## Ft

Spectrum with "sine wiggles;

Truncated FID

### Figure 9.48

the FID by adding zeros at the end (sort of like extending soup by adding water). If we extend our FID of 256 complex data points to 1024 by adding 768 pairs of zeros, we will have 1024 complex pairs in our spectrum after the Fourier transform and 1024 data points in our real spectrum after phase correction. The digital resolution is now 3.5 Hz (0.006 ppm) per data point; not as good as a 1D spectrum but plenty of detail for showing crosspeaks and even some of the larger J couplings. The amount of zero filling is determined by the Varian parameters fn (Fourier number) and fn1 (Fourier number in F1) and the Bruker parameters si(F2) and si(F1) for "size." In either case, the FID is zero filled from the original number of complex data points acquired (Bruker: td/2; Varian: np/2 in F2 and ni in F1) to the final matrix size (Bruker: si/2; Varian fn/2 in F2 and fn1 /2 in F1). By now you can see that Bruker usually uses the same parameter names for F2 and F1 and identifies them by placing them in different columns of a parameter display; Varian adds a "1" to the F2 parameter name to generate the F1 parameter name.

The second problem is that the t1 FID makes a large "jump" from a finite signal at 71 ms to zero because we ran out of time (or patience) sampling it. This sudden discontinuity in the time domain data can be viewed as multiplying our FID by a rectangular window function that is one while we are sampling the FID and falls suddenly to zero after 71 ms. The effect on the spectrum, in frequency domain, is that our peaks get "wiggles" at the base that extend far out in either direction, upfield and downfield from the peak. In a 2D spectrum, these wiggles will appear as intense streaks of alternating positive and negative intensity extending above and below the crosspeaks and diagonal peaks (Fig. 9.49). The sinc artifacts extend far away from the crosspeak because the sinc function (sin v/v) decays as 1/v, just like the dispersive peak and the magnitude mode peak. The Fourier transform does not like sudden and radical changes in time domain!

We can understand this effect precisely by applying the convolution theorem, which says that multiplying the FID by a function has the effect of "convoluting" the spectrum with the Fourier transform of that function. Convolution is the process of moving a multiplier function from left to right through a digital dataset, stopping at each alignment of the data points, multiplying the data by the multiplier function and adding up all the products to get a single number at each stop. This set of numbers is the result of the calculation: the "convolution" of the multiplier function and the data (Fig. 9.50). The multiplier function in this case is the Fourier transform of a rectangular "pulse" function ("on" from t1 = 0 to t1 = 71 ms). We saw in Chapter 8 that the result is a "sinc" function (sin v/v) that has a separation of 1/0.071 s = 14 Hz at the base of the central peak. Now we slide this function by our spectrum, which might be a single NMR peak. As the wiggles pass by the peak, we will get alternating positive and negative intensities that increase as the central peak of

the sine function approaches the NMR peak. As it passes through the NMR peak we get a large positive intensity, and after that we get alternating positive and negative intensity of decreasing amplitude (Fig. 9.48).

What we need to do is to smooth the transition from a finite FID to zero, which will have the effect of "calming down" the wiggles in frequency domain. For this purpose we need a multiplier function that goes smoothly to zero at the end of the FID data. Two commonly used window functions that accomplish this are the sine-bell and the cosine-bell functions (Fig. 9.51). The cosine-bell (or "90o-shifted sine-bell") function starts at the maximum (sine of 90o) at the beginning of the FID and goes smoothly to zero (sine of 180o) at the end of the acquired data. This window function is commonly used for 2D experiments with low-intensity crosspeaks that require sensitivity enhancement such as NOESY and ROESY. Because the function gives greater weight to the beginning of the FID where the signal-to-noise ratio is greater, the sensitivity is enhanced at the expense of the resolution. The sine-bell (or "unshifted sine-bell") function starts at the zero point of the sine function (sine of 0o) at the start of the FID data, reaches a maximum halfway through the FID (sine of 90o) and falls back smoothly to zero by the end of the acquired FID data (sine of

