This report compares various simulation and data analysis methods for free induction decay (FID) signals in Nuclear Magnetic Resonance (NMR) Spectroscopy. The methods discussed include discrete fast Fourier transformation (FFT), least squares fitting (LSF), short-time Fourier transformation (STFT), and wavelet transformation. NMR is a widely used technique for elucidating the chemical composition and structure of molecules. It measures the nuclear magnetic spin frequencies of different atoms in a sample, producing signals that are waves over the time domain. This report employs Gaussian wave frequency simulations to model the interference and dephasing among adjacent frequencies of each Gaussian chemical shift. Typically, FIDs are analysed using the FFT algorithm, although least squares fitting and machine learning algorithms are also employed. The simulations indicate that STFT can generate spectrograms that are useful for further neural network training, while LSF is effective in processing signals around or even below one wave cycle.