond to zero-crossings of the sine function, as shown in Fig. 3.8 hence the sidelobes are not observable. Doubling the number of points by zero-filling also places points only at zero-crossings for the absorption mode. Only when the number of points is quadrupled are the sidelobes observable. 

If we take the original FID and add an equal number of zeroes to it, the corresponding spectrum has double the number of points and so the line is represented by more data points. This is illustrated in Fig. 4.14 (b). Adding a set of zeroes equal to the number of data points is called one zero filling . 

We can carry on with this zero filling process. For example, having added one set of zeroes, we can add another to double the total number of data points ( two zero fillings ). This results in an even larger number of data points defining the line, as is shown in Fig. 4.14 (c). 

In order to create more points defining the lines without any more real resolution, we can use zero filling. For example, if 8K zeroes were appended to the extant 8K points of an FID, a subsequent 16K point transform will result in 8K points in the real spectrum and they will consist of the original 4K points with an additional 4K points interpolated between them. This doubles the number of points and makes the spectrum smoother. Note that you could have digitized the FID at 16K points in the first place, but without severe apo-dization the last 8K points would have consisted only of noise. Zero filling does the same job without the noise from the last 8K points. 

Fig. 7 Comparison of alternative processing on a 3D N-NOESY-HSQC spectrum of human translation initiation factor eIF4e. (a) Uniformly sampled reference. The time domain data were acquired as 6,400 hyper-complex points sampled in the two indirect dimensions [128 (Hjndir) x 50 ( N)]. The spectra were measured on a 700-MHz spectrometer with sweep widths of 9765 Hz and 2270 Hz, respectively. The t ,ax hence were 0.013 and 0.022 s each for the indirect proton and nitrogen dimensions, respectively, representing nearly an optimal situation for the nitrogen dimension, but not for the indirect proton dimension. Data were transformed with the standard KFT procedure after cosine apodization and doubling the time domain by zero filling, (b) Reducing the number of complex points to 42 (32%) (bl) and 13 (10%) (b2) in the indirect proton dimension, cosine apodization, and zero filling result in low resolution spectra in the indirect...

It has been shown [2] that by doubling the number of data points in the time domain by appending zeros (a single zero-fill ), it is possible to improve the frequency resolution in the spectrum. The reason for this gain stems from the fact that information in the FID is split... 

Consider a microchannel filled with an aqueous solution. There is an eleetrieal doubly layer field near the interface of the channel wall and the liquid. If an electric field is applied along the length of the channel, an electrical body force is exerted on the ions in the diffuse layer. In the diffuse layer of the double layer field, the net charge density, pe is not zero. The net transport of ions is the excess counterions. If the solid surface is negatively eharged, the counterions are the positive ions. These excess counterions will move under the influenee of the... 

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