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4 2 0 -2 -4 4 2 0 -2 -4 4 2 0 -2 -4 4 2 0 -2 -4 Hz Hz Hz Hz

4 2 0 -2 -4 4 2 0 -2 -4 4 2 0 -2 -4 4 2 0 -2 -4 Hz Hz Hz Hz

Figure 3.35

window functions is very important for processing 2D NMR data. Figure 3.35 shows the result of Fourier transformation of a single-frequency FID with noise, after multiplication with the window functions of Figure 3.34. The Gaussian window actually increases S/N in this example because it decays during the last three quarters of the FID, but it narrows the peak because it grows during the first quarter. The unshifted sine-bell window narrows the peak even more (from 1.0 to 0.60 Hz), but the peak shape is distorted (prominent negative "ditches" appear on either side) and the S/N is cut in half. The exponential multiplier (LB = 1.0) gives a doubling of signal-to-noise ratio in this example (Fig. 3.35, right). The effect of these window functions on S/N depends greatly on the decay rate of the signal in the raw FID and the acquisition time: If AQ (AT) is long relative to the FID decay, we are acquiring mostly noise in the later part of the FID, and any window that significantly reduces this part of the FID will result in a dramatic S/N improvement. In this case it might be better, however, to just reduce the acquisition time.

Bruker uses the command EM (exponential multiplication) to implement the exponential window function, so a typical processing sequence on the Bruker is EM followed by FT or simply EF (EF = EM + FT). Varian uses the general command wft (weighted Fourier transform) and allows you to set any of a number of weighting functions (lb for exponential multiplication, sb for sine bell, gf for Gaussian function, etc.). Executing wft applies the window function to the FID and then transforms it.

3.12 DATA MANIPULATION AFTER THE FOURIER TRANSFORM 3.12.1 Phase Correction

After you Fourier transform your FID, you get a frequency-domain spectrum with peaks, but the shape of the peaks may not be what you expected. Some peaks may be upside down, whereas others may have a "dispersive" (half up-half down) lineshape (Fig. 3.36). The shape of the peak in the spectrum (+ or — absorptive, + or — dispersive) depends on the starting point of the sine function in the time-domain FID (0° or 180°, 90° or —90°). The starting point of a sinusoidal function is called its "phase." Phase errors come in all possible angles, including those intermediate between absorptive and dispersive (Fig. 3.37). The spectrum has to be phase corrected ("phased") after the Fourier transform to obtain the

Figure 3.36

desired "absorptive" (0o error) peak shape. Phasing corrects certain unavoidable instrumental errors involved in acquiring the FID.

Recall that the raw NMR data (FID) consists of two numbers for each data point: one real value and one imaginary value. After the Fourier transform, there are also two numbers for each frequency point: one real and one imaginary. In a perfect world, the real spectrum would be in pure absorptive mode (normal peak shape) and the imaginary spectrum would be in pure dispersive (up/down) mode. In reality, each spectrum is a mixture of absorptive and dispersive modes, and the proportions of each can vary with chemical shift (usually in a linear

Phase errors Figure 3.37

Phase correction

Dispersive i L Imaginary

Real Figure 3.38

fashion). To correct for this, we calculate a linear combination of the real and the imaginary spectra, and use this for our "phased" spectrum (Fig. 3.38). For the mathematically inclined, the actual linear combination is

Absorptive spectrum = (real spectrum) x cos(0) + (imaginary spectrum) x sin(0)

The angle 0 can be thought of as a rotation of the two mutually perpendicular vectors representing the real and imaginary spectra. The problem of phase correction boils down to finding the correct phase rotation angle 0. Well, actually it is a little more complicated because the phase correction 0 is usually a linear function of the chemical shift (8). Defining the line

0(8) = (m x 8) + b requires that you determine two parameters: the intercept b (called the zero-order phase correction) and the slope m (called the first-order phase correction). All phasing routines are based on optimizing these two numbers.

Consider a hypothetical spectrum with six equally spaced peaks (Fig. 3.39, top). There is a chemical shift dependent (linear) phase error that makes the phase error grow by 450 with each peak as we move from left to right. The phase correction process starts with choosing a large peak at one end of the spectrum as the "pivot" peak: this is the peak that is defined as 8 = 0 for the purposes of phase correction. In Figure 3.39, we choose the rightmost peak, which has a phase error of 1350. The phase of this pivot peak is optimized by varying the intercept (b) value (the zero-order phase correction) until the pivot peak is perfectly absorptive. This correction applies equally to all of the peaks in the spectrum, regardless of chemical shift, subtracting 1350 from the phase error of each of the six peaks. Then another peak is chosen (without moving the pivot) at the other end of the spectrum and its phase is optimized by adjusting the slope (m) parameter (Fig. 3.39, center). The phase correction applied to each peak is determined by the vertical position of the line as it goes through that chemical shift. The key to this method is that changing the slope has no effect on the "pivot" peak because the line goes through zero at this chemical shift. The actual zero of the chemical shift scale is not important—this is just for the purpose of the phase calculation. With both parameters set, the line is defined and all peaks in between should also be correctly phased (Fig. 3.39, bottom). Because the dependence on chemical shift should be linear, correcting both ends of the spectrum should make the whole spectrum

Dispersive i L Imaginary

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