doc update

docs/processor_api/core.html

@@ -424,6 +424,7 @@ <h1>grafx.processors.core<a class="headerlink" href="#grafx-processors-core" tit
 </p>
 <p>where <span class="math notranslate nohighlight">\(h[n]\)</span> is the true infinite impulse response (IIR). Clearly, increasing the number of samples <span class="math notranslate nohighlight">\(N\)</span> reduces the error.</p>
 <p>The third one, <code class="code highlight python docutils literal highlight-python"><span class="s2">&quot;ssm&quot;</span></code>, is based on the diagonalisation of the state-space model (SSM) of the biquad filter so it only works for the second-order filters.
+This idea is based on <a class="reference external" href="https://benhayes.net/">Ben Hayes</a>’s derivation of associative scan for parallel IIR filter computation and implemented by <a class="reference external" href="https://yoyololicon.github.io/">Chin-Yun Yu</a>.
 The direct form II implementation of the biquad filter can be written in state-space form <span id="id6">[<a class="reference internal" href="../references/reference.html#id160" title="Julius Orion Smith. Introduction to digital filters: with audio applications. Volume 2. Julius Smith, 2007.">Smi07b</a>]</span> as
 <div class="math-wrapper docutils container">
 <div class="math notranslate nohighlight">
@@ -433,6 +434,7 @@ <h1>grafx.processors.core<a class="headerlink" href="#grafx-processors-core" tit
 \end{split}\]</div>
 </div>
 
+where the state transition transition matrices are given as
 <div class="math-wrapper docutils container">
 <div class="math notranslate nohighlight">
 \[\begin{split}

docs/processor_api/core.html~

@@ -424,6 +424,7 @@ h_N[n] = \sum_{m=0}^\infty h[n+mN].
 </p>
 <p>where <span class="math notranslate nohighlight">\(h[n]\)</span> is the true infinite impulse response (IIR). Clearly, increasing the number of samples <span class="math notranslate nohighlight">\(N\)</span> reduces the error.</p>
 <p>The third one, <code class="code highlight python docutils literal highlight-python"><span class="s2">&quot;ssm&quot;</span></code>, is based on the diagonalisation of the state-space model (SSM) of the biquad filter so it only works for the second-order filters.
+This idea is based on <a class="reference external" href="https://benhayes.net/">Ben Hayes</a>’s derivation of associative scan for parallel IIR filter computation and implemented by <a class="reference external" href="https://yoyololicon.github.io/">Chin-Yun Yu</a>.
 The direct form II implementation of the biquad filter can be written in state-space form <span id="id6">[<a class="reference internal" href="../references/reference.html#id160" title="Julius Orion Smith. Introduction to digital filters: with audio applications. Volume 2. Julius Smith, 2007.">Smi07b</a>]</span> as
 <div class="math-wrapper docutils container">
 <div class="math notranslate nohighlight">

src/grafx/processors/core/iir.py

@@ -1,4 +1,5 @@
 import warnings
+from functools import partial, reduce
 
 import numpy as np
 import torch
@@ -7,7 +8,6 @@
 import torch.nn.functional as F
 import torchaudio
 from torchaudio.functional import lfilter
-from functools import reduce, partial
 from torchlpc import sample_wise_lpc
 
 from grafx.processors.core.convolution import FIRConvolution
@@ -67,6 +67,7 @@ class IIRFilter(nn.Module):
         x_i[n+1] &= A_i x_i[n] + B_i s[n], \\
         y_i[n] &= C_i x_i[n] + \bar{b}_{i, 0} s[n],
         $$
+        where the state transition transition matrices are given as
         $$
         A_i = \begin{bmatrix}-\bar{a}_{i, 1} & -\bar{a}_{i, 2} \\ 1 & 0 \end{bmatrix}, \quad 
         B_i &= \begin{bmatrix}1 \\ 0 \end{bmatrix}, \quad