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<h1 id="introduction-to-fixed-point-signal-processing">Introduction to Fixed Point Signal Processing</h1>
<p>
Posted on Wed 09 December 2020 in <a href="./category/signal-processing.html">signal-processing</a>
</p>
</header>
<div>
<div class="section" id="introduction">
<h2>Introduction</h2>
<p>In 1927, Alan Turing’s paper discussed a class of numbers called <em>finite
and computable numbers</em>. This allowed for the construction of digital
computer. Signal processing, in essence, is the study of converting
real-world signal into finite and computable numbers, and processing
them with the fewest bits possible.</p>
</div>
<div class="section" id="what-are-fixed-point-numbers">
<h2>What are fixed-point numbers?</h2>
<ul class="simple">
<li>Over the years, computers had different encoding schemes for
representing and processing numerical data<ul>
<li><strong>Binary coded decimal</strong>: Still used today for financial
calculations, inefficient.</li>
<li><strong>Binary one’s complement</strong>: Used by early computers like <a class="reference external" href="https://en.wikipedia.org/wiki/Apollo_Guidance_Computer">Apollo
Guidance
Computer</a>.
Very rare to see a computer with this numeric encoding for
integers. However, <a class="reference external" href="https://en.wikipedia.org/wiki/IEEE_754">IEEE
754</a> floating point
encoding are indeed one’s complement.</li>
<li><strong>Binary two’s complement</strong>: Most common implementation for
integers for all major architectures, including x86, ARM, RISC-V
and many DSP</li>
<li><strong>Floating point</strong>: Makes writing math expressions easy, there are
several standards and extension. <a class="reference external" href="https://en.wikipedia.org/wiki/IEEE_754">IEEE
754</a>, <a class="reference external" href="https://en.wikipedia.org/wiki/Unum_(number_format)#Unum_III">Posit Unum
III</a></li>
<li><strong>Bignum</strong>: Arbitrary precission arithmetic, used for
cryptography, mathmatics, etc.</li>
<li>There are otheres: Logarithmic numbers, Computer algebra system
(CAS), etc.</li>
</ul>
</li>
<li><strong>Fixed-point</strong> in this context means binary two’s complement numbers
with scaling factor for fractions.</li>
</ul>
</div>
<div class="section" id="its-the-2020s-do-we-still-need-fixed-point-arithmetic-for-signal-processing">
<h2>It’s the 2020’s, do we still need fixed point arithmetic for signal processing?</h2>
<ul class="simple">
<li>Floating point machines are more common now, but that was not the
case all the time.</li>
<li>FPUs (a.k.a math coprocessor) were expensive and slow!<ul>
<li>Intel 8087 costed around <a class="reference external" href="https://books.google.com/books?id=0C4EAAAAMBAJ&lpg=PA27&ots=z8RRGv5UvJ&dq=intel%20personal%20computer%20enhancement%208087%20price&pg=PA27#v=onepage&q=intel%20personal%20computer%20enhancement%208087%20price&f=false">$375 in
1985</a>,
<a class="reference external" href="https://www.usdinflation.com/amount/375">$915.15 in 2020 USD</a></li>
<li>FPU was uncommon in desktop computers (extra headache for
programmers)</li>
</ul>
</li>
<li>Most programs were written without the support of FPU</li>
<li>In the past, fixed point signal processing was mainstream<ul>
<li>Speak & Spell used
<a class="reference external" href="https://en.wikipedia.org/wiki/Texas_Instruments_LPC_Speech_Chips">TMS5100</a>
for speech synthesis (70’s)</li>
<li>AT&T’s used
<a class="reference external" href="https://www.computerhistory.org/siliconengine/single-chip-digital-signal-processor-introduced/">DSP1</a>
for call routing (80’s)</li>
<li>NeXTcube and ProTools used <a class="reference external" href="https://www.computerhistory.org/siliconengine/single-chip-digital-signal-processor-introduced/">Motorola
56001</a>
DSP to process image and audio (90’s)</li>
</ul>
</li>
<li>To answer the question: It depends on various costs and benefits</li>
</ul>
</div>
<div class="section" id="pros-and-cons">
<h2>Pros and cons</h2>
<ul class="simple">
<li>On one hand, doing signal processing in floating point offers:<ul>
<li>Greater dynamic range</li>
<li>Less development time</li>
<li>Precision</li>
</ul>
</li>
<li>Fixed point is still best at:<ul>
<li>Minimizing cost of parts: Less logic gates are needed to operate on
integers than floating point</li>
<li>Efficient power utilization</li>
<li>More bandwidth utilization (less bits)</li>
</ul>
</li>
</ul>
</div>
<div class="section" id="where-are-fixed-point-arithmetic-done-today">
<h2>Where are fixed point arithmetic done today?</h2>
<ul class="simple">
<li>Your phones modem: 3G, LTE, and now 5G, where FPGAs are involved for
