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Real Imaginary Real Imaginary

We will see that in 2D NMR, the sampling in the second dimension can also be done either way, except that this choice is up to the user and is not "hard wired." The alternate ("Bruker-like") sampling method is called "TPPI" (time proportional phase incrementation), and the simultaneous ("Varian-like") method is called "States" or "StatesHaberkorn" (after the originators of the technique). The consequences for processing and interpretation of the data are the same in the second dimension of 2D spectra as they are in 1D NMR.

This leads to confusion over two parameters: the number of points collected (do you mean the total number of data points, or the number of real/imaginary pairs or "complex" pairs?) and the time spacing between data points. Both Bruker and Varian list the number of data points (Varian: NP, Bruker: TD) as the total number of points, counting both real and imaginary. Some independent NMR software packages (e.g., Felix) count points as "complex pairs": One "point" corresponds to one pair of numbers (real and imaginary). The time spacing between successive data points sampled in the FID is called the dwell time (Bruker: DW). In the above example, the dwell time is clearly equal to 80 ^s between samples for the Bruker data, but in the Varian case we need to think of the dwell time as the average time per sample, which is still 80 ^s because two samples are collected in a period of 160 ^s. Varian does not have a parameter corresponding to dwell time, leaving the sampling process hidden from the user. The two types of data (alternate and simultaneous) must be processed by a different Fourier transform algorithm, but this is transparent as long as you process the data on the instrument that acquired it. If you transfer the data to another computer and use "third party" software (e.g., Felix, MestRec, Acorn-NMR, NMR-pipe, etc.) to process it, you need to choose the correct Fourier transform method for the type of FID data (alternating or simultaneous) being processed.

How rapidly do we need to sample the data? Clearly this is limited by how fast the hardware can convert analog to digital, but in most cases this limitation is not serious. It turns out that the rate of sampling is determined by the highest frequency signal you need to describe by the digitized data. In other words, what peak in your spectrum is farthest from the center frequency (the reference frequency)? For the sake of simplicity, consider a spectrometer from the middle ages that does not use quadrature detection, so that the audio frequency scale runs from zero on the right to the maximum detectable audio frequency (Fmai) on the left. The highest frequency signal needs to be sampled at least twice during each cycle of its sine wave, meaning that the number of samples has to be twice the number of cycles in the highest frequency signal allowed.

Number of samples in 1 s = 1 /DW = 2 x number of cycles in 1s 1/DW = 2 X Fmax

Once we have chosen a particular dwell time DW, the maximum frequency we can accurately determine (since the computer does not know anything about the what the signal does in between the samples) is 1/(2 x DW). What happens if the frequency of a signal exceeds 1/(2 x DW)? The signal will not simply disappear; instead it is misinterpreted as a signal of lower frequency. For example, if the dwell time is 1/6 of a second, we will get six samples in 1 s from a signal of 1.5 Hz for a total of four samples per cycle, describing the sine wave quite accurately (Fig. 3.14, top). If we keep the sampling rate constant and increase the frequency to 3 Hz, we now have two samples per cycle, which is the minimum to describe its frequency: one sample at each trough and one sample at each crest of the wave (Fig. 3.14, middle). The frequency is now equal to Fmax = 1/(2 x DW) = 1/(2 x (1/6)) = 3 Hz, and the peak will appear in the spectrum at the left edge of the spectral window. If we now increase the frequency of the FID signal to 4.5 Hz, we have 1.33 samples per cycle, which is not sufficient to describe the sine wave and accurately determine its frequency (Fig. 3.14, bottom). Instead, a simpler interpretation would be to connect the dots to reveal a different sine wave of frequency 1.5 Hz, since we do not know what is going on between samples. Thus the peak would appear in the center of the spectrum, at Fobs = Fmax — (F — Fmax) = 3 — (4.5 — 3) = 3 — 1.5 = 1.5 Hz. This process is called "aliasing" or "folding" because the peak appears at the wrong position in the NMR spectrum. Anyone who has watched Western movies or television shows has seen the phenomenon of aliasing. A film (or videotape) of a moving stagecoach will often show the wheels slowing, coming to a

stop, or reversing direction even though the stagecoach is still obviously moving forward at full speed. The film is sampling the position of the spokes at a rate of 30 frames (samples) per second. If the wheels move fast enough, the motion of the spokes exceeds the sampling rate and we interpret the motion as being at a lower frequency than it really is. If this occurs in an NMR spectrum, we need to increase the sampling rate (decrease the dwell time DW) until we have two or more samples per cycle of the aliased frequency. Usually the aliased peak can be identified because it is lower in intensity and cannot be correctly phased.

The limits of frequency imposed by a fixed sampling rate lead directly to the concept of the "spectral window" (Fig. 3.15). In the case of quadrature detection, the center of the window is the zero point of audio frequency, which is determined by the reference frequency. The width of the spectral window is called the spectral width (SW), which is determined by the sampling rate and corresponds to Fmax in the nonquadrature example. The extremes of the spectral window are +SW/2 at the left edge and — SW/2 at the right edge, and we can replace Fmax with SW in the equation: SW = 1/(2 x DW). The spectral window can be moved to the left or right by adjusting the offset (Bruker: O1; Varian: TO), which changes the exact value of the reference frequency. The offset frequency (in hertz) is added to the fundamental resonance frequency for the nucleus of interest to obtain the reference frequency. For example, a 250 MHz instrument set up for proton acquisition might have a fundamental 1H frequency of 250.13 MHz. Adding an offset (O1) of 10,000 Hz (0.01 MHz) would yield a reference frequency of 250.14 MHz. To move the spectral window downfield by 1 ppm (250 Hz), one would simply add 250 Hz to the offset value (O1), changing the value of this parameter from 10,000 to 10,250.

