From 911a15a75a7905d1cbe92cded006986948f689aa Mon Sep 17 00:00:00 2001
From: ascpixi <44982772+ascpixi@users.noreply.github.com>
Date: Mon, 23 Dec 2024 01:23:22 +0100
Subject: [PATCH] fix(docs): fix LaTeX rendering in README

---
 readme.md | 44 ++++++++++++++++++++++----------------------
 1 file changed, 22 insertions(+), 22 deletions(-)

diff --git a/readme.md b/readme.md
index f96b7e7..823b746 100644
--- a/readme.md
+++ b/readme.md
@@ -8,43 +8,43 @@ In an HFTNN forward transform, the following steps are performed:
 
 1. Given a discrete signal $x[n]$, the signal is partitioned into non-overlapping, windowed segments of length $N$. Each segment is denoted as $x_{\text{chunk}}[k]$, where $k$ is the segment index.
 
-   $$
-   x_{\text{chunk}}[k] = x[n], \quad \text{for } n = kN \text{ to } (k+1)N - 1
-   $$
+$$
+x_{\text{chunk}}[k] = x[n], \quad \text{for } n = kN \text{ to } (k+1)N - 1
+$$
 
 2. The Fourier transform of each segment $x_{\text{chunk}}[k]$ is denoted as $X_{\text{chunk}}[k]$.
 
-   $$
-   X_{\text{chunk}}[k] = \mathcal{F}\left(x_{\text{chunk}}[k]\right)
-   $$
+$$
+X_{\text{chunk}}[k] = \mathcal{F}\left(x_{\text{chunk}}[k]\right)
+$$
 
 3. For each Fourier transform bin, harmonics are selected within a specified range of octaves, including neighboring bins. Let $\text{HFT}$ be the resulting array.
 
-   $$
-   \text{HFT}[j] = \left[\text{magnitude}, \text{phase}\right]
-   $$
+$$
+\text{HFT}[j] = \left[\text{magnitude}, \text{phase}\right]
+$$
 
    Here, $j$ represents the index iterating over the selected harmonics and neighboring bins.
 
 4. The frequency corresponding to each harmonic is calculated using the function $\text{freq}(j)$. The corresponding FFT bin index is denoted as $\text{binIdx}(j)$.
 
-   $$
-   \text{freq}(j) = \text{intervalToFreq}\left(j, \text{fundamental}, \text{frequenciesInOctave}\right)
-   $$
+$$
+\text{freq}(j) = \text{intervalToFreq}\left(j, \text{fundamental}, \text{frequenciesInOctave}\right)
+$$
 
-   $$
-   \text{binIdx}(j) = \text{freqToFftBin}\left(\text{freq}(j), \text{sampleRate}, \text{signalLength}\right)
-   $$
+$$
+\text{binIdx}(j) = \text{freqToFftBin}\left(\text{freq}(j), \text{sampleRate}, \text{signalLength}\right)
+$$
 
 5. To handle edge cases where the computed bin index $\text{binIdx}$ and neighboring indices $\text{idx}$ may exceed array bounds, a conditional statement is used.
 
-   $$
-   \text{HFT}[j] = 
-   \begin{cases} 
-   [0, 0], & \text{if } \text{idx} < 0 \text{ or } \text{idx} \geq \text{bins.length} \\
-   [\text{magnitude}, \text{phase}], & \text{otherwise}
-   \end{cases}
-   $$
+$$
+\text{HFT}[j] = 
+\begin{cases} 
+[0, 0], & \text{if } \text{idx} < 0 \text{ or } \text{idx} \geq \text{bins.length} \\
+[\text{magnitude}, \text{phase}], & \text{otherwise}
+\end{cases}
+$$
 
 A similar algorithm is employed for the inverse temporal transform.