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| -rw-r--r-- | content/blog/fourier-noise.md | 8 | ||||
| -rw-r--r-- | layouts/partials/katex.html | 6 |
2 files changed, 8 insertions, 6 deletions
diff --git a/content/blog/fourier-noise.md b/content/blog/fourier-noise.md index e17c119..8e1a114 100644 --- a/content/blog/fourier-noise.md +++ b/content/blog/fourier-noise.md @@ -83,11 +83,11 @@ $$\begin{aligned}dx_rdx_i &= \|{J_{x_r, x_i}(\alpha, \theta)}\| d\alpha d\theta Therefore, the integral now becomes the following. Note that for the distribution of $A$ we allow $\theta$ across its entire range. The value of $a$ also has to remain positive, therefore, the lower limit is changed from $- \infty$ to $0$. -$$\begin{equation}P_A(a) = \int_0^{2\pi} \int_0^a \frac{1}{2\pi\sigma_n^2} exp(\frac{-\alpha^2}{2\sigma_n^2}) \alpha d\alpha d\theta\end{equation}$$ +$$P_A(a) = \int_0^{2\pi} \int_0^a \frac{1}{2\pi\sigma_n^2} exp(\frac{-\alpha^2}{2\sigma_n^2}) \alpha d\alpha d\theta$$ I leave the simplification of this to the reader but this results in the following distribution: -$$\begin{equation}P_A(a) = 1 - exp(\frac{-a^2}{2\sigma_n^2}) \end{equation}$$ +$$P_A(a) = 1 - exp(\frac{-a^2}{2\sigma_n^2})$$ This makes the Probability Distribution Function (PDF) of $A$ to be: @@ -99,10 +99,10 @@ This is a well know distribution called the [Rayleigh Distribution](https://en.w The similar derivation for the phase can be done and shows that the Cummulative Distribution Function (CDF) is: -$$\begin{equation}P_\Phi(\phi) = \frac{\phi}{2\pi}\end{equation}$$ +$$P_\Phi(\phi) = \frac{\phi}{2\pi}$$ which makes the Probability Distribution Function (PDF): -$$\begin{equation}p_\Phi(\phi) = \frac{1}{2\pi}\end{equation}$$ +$$p_\Phi(\phi) = \frac{1}{2\pi}$$ This means that the phase is uniformly distributed across the the interval $(0, 2\pi)$. diff --git a/layouts/partials/katex.html b/layouts/partials/katex.html index 69c9302..57e5c1a 100644 --- a/layouts/partials/katex.html +++ b/layouts/partials/katex.html @@ -2,11 +2,13 @@ <style> .katex-display > .katex { max-width: 100%; - overflow-x: scroll; + overflow-x: auto; + overflow-y: hidden; } </style> <script defer src="https://cdn.jsdelivr.net/npm/katex@latest/dist/katex.min.js"></script> -<script defer src="https://cdn.jsdelivr.net/npm/katex@latest/dist/contrib/auto-render.min.js" onload="renderMathInElement(document.body);"> +<script defer src="https://cdn.jsdelivr.net/npm/katex@latest/dist/contrib/auto-render.min.js" onload="renderMathInElement(document.body);"></script> +<script> document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: [ |