## The Fourier Transform

The raw data or FID is a series of intensity values collected as a function of time: timedomain data. A single proton signal, for example, would give a simple sine wave in time with a particular frequency corresponding to the chemical shift of that proton. This signal dies out gradually as the protons recover from the pulse and relax. To convert this time-domain data into a spectrum, we perform a mathematical calculation called the Fourier transform (FT), which essentially looks at the sine wave and analyzes it to determine the frequency. This frequency then appears as a peak in the spectrum, which is a plot in frequency domain of the same data (Fig. 3.27). If there are many different types of protons with different chemical shifts, the FID will be a complex sum of a number of decaying sine waves with different frequencies and amplitudes. The FT extracts the information about each of the frequencies: their intensities, phases and even the rate at which they decay, which determines the linewidth of each peak in the spectrum (signals that decay quickly transform into broad peaks, whereas signals that last a long time transform into sharp peaks). The method of data collection ("Bruker" sequential vs. "Varian" simultaneous) will affect the type of Fourier transform calculation you must perform. This difference is invisible if you process your data on the instrument on which it was acquired, but if you transfer data to a separate workstation and use independent processing software, you need to tell the software which kind of data you have. For example, with the Felix software package you will have to specify Bruker Fourier transform (bft) or complex Fourier transform (ft) for Bruker or Varian data, respectively.

### 3.10.1 How the FT Works

It is actually very easy to visualize how the Fourier transform works. Consider an FID with a single frequency (one peak in the spectrum). The goal of the Fourier transform is to determine the value of that frequency. First, we pick a "guess" frequency v and multiply the FID by the "test function" sin(2^vt). At each point in time we multiply the value of the FID with the value of the test function, and then we measure the area under the curve of the product:

Suppose, first of all, that we guessed right and the test function has exactly the same frequency as the FID (v = vo = 2.5 Hz). The two functions (Fig. 3.28) are completely "in sync": wherever the FID is positive the test function is positive and wherever the FID is negative the test function is also negative. The product of these two functions is thus always positive (positive x positive = positive; negative x negative = positive). For our spectrum, Product: area = 1.00

—i—i—i—i—i—i—i—i—i—i—i—i—i—i—|— 0 0.5 s 1.0 1.5 we take the area under the curve of this product function (the integral) as the intensity value for the spectrum at frequency v = vo. This maximum positive intensity (1.00) falls right at the top of our peak in the spectrum. Now consider what happens if we pick a guess frequency that is a bit lower than vo: v = 2.0 Hz (Fig. 3.29). The test function is "slower" than the FID and begins to fall "out of sync" as time progresses, so the product function starts out positive (p x p or n x n) and then goes negative (p x n or n x p). As the test function "outruns" the oscillations of the FID, the product function jumps back and forth between positive and negative. Because of the decay of the FID, greater weight is given to the earlier part, and the positive swing outweighs the negative swing, leading to a small positive total area (0.289). In frequency domain, this is down the right side of our peak. An even lower guess frequency (1.66 Hz, Fig. 3.30) leads to a faster oscillation of the product function and better cancelation of the positive and negative areas. This point (intensity 0.124) is farther down the right-hand side of the peak in frequency domain, close to the baseline. Test frequencies still farther from the FID frequency will lead to even more rapid oscillation of the product function and nearly perfect or perfect cancelation of the positive and negative areas: here we are far from the peak in frequency domain, and the intensity of spectrum(v) is zero.

The real power of the Fourier transform is the linear nature of the calculation. If we have an FID that is a sum of two different pure frequencies (like Fig. 3.16), the spectrum function looks like this:

spectrum(v) = J [FIDa (t ) + FID^ (t )]sin(2^vt )dt We can multiply the terms and separate to obtain spectrum(v) = / FIDfl(t)sin(2nvt)dt + FID^(t)sin(2nvt)dt —I—I—I—I—I—I—I—I—I—I—I—I—I—I—r~

Figure 3.30

—I—I—I—I—I—I—I—I—I—I—I—I—I—I—r~

Figure 3.30

Thus, the Fourier transform of the sum of two pure signals is just the sum of the Fourier transforms of the individual signals. The first term above (using FIDa(t)) will be nonzero only when the test frequency v is at or near va (the frequency of FIDa), and the second term will only be nonzero only when the test frequency v is at or near vb (the frequency of FIDb). This is how the Fourier transform "pulls apart" the individual frequencies that are all mixed up in the time-domain data (the FID).

