piptouque / atiam_project_audio

Normalising a gaussian

Not actually used, doesn't work.

Our discretised Gaussian $(g_k)_{k \in [|0, n|]}$, $\sim N(\mu_d, \sigma_d^2)$.

We want: $$ \begin{equation} \begin{aligned} g_{\frac{n}{2}} &= \frac{1}{\sigma_d \sqrt{2 \pi}}, \ g_0 &= \frac{\epsilon_n}{\sigma_d \sqrt{2 \pi}}. \end{aligned} \end{equation} $$

With $\epsilon_n \neq 0$ chosen such that $\epsilon_n \to_{n \to \infty} 0$.

$$ \begin{equation} \begin{aligned} e^{-\frac{(\frac{n}{2} - \mu_d)^2}{2\sigma_d^2}} &= 1, \\ e^{-\frac{\mu_d^2}{2\sigma_d^2}} &= \epsilon_n. \end{aligned} \end{equation} $$

We find: $$ \begin{equation} \begin{aligned} \mu_d &= \frac{n}{2}, \ \sigma_d^2 &= \frac{n^2}{4 \ln(\epsilon_n)}. \end{aligned} \end{equation} $$

We can then normalise the gaussian with $(\mu_d, \sigma_d^2)$.