## An illustration of the performance of the FPT for a clinical MRS signal

We now present an illustration of the performance of the FPT for a clinical MRS signal obtained from a recording of the brain of a healthy volunteer. We use the measured time domain data (FID) acquired at the static magnetic field strength of 4T. These data of full signal length N = 2048 encoded by the group at the Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, USA [17] have been kindly made available to us. In Figure 18.1, we present absorption spectra at three signal lengths, comparing the FFT (left column) and the FPT (right column). At the top of Figure 18.1, the most dramatic difference between the FFT and FPT is seen at the shortest signal length (N/16 = 128). Here, the FFT essentially presents no meaningful spectroscopic information. In contrast, with the FPT, at (N/16 = 128) nearly 90% of the NAA concentration is predicted by the peak at around 2.0 ppm.

In the middle panel it is seen that at N/4 = 512, the FFT has still not predicted even 70% of the NAA concentration at 2.0 ppm, and the ratio between creatine and choline (3.0 and 3.3 ppm) appears to be nearly equal, and thus wrong. In contrast, with the FPT at N/4 = 512, these three major peaks are now practically identical to those at full signal length. At half signal length (N/2 = 1024) at the bottom panel, the FFT has still not demonstrated the accurate ratio between creatine and choline at 3.0 and 3.3 ppm, respectively; these two metabolites are still incorrectly appearing as almost equal. Moreover, the triplet of glutamine and glutamate near 2.3 ppm can be discerned at half signal length only by the FPT, and not by the FFT. By contrast, it is seen that at half signal length (N/2 = 1024) the FPT resolves with fidelity more than twenty metabolites, in which all peak parameters are accurately extracted, including the overlapping resonances. Furthermore, while the FFT demands the total signal length (N = 2048) to fully resolve all the metabolites, the difference between the two FPT spectra at N = 1024 and N = 2048 is buried entirely in the background noise [18]. In other words, the FPT spectra at half-signal length can be treated as fully converged.

Most importantly, as is clear from Figure 18.1, the FPT produces no spurious metabolites or other spectral artefacts in the process of converging in a strikingly steady fashion as a function of the increased signal length. Moreover, it is obvious that the FPT exhibits a much faster convergence rate than that in the FFT (for further illustrations see Ref. [6, 7, 18].

Figure 18.1 Fourier and PadÃ© absorption spectra computed using the time signal (divided by 10 000) at 3 truncated signal lengths (N/16 =1 28, N/4 = 512, N/2 = 1024), where the full signal length is N = 2048, as encoded in Ref. [17] at 4T from occipital grey matter of a health volunteer. From Ref. [11], with permission.

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Chemical shift (ppm) PADE 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) 4 3 2 Chemical shift (ppm) FOURIER The abscissa represents chemical shift in dimensionless units (ppm). The ordinates are intensities in arbitrary units (a.u.). The symbols K(F) and K (P/Q) denote the real part of the complex Fourier and Pade spectrum F and P/Q, respectively. |

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