Algorithm 1 Fast FrFT Algorithm with Plotting for EEG Signals.

Define Parameters:

$\alpha$: The desired fractional Fourier transform angle. N: The length of EEG signal $x\left(n\right)$. K: Scaling factor.

Algorithm Steps:

1. Input: EEG signal $x\left(n\right)$ of length N and desired fractional Fourier transform angle $\alpha$ for alcoholism detection. 2. Output: EEG signal $y\left(n\right)$ in the fractional Fourier domain and a scatter plot of the real and imaginary parts of the fast FrFT coefficients. 3. Set the initial value of $y\left(n\right)$ to $x\left(n\right)$. 4. Calculate the scaling factor $K=exp(-j\pi \alpha /2)$. 5. Initialize k to 1. 6. While $k\le {log}_2\left(N\right)$ do: (a) Set FFT length to $2^k$. (b) Calculate the filter coefficients $f_k\left(m\right)$: $$f_k\left(m\right)=\sqrt{m+\frac{1}{2}}·exp\left(j\pi K\frac{{(m-N/2)}^2}{2^k}\right)$$ (2) (c) Filter the EEG signal $y\left(n\right)$ using the filter coefficients $f_k\left(m\right)$ and perform an FFT to obtain the signal in the fractional Fourier domain. (d) Multiply the signal in the fractional Fourier domain by the scaling factor $K^{2^{k-1}}$. (e) Inverse FFT the signal to obtain the EEG signal in the time domain. (f) Set $y\left(n\right)$ to the filtered EEG signal obtained in the previous step. (g) Increment k by 1. 7. Output EEG signal $y\left(n\right)$ in the fractional Fourier domain. 8. Calculate the fast FrFT coefficients $c_k$: $$c_k=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}y\left(n\right)·exp\left(-j\pi \alpha \frac{{(n-N/2)}^2}{N}\right)·exp\left(-j2\pi \frac{nk}{N}\right)$$ (3) 9. Generate a scatter plot of the real and imaginary parts of the fast FrFT coefficients $c_k$ for alcoholism detection.