The actual Fourier transform is a digital calculation, so not all frequencies are tested. In fact, the number of frequencies tested is exactly equal to the number of time values sampled in the FID. If we start with 16,384 complex data points in our FID (16,384 real data points and 16,384 imaginary data points), we will end up with 16,384 data points in the real spectrum (the imaginary spectrum is discarded). Another difference from the above description is that the actual Fourier transform algorithm used by computers is much more efficient than the tedious process of multiplying test functions, one by one, and calculating the area under the curve of the product function. This fast Fourier transform (FFT) algorithm makes the whole process vastly more efficient and in fact makes Fourier transform NMR possible.

3.11 DATA MANIPULATION BEFORE THE FOURIER TRANSFORM 3.11.1 Zero Filling

Before performing the FT, there are two things we can do to enhance the quality of the spectrum. First, the size of the data set can be artificially increased by adding zeroes to the end of the list of FID data. This process of zero filling has no effect on the peak positions, intensities, or linewidths of the spectrum, but it does increase the digital resolution (fewer hertz per data point) in the spectrum (Fig. 3.31). This can be useful to give better definition 18 12 6 0 -6 -12 -18 Hz

Low digital resolution 3.0 Hz/pt.

High digital resolution 0.3 Hz/pt.

### Figure 3.31

of peak shapes for sharp peaks. For example, you might have an FID that contains 3276 total data points (1638 pairs of real, imaginary). If you transform it directly, you will have 1638 points in your spectrum (i.e., the real spectrum). If your spectral width (SW) was 4915 Hz when you acquired the data, your spectrum will have a digital resolution of 4915/1638 or 3.00 Hz per point. A doublet with a splitting of 8 Hz would be described by only three points, so the measurement of the splitting would be very inaccurate due to the "graininess" of the spectrum (Fig. 3.31, left). If, on the other hand, you zero fill the acquired data by adding 14,746 pairs of zeroes to the data list before FT (16,384 total complex pairs), you will get a spectrum with 16,384 (16 K) data points describing the full 4915 Hz spectral window. The digital resolution is much greater (4915/16,384 = 0.300 Hz per point), and the same doublet would be described by 39 data points (Fig. 3.31, right). Zero filling is accomplished by simply defining the final data size before FT (Bruker SI, Varian FN) to a larger number than the acquired number of data points (Bruker TD, Varian NP). In the above example, you would set TD (NP) to 3276 and SI (FN) to 16,384.

### 3.11.2 Weighting or Window Functions

A more common pre-FT massaging of data is the application of a window function or weighting function. The idea is to emphasize ("weight") certain parts of the FID at the expense of others. For example, suppose that your FID signal disappears into the noise after 0.2 s, even though you acquired data up to 1.0 s. The noise from 0.2 to 1.0 s in your FID only increases the noise in your spectrum and does not contribute to the peak height, so your signal-to-noise ratio is reduced. One solution would be to simply set all the data after 0.2 s. to zero, but this introduces a sharp discontinuity in the FID at 0.2 s, which could introduce artifacts into the spectrum. A smoother method is to multiply the FID by an exponential decay function that emphasizes the early data in the FID and deemphasizes the later (mostly noise) data (Fig. 3.32). In the figure, a signal of 1 Hz linewidth is buried in noise after 0.5 s of the 2.6-s acquisition time. The "steepness" of this exponential multiplier (line-broadening parameter LB) can be varied so that it matches the natural decay of the signal (LB = 1 Hz). The net effect on the spectrum (Fig. 3.33) is that the signal-to-noise ratio is increased (from 27.8 to 56.1) and the peak is 1 Hz broader (2 vs. 1 Hz), because faster decay of the signal leads to a broader peak. The line broadening actually reduces the absolute peak height, but the reduction in noise level more than compensates for this effect. If the line-broadening effect is not a problem, the increase in S/N is usually worth the price, especially for carbon —I—I—I—1—I—I—I—I—1—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—r

Figure 3.32

—I—I—I—1—I—I—I—I—1—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—r

Figure 3.32

spectra where signal is always weak. I usually use an LB value of 0.2 Hz for proton spectra and 1.0 Hz for carbon spectra. If you have a very weak signal carbon spectrum and just want to see if there are peaks, you can use an LB of 3.0 or 5.0 Hz.

Other window functions can be used for the opposite effect: resolution enhancement (Fig. 3.34). By deemphasizing the beginning of the FID and amplifying the later part, the

 1 Hz MAT v-^/vv^j Peak height: 10.0 Noise height: 0.36 S/N ratio: 27.8 No window function 1.0 Hz linewidth
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