180°). Because data later in the FID (in the center) is emphasized over data early on in the FID, this window function leads to resolution enhancement at the expense of sensitivity (signal-to-noise ratio). It is commonly used for COSY data where the peaks are antiphase and will "self-cancel" if they are broad (Fig. 9.37). This radical resolution enhancement was encountered in Chapter 2 for 1D spectra, where we saw the resulting "ditches" on either side of the peaks (Fig. 2.9). In the DQF-COSY spectrum of sucrose these ditches are clearly visible in the F2 slices (Fig. 9.28). In Figure 9.35 the DQF-COSY spectrum was processed with an unshifted sine-bell (left side), but the COSY-35 was processed with a simple exponential multiplier to facilitate accurate curve-fitting (right side). The difference in peak resolution is clearly visible in the 2D spectra, and in the F2 slices of the COSY-35 we see no ditches. Less radical resolution enhancement can be achieved by shifting the sine bell by 30° or 45° (Fig. 9.51), always keeping the 180° point of the sine function at the end of the FID. These window functions are commonly used for 2D experiments with strong crosspeaks (efficient magnetization transfer) such as 2D TOCSY.

The size of the window must be carefully fit to the FID being processed. Varian uses the parameter sb to describe the width (in seconds) of the sine-bell window from the 0° point to the 90° point. Thus for an unshifted sine-bell function, we want the 0° to 180° portion of the sine function (2 sb) to just fit over the time duration of the FID (at). This is accomplished by setting the value of sb to one-half the acquisition time: sb = at/2. Since the sine-bell is not shifted, the "sine-bell shift" (sbs) is set to zero. For a cosine-bell or 90° shifted sine-bell window, we want the portion of the sine function from 90° to 180° (or sb, since the 0° to 90° portion is of the same duration as the 90° to 180° portion) to just fit over the FID (duration at): sb = at. In addition, the whole sine function is shifted to the left side by the duration of the FID, so we set the parameter sbs (sine-bell shift) equal to -at (left shift corresponds to a negative number). In F1 we do not have a parameter for acquisition time (at) in t1, but we know that the maximum t1 value is just the number of data points times the sampling delay:

i1(max) = (ni x 2) x At1 = (ni x 2) x (1/(2 x sw 1)) = ni/sw1

So you can just set sbl = ni/sw1 and sbsl = -sbl for a 90o-shifted sine-bell, and sb1 = ni/(2 x swl) and sbsl = 0 for an unshifted sine-bell. Bruker uses the parameter wdw (in both F1 and F2) to set the window function (SINE = sine-bell, QSINE = sine-squared, etc.) and ssb for the sine-bell shift. For example, if ssb = 2, the sine function is shifted 90o (180o/ssb) and we get a simple cosine-bell window. For an unshifted sine-bell, use ssb = 0.

### 9.7.3 Phase Correction in Two Dimensions

Phase errors appear in 2D spectra as "streaks" with negative intensity on one side and positive intensity on the other side. Vertical streaks correspond to Fi phase errors and horizontal streaks to F2 phase errors (Fig. 9.52). For example, if positive intensity is color-coded red and negative intensity blue, an F2 phase error will appear as a crosspeak or diagonal peak with a red streak extending out to the left side and a blue streak extending out to the right side, or vice versa. Sometimes there are severe phase errors in both dimensions, leading to a pattern of horizontal and vertical streaks (Fig. 9.52, upper right).

The complex FID, consisting of real and imaginary parts, is converted by a complex Fourier transform to a complex spectrum, consisting of a real spectrum and an imaginary spectrum. Phase correction involves "rotating" the complex spectrum in the complex plane until the real spectrum is absorptive and the imaginary spectrum is dispersive (Chapter 3, Fig. 3.38). The imaginary spectrum is then discarded and we use the real (absorptive) spectrum. In 2D processing there are two Fourier transforms: one in t2/F2 and one in t1/F1. Each one generates two spectra, so we can potentially end up with four 2D matrices (Fig. 9.53). Phasing a 2D matrix would then involve forming a linear combination of all four final 2D spectra to get absorptive lineshape in both dimensions. Regardless of the software you are using, you are looking for four numbers: the phase correction parameters in F2 (zero-order and first-order) and the phase correction parameters in F1 (zero-order and first-order). The zero-order correction is applied equally to all peaks in the spectrum and the first-order parameter is a linear function of chemical shift, going through zero at the "pivot peak." The process is based on phase correction of 1D "slices": make an F2 (horizontal) slice through a peak (diagonal or crosspeak) in the 2D spectrum and phase correct it as a 1D spectrum to generate the F2 phase correction parameters (Fig. 9.52). Then make an F1 (vertical slice)

F(FuF2) Real,Real

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