software defined radios</li>
<li>Wearable electronics: BlueTooth earbuds, smart watch, smart glasses,
etc.</li>
<li>Physical layer network communication: Fiber-optic, Satellite, etc.</li>
<li>Special purpose accelerators: Convolutional neural network engines,
audio/video codec, image processors</li>
<li>Operating systems kernel (i.e. Linux, XNU): Cannot execute floating
point math in kernel-mode in the Linux Kernel (Well, you can, but
it's a lot harder)</li>
</ul>
</div>
<div class="section" id="getting-started-with-fixed-point-arithmetic">
<h2>Getting Started with Fixed-point Arithmetic</h2>
<ul class="simple">
<li>Know the Q notation</li>
<li>The Q notation convention describes binary fixed point number scaling
and bits</li>
<li>Written as Q<strong>m</strong>.<strong>n</strong>, where <strong>m</strong> is the number of bits
reserved for integer portion and **n* is the number of bits
reserved for fractional portion.</li>
<li>Sometimes <strong>m</strong> is not indicated in the notation if there are no bits
reserved for integer bits or sign bit<ul>
<li>For example <tt class="docutils literal">Q15</tt> is used to decribe the format of 16-bit PCM
format used by CDs (you know… those shiny disks). <tt class="docutils literal">Q15</tt> can also
be written as <tt class="docutils literal">Q1.15</tt>, where there are 15 bits reserved for
fraction and 1 bit for sign.</li>
<li>CD audio has a range of <tt class="docutils literal"><span class="pre">[-1.0,</span> +0.999969482421875]</tt></li>
<li>What happened to +1.0?</li>
</ul>
</li>
<li>Sometimes <strong>U</strong> is used to describe unsigned<ul>
<li>8-bit PCM samples tend have a bias (<span class="math">\(s + 127\)</span>) in their
format, typically <tt class="docutils literal">U8</tt> is used to describe that format.</li>
</ul>
</li>
<li>Sometimes <tt class="docutils literal">S16</tt> is used to represent <tt class="docutils literal">Q1.15</tt> as well</li>
</ul>
<div class="section" id="how-to-get-the-range-of-values-for-a-certain-fixed-point-type">
<h3>How to get the range of values for a certain fixed-point type:</h3>
<div class="math">
\begin{equation*}
Q(m,n) \in \mathbb{R} : -2^{m-1} \leq Q(m,n) \leq 2^{m-1} - 2^{-n}
\end{equation*}
</div>
</div>
<div class="section" id="what-is-the-resolution-of-a-certain-fixed-point-type">
<h3>What is the resolution of a certain fixed-point type:</h3>
<div class="math">
\begin{equation*}
2^{-n}
\end{equation*}
</div>
<p>What is the resolution for a 32-bit floating point type? Extra credit.</p>
<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">qrange</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="sd">"""Get numeric range of _signed_ Qm.n integers</span>
<span class="sd"> Args:</span>
<span class="sd"> m: Number of bits for integer portion</span>
<span class="sd"> n: Number of bits for fraction portion</span>
<span class="sd"> Returns:</span>
<span class="sd"> tuple(min_value, max_value)</span>
<span class="sd"> """</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">ldexp</span>
<span class="n">min_value</span> <span class="o">=</span> <span class="n">ldexp</span><span class="p">(</span><span class="o">-</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">max_value</span> <span class="o">=</span> <span class="n">ldexp</span><span class="p">(</span><span class="o">+</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">m</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">-</span> <span class="n">ldexp</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span> <span class="o">-</span><span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">min_value</span><span class="p">,</span> <span class="n">max_value</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="how-to-convert-decimal-to-fixed-point">
<h2>How to convert decimal to fixed point</h2>
<p>Given <span class="math">\(Q(m, n)\)</span>, for a given value <span class="math">\(x\)</span>, quantized fixed
point value, <span class="math">\(y\)</span> is:</p>
<div class="math">
\begin{equation*}
y = \lceil2^{n}x\rfloor
\end{equation*}
</div>
<p>May want to saturate the results so they don’t overflow in fixed point:</p>
<div class="math">
\begin{equation*}
\text{clip}\left(x, m, n\right) := \max\left(-2^{m - 1}, \min\left(x, 2^{m - 1} - 2^{-n}\right)\right)
\end{equation*}
</div>
<p>So final expression:</p>
<div class="math">
\begin{equation*}
y = \lceil2^{n}\text{clip}\left(x, m, n\right)\rfloor
\end{equation*}
</div>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">clip</span><span class="p">,</span> <span class="n">ldexp</span><span class="p">,</span> <span class="n">rint</span>