Why would you need to move the spectral window upfield or downfield? The lock system changes the magnetic field strength of the spectrometer (Bo) slightly to center the 2H frequency of the solvent at the null point of the lock feedback circuit. Changing the field changes all of the resonant frequencies of the spectrum by the same amount, effectively moving the whole spectrum upfield or downfield by as much as 5 ppm when you

change from one deuterated solvent (e.g., CDCI3) to another (e.g., d6-acetone). If this is not corrected by changing the offset by an equal and opposite amount, the spectrum will move out of the spectral window and some peaks will be aliased. For routine work, this hassle has been removed in two ways. On the old Bruker (AM) instruments, you had to have a list of O1 values for different solvents in order to keep the center of the spectral window at the same value (e.g., 5 ppm) for all solvents. On the Varian, the correction is made automatically by entering the lock solvent as the parameter "SOLVNT." This changes the fundamental resonance frequency so that the offset (TO) is always the same for a given ppm value at the center of the window. This can be frustrating if you neglect to change the SOLVNT parameter for solvents other than the default setting (e.g., CDCl3). The newer Bruker instruments (DRX) use a parameter in the lock system called lock shift, which is the ppm value of the lock solvent (for example, 7.24 for CDCl3), and this corrects the reference frequency internally. If you use the automatic lock and specify the lock solvent, this parameter is automatically set to the correct value. Sometimes the spectral window needs to be changed for unusual samples with chemical shifts outside the standard (for example, 11 ppm to -1 ppm for *H) spectral window. If you have a carboxylic acid with an OH resonance at 13 ppm, you would like to have a spectral window from -1 to 17 ppm. That means you need to increase the spectral width by 6 ppm (from 12 ppm to 18 ppm) and move the center of the spectral window downfield (to higher frequency) by 3 ppm (from 5 ppm to 8 ppm). On a 200 MHz instrument that would mean adding 6 x 200 = 1200 Hz to the spectral width (SW) parameter and adding 3 x 200 = 600 Hz to the transmitter offset (TO or O1) parameter. You will have to repeat the acquisition, of course, because these parameters have no effect, except at the time that the FID is acquired.

With quadrature detection, the range of audio frequencies detected runs from +SW/2 to —SW/2, with zero in the center. The same relationship exists between the maximum frequency detectable and the dwell time, except that we substitute SW for Fmax:

The last equation tells us what value of the dwell time we have to use to establish a particular spectral width. In practice, the user enters a value for SW and the computer calculates DW and sets up the ADC to digitize at that rate. It is important to understand that with the simultaneous (Varian-type) acquisition mode, there is a wait of 2 x DW between acquisition of successive pairs of data points. The average time to acquire a data point (DW) is the total time to acquire a data set divided by the number of data points acquired whether they are acquired simultaneously or alternately. The spectral window is fixed once the sampling rate and the reference frequency have been set up. The spectral window must not be confused with the "display window," which is simply an expansion of the acquired spectrum displayed on the computer screen or printed on a paper spectrum (Fig. 3.15, bottom). The display window can be changed at will but the spectral window is fixed once the acquisition is started.

Any peak outside the spectral window will be aliased ("folded") into the spectral window at a position the same distance from the edge of the window. Aliased peaks are usually reduced in intensity (by the audio filter) and impossible to correctly phase; increasing the spectral width will eliminate them and reveal the peak in its correct position. The manner of aliasing depends on the type of acquisition. With the "Bruker-type" acquisition (alternating acquisition of real and imaginary data samples), aliased peaks appear reflected at equal distance from the same edge of the spectral window ("folded"), as shown in Figure 3.15 (upper left). With the "Varian-type" acquisition (simultaneous acquisition of a real, imaginary pair of samples), aliased peaks appear at equal distance inside the opposite edge of the spectral window ("aliased"). The terms "folding" and "aliasing" are often used interchangeably, but it would be more accurate to use "folding" for the alternating mode (reflecting across the nearest edge of the spectral window) and "aliasing" for the simultaneous mode (appearing at the same distance inside the other edge of the spectral window as the frequency is from the nearest edge).

The same phenomenon applies to aliasing in the second dimension of a 2D spectrum: Alternating (TPPI) acquisition in the second dimension will lead to aliasing on the same side of the spectral window ("folding"); simultaneous (States) acquisition will lead to aliasing from the opposite edge of the spectral window ("aliasing").

An example of the real part of an actual audio frequency FID of a sample of chloroform and dichloromethane (recorded on a Varian Gemini-200) is shown in Figure 3.16. The full real FID (acquisition time (AT) = 2.9 s) is shown in the inset at the upper right: a decaying oscillating signal is clearly visible with a frequency of about 34 cycles per 2.9 s = 11.7 Hz. A horizontal expansion of the first 0.43 s (below the inset) makes it clear that there are two different frequencies, with their oscillating signals added together in the FID. The slower (lower frequency) oscillation completes one cycle in 87.5 ms, corresponding to a CH2Cl2 frequency of 11.5 Hz (1/0.0875 s), whereas the faster (higher frequency) signal completes a cycle in 2.67 ms, corresponding to a CHCl3 frequency of 375 Hz (1/0.00267 s). Figure 3.17 shows a horizontal expansion of the first 36 ms of the real FID, capturing one-half cycle of the 11.5 Hz signal and 13.5 cycles of the 375 Hz signal. Now we can see the "grain" of the actual digital samples: one every 360 |xs (0.36 ms) for a dwell time of 180 |xs, since there are two data points, real and imaginary, for each point shown. The NMR software draws straight lines to connect the data points (□), but in fact, we know nothing about the signal in between these samples. One cycle of the highest frequency signal (375 Hz) corresponds to

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