Relation between the FID and the NMR spectrum

NMR users can deal with spectrum evaluation in the daily work, but how is the spectrum information stored in the time domain (FID)? The origin of the NMR signal comes from the application of a radiofrequency pulse that moves the net magnetization from the equilibrium (along the main magnetic field) to the xy plane. We will consider that the magnetization follows a cosine trajectory (Figure 1a) but with a reduction in intensity over time (Figure 1b) due to the relaxation processes (T₁ and T₂).

Figure 1: (a) Representation of a cosine wave; (b) Representation of an FID.

Esvan and Zeinyeh^1,2 published an interactive way to understand how the Fourier transformation works, converting the FID information into the spectrum. We will use this excellent teaching tool to demonstrate an FID composed of only two frequencies (2 non-equivalent protons) for simplicity, with values of 1 and 5 Hz (Figure 2). The FID will be the sum of all frequencies, and the Fourier transformation will need to deal with it.

Figure 2: (a) Representation of both cosine waves (1 and 5 Hz); (b) Representation of the FID containing both frequencies (1 and 5 Hz). This figure was reproduced from the Esvan and Zeinyeh² teaching tool with author's permission to be used in this blog.

To extract the frequencies embedded in the FID, the computer will run the Fourier transformation (FT), which is a multiplication of the FID by cosine functions with increasing frequencies, covering all the frequency range of your spectral width. Esvan and Zeinyeh call these cosine functions "Trial Frequencies". 

Case 1: Using a trial frequency that matches with a frequency in the FID, such as 1 Hz: the multiplication of both functions will always lead to a positive value since the trial frequency, and the FID will be positive or negative at the same time, as both have the same frequency. In this case, the total area under the multiplication curve will always be positive (Figure 3a). Also, as the trial frequency has an amplitude equal to 1, the intensities of the original FID are preserved (Figure 3b).

Figure 3: a) The multiplication of the trial frequency (pink) with the same frequency embedded in the FID (blue) will lead to an area with positive values, and b) as the trial frequency has an amplitude equal to 1, the intensity information from the FID is preserved. This figure was reproduced from the Esvan and Zeinyeh² teaching tool with author's permission to be used in this blog.

Case 2: If the trial frequency doesn't correspond to any embedded frequency in the FID, the multiplication curve will regularly oscillate between positive and negative values. The total area under the curve will be close to zero (Figure 4).

Figure 4: The multiplication of trial frequency (pink) with a frequency not embedded in the FID (blue) will lead to a curve containing positive and negative values. The area under the curve will be close to zero. This figure was reproduced from the Esvan and Zeinyeh² teaching tool with author's permission to be used in this blog.

Each of these multiplications (trial frequency and FID) corresponds to a point in the spectrum with its respective intensity. This process is performed very quickly by the computer and will originate the well-known NMR spectrum. The animation in Figure 5 is a visual way to understand the whole process; the integral of the multiplication area will vary according to the region in the spectrum, and its point-by-point plot will generate the final spectrum.

Figure 5: Animation of the multiplication of many trial functions by the FID and generation of the spectrum. The spectral width used for this purpose was in the range from 0 to 11 Hz. This figure was reproduced from the Esvan and Zeinyeh² teaching tool with author's permission to be used in this blog.

[1] Esvan, Y. J. and Zeinyeh, W. J. Chem. Educ. 2020, 97, 263–264
[2] https://sites.google.com/view/chimie-analytique-pharma-lyon/web-apps
*Authors of the web-application have no affiliation with Nanalysis
[3] Keeler, James, Understanding NMR spectroscopy, 2^nd edition, chapter 5, pp. 78-80.