<span class="k">def</span> <span class="nf">ftoq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="sd">"""Convert a float value to fractional integer</span>
<span class="sd"> Args:</span>
<span class="sd"> x: A double/float value</span>
<span class="sd"> m: Bits for integer portion</span>
<span class="sd"> n: Scaling factor from Q notation</span>
<span class="sd"> Returns:</span>
<span class="sd"> np.array_like(int)</span>
<span class="sd"> """</span>
<span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span> <span class="o">=</span> <span class="n">qrange</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">rint</span><span class="p">(</span><span class="n">ldexp</span><span class="p">(</span><span class="n">clip</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">),</span> <span class="n">n</span><span class="p">)))</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">qtof</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="sd">"""Convert a fractional integer to float</span>
<span class="sd"> Args:</span>
<span class="sd"> x: A double/float value</span>
<span class="sd"> m: Number of bits for integer portion</span>
<span class="sd"> n: Number of bits for fraction portion</span>
<span class="sd"> Returns:</span>
<span class="sd"> np.array_like(float)</span>
<span class="sd"> """</span>
<span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span> <span class="o">=</span> <span class="n">qrange</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
<span class="k">return</span> <span class="n">clip</span><span class="p">(</span><span class="n">ldexp</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="o">-</span><span class="n">n</span><span class="p">),</span> <span class="n">vmin</span><span class="p">,</span> <span class="n">vmax</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">vectorize</span>
<span class="n">vftoq</span> <span class="o">=</span> <span class="n">vectorize</span><span class="p">(</span><span class="n">ftoq</span><span class="p">)</span>
<span class="n">vqtof</span> <span class="o">=</span> <span class="n">vectorize</span><span class="p">(</span><span class="n">qtof</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">vquantize</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="n">vqtof</span><span class="p">(</span><span class="n">vftoq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">),</span> <span class="n">m</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">n</span><span class="si">:</span><span class="s2"> 1.08f</span><span class="si">}</span><span class="s2"> -> </span><span class="si">{</span><span class="n">ftoq</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span><span class="si">:</span><span class="s2"> 6d</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">(</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.707</span><span class="p">,</span> <span class="mf">0.99999999</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">"</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">n</span><span class="si">:</span><span class="s2"> 6d</span><span class="si">}</span><span class="s2"> -> </span><span class="si">{</span><span class="n">qtof</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span><span class="si">:</span><span class="s2"> 1.08f</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">(</span><span class="mi">16384</span><span class="p">,</span> <span class="mi">23168</span><span class="p">,</span> <span class="mi">32767</span><span class="p">)))</span>
</pre></div>
<pre class="literal-block">
0.50000000 -> 16384
0.70700000 -> 23167
0.99999999 -> 32767
16384 -> 0.50000000
23168 -> 0.70703125
32767 -> 0.99996948
</pre>
</div>
<div class="section" id="converting-from-to-fixed-point-in-c-c">
<h2>Converting from/to fixed-point in C, C++</h2>
<p>Interface:</p>
<div class="highlight"><pre><span></span><span class="cm">/* frac_convert.h */</span><span class="w"></span>
<span class="cp">#ifndef FRAC_CONVERT_H</span>
<span class="cp">#define FRAC_CONVERT_H</span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qminv</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">);</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qmaxv</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">);</span><span class="w"></span>
<span class="kt">long</span><span class="w"> </span><span class="nf">ftoq</span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">);</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qtof</span><span class="p">(</span><span class="kt">long</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">);</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">clip</span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">vmin</span><span class="p">,</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">vmax</span><span class="p">);</span><span class="w"></span>
