I have not read the whole article. But, what is shown at the beginning is not the Fourier Transform, it is the Discrete Fourier Transform (DFT).

Though the DFT can be implemented efficiently using the Fast Fourier Transform (FFT) algorithm, the DFT is far from being the best estimator for frequencies contained in a signal. Other estimators (like Maximum Likelihood [ML], [Root-]MUSIC, or ESPRIT) are in general far more accurate - at the cost of higher computational effort.