<span class="cp">#endif </span><span class="cm">/* FRAC_CONVERT_H */</span><span class="cp"></span>
</pre></div>
<p>Implementation:</p>
<div class="highlight"><pre><span></span><span class="cm">/* frac_convert.c */</span><span class="w"></span>
<span class="cp">#include</span><span class="w"> </span><span class="cpf"><math.h></span><span class="cp"></span>
<span class="kt">long</span><span class="w"> </span><span class="nf">ftoq</span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="kt">long</span><span class="p">)(</span><span class="n">rintl</span><span class="p">(</span><span class="n">ldexp</span><span class="p">(</span><span class="n">clip</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">qminv</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">),</span><span class="w"> </span><span class="n">qmaxv</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">)),</span><span class="w"> </span><span class="n">n</span><span class="p">)));</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qtof</span><span class="p">(</span><span class="kt">long</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">clip</span><span class="p">(</span><span class="n">ldexp</span><span class="p">((</span><span class="kt">double</span><span class="p">)</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="n">n</span><span class="p">),</span><span class="w"> </span><span class="n">qminv</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">),</span><span class="w"> </span><span class="n">qmaxv</span><span class="p">(</span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="n">n</span><span class="p">));</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qmaxv</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">one</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mf">1.0</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">ldexp</span><span class="p">(</span><span class="n">one</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">ldexp</span><span class="p">(</span><span class="n">one</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="n">n</span><span class="p">);</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">qminv</span><span class="p">(</span><span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="p">,</span><span class="w"> </span><span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="p">(</span><span class="kt">void</span><span class="p">)(</span><span class="n">n</span><span class="p">);</span><span class="w"> </span><span class="cm">/* unused parameter */</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">ldexp</span><span class="p">(</span><span class="mf">-1.0</span><span class="p">,</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="kt">double</span><span class="w"> </span><span class="nf">clip</span><span class="p">(</span><span class="kt">double</span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">vmin</span><span class="p">,</span><span class="w"> </span><span class="kt">double</span><span class="w"> </span><span class="n">vmax</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">fmax</span><span class="p">(</span><span class="n">vmin</span><span class="p">,</span><span class="w"> </span><span class="n">fmin</span><span class="p">(</span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="n">vmax</span><span class="p">));</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
</pre></div>
</div>
<div class="section" id="exercise">
<h2>Exercise</h2>
<p>Suppose you are working on a thermostat for ACME Corporation. In order
to reduce production and power costs, they have decided to choose a
microprocessor without an FPU. Your job is to take temperature data from
the sensor, which outputs in Celsius, and convert it to Fahrenheit for
the user interface team. They want to be able to display up to 1 decimal
place accurately. You are free to decide on the appropriate <tt class="docutils literal">QM.N</tt>
format. Note, the formula for convertion is:</p>
<div class="math">
\begin{equation*}
F = \frac{9}{5} C + 32
\end{equation*}
</div>
<ol class="arabic simple">
<li>How many bits are needed for fractional portion in order to achieve
one decimal place of accuracy?</li>
<li>Given temperature sensor has a range of -100° to 155° in Celsius,
which corresponds to roughly -150° to +310° degrees Fahrenheit. How
many bits do we need for the integer portion?</li>
</ol>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">log2</span><span class="p">,</span> <span class="n">ceil</span>
<span class="c1"># Question: How many bits do we need to compute this result?</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span> <span class="c1"># Decimal places</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Numbers of bits needed for </span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s2"> decimal places accuracy: </span><span class="si">{</span><span class="n">ceil</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">log2</span><span class="p">(</span><span class="mi">10</span> <span class="o">**</span> <span class="p">(</span><span class="o">-</span><span class="n">n</span><span class="p">))))</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="c1"># Question: Given temperature sensor has a range of -100 to 155 in Celsius,</span>
<span class="c1"># which corresponds to roughly -150 to +310 degrees Fahrenheit. How many bits</span>
<span class="c1"># do we need for the integer portion?</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">310</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Bits needed to store up to </span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s2">: </span><span class="si">{</span><span class="n">ceil</span><span class="p">(</span><span class="n">log2</span><span class="p">(</span><span class="n">n</span><span class="p">))</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="c1"># Let's give it some room and use an extra two bits for quantization error.</span>
<span class="c1"># Also, one extra bit is needed for sign, thus a format of Q10.6 is needed.</span>
<span class="c1"># The computation can be done using 16 bit signed integers at Q10.6 format</span>
<span class="c1"># Note the formula: F = 9/5 * C + 32</span>
</pre></div>
<pre class="literal-block">
Numbers of bits needed for 1 decimal places accuracy: 4.0
Bits needed to store up to 310: 9.0
</pre>
<div class="highlight"><pre><span></span><span class="n">ftoq</span><span class="p">(</span><span class="mi">9</span><span class="o">/</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">ftoq</span><span class="p">(</span><span class="mf">32.0</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">),</span> <span class="n">qtof</span><span class="p">(</span><span class="o">-</span><span class="mi">7404</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
</pre></div>
<pre class="literal-block">
(115, 2048, -115.6875)
</pre>
<div class="highlight"><pre><span></span><span class="c1"># Reference function for comparison</span>
<span class="k">def</span> <span class="nf">ctof_f64</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
<span class="sd">"""Convert Celsius to Fahrenheit</span>
<span class="sd"> Args:</span>
<span class="sd"> c: Celsius</span>
<span class="sd"> Return:</span>
<span class="sd"> Fahrenheit</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="mi">9</span><span class="o">/</span><span class="mi">5</span> <span class="o">*</span> <span class="n">c</span> <span class="o">+</span> <span class="mf">32.0</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">ctof_q10x6</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
<span class="sd">"""Convert Celsius to Fahrenheit in Q10.6 format</span>
<span class="sd"> Args:</span>
<span class="sd"> c: Celsius in Q10.6 format</span>
<span class="sd"> Return:</span>
<span class="sd"> int16(Q10.6)</span>
<span class="sd"> """</span>
<span class="n">C0</span> <span class="o">=</span> <span class="n">int16</span><span class="p">(</span><span class="n">ftoq</span><span class="p">(</span><span class="mi">9</span><span class="o">/</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="n">C1</span> <span class="o">=</span> <span class="n">int16</span><span class="p">(</span><span class="n">ftoq</span><span class="p">(</span><span class="mi">32</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="k">return</span> <span class="n">q10x6mul</span><span class="p">(</span><span class="n">C0</span><span class="p">,</span> <span class="n">c</span><span class="p">)</span> <span class="o">+</span> <span class="n">C1</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">int32</span><span class="p">,</span> <span class="n">int16</span><span class="p">,</span> <span class="n">clip</span>
<span class="k">def</span> <span class="nf">qround</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="sd">"""Round an integer `a` by `b`"""</span>
<span class="k">if</span> <span class="n">a</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span>
<span class="k">return</span> <span class="n">a</span>
<span class="k">def</span> <span class="nf">q10x6mul</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">):</span>
<span class="sd">"""Multiply two Q10.6 integers"""</span>
<span class="n">a</span> <span class="o">=</span> <span class="n">int32</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="n">b</span> <span class="o">=</span> <span class="n">int32</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span> <span class="o">*</span> <span class="n">b</span> <span class="c1"># c: Q20.12</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">qround</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mb">0b10000</span><span class="p">)</span> <span class="c1"># Round to nearest b0.000001 bit</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">c</span> <span class="o">>></span> <span class="mi">6</span> <span class="c1"># Rescale c to Q10.6</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">clip</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="o">-</span><span class="mi">32768</span><span class="p">,</span> <span class="mi">32767</span><span class="p">)</span> <span class="c1"># Saturate result</span>
<span class="k">return</span> <span class="n">int16</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
</pre></div>
<p>Same thing can be implemented in C as:</p>
<div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><stdint.h></span><span class="cp"></span>
<span class="k">typedef</span><span class="w"> </span><span class="kt">int16_t</span><span class="w"> </span><span class="n">q10x6_t</span><span class="p">;</span><span class="w"></span>
<span class="k">typedef</span><span class="w"> </span><span class="kt">int32_t</span><span class="w"> </span><span class="n">q20x12_t</span><span class="p">;</span><span class="w"></span>
<span class="n">q10x6_t</span><span class="w"> </span><span class="nf">celsius_to_fahrenheit</span><span class="p">(</span><span class="n">q10x6_t</span><span class="w"> </span><span class="n">celsius</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q10x6_t</span><span class="w"> </span><span class="n">C0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">115</span><span class="p">;</span><span class="w"> </span><span class="cm">/* 9/5 in Q10.6 */</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q10x6_t</span><span class="w"> </span><span class="n">C1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">2048</span><span class="p">;</span><span class="w"> </span><span class="cm">/* 32 in Q10.6 */</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Multiplying Q10.6 by Q10.6 -> Q20.12 */</span><span class="w"></span>
<span class="w"> </span><span class="n">q20x12_t</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">((</span><span class="n">q20x12_t</span><span class="p">)</span><span class="n">celsius</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">((</span><span class="n">q20x12_t</span><span class="p">)</span><span class="n">C0</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Rounding to nearest 6 bits */</span><span class="w"></span>
<span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">y</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="mh">-0x10</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">+</span><span class="mh">0x10</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">>>=</span><span class="w"> </span><span class="mi">6</span><span class="p">;</span><span class="w"> </span><span class="cm">/* Q20.6 */</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Add 32 to result */</span><span class="w"></span>
<span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="n">C1</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">q10x6_t</span><span class="p">)</span><span class="n">y</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
</pre></div>
<div class="section" id="plotting-the-errors">
<h3>Plotting the errors</h3>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">linspace</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Celsius </span><span class="se">\t</span><span class="s2">Q10.6 </span><span class="se">\t\t</span><span class="s2">F64 </span><span class="se">\t\t</span><span class="s2">Error"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">155</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">15</span><span class="p">):</span>
<span class="n">cq10x6</span> <span class="o">=</span> <span class="n">ftoq</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="n">fq10x6</span> <span class="o">=</span> <span class="n">ctof_q10x6</span><span class="p">(</span><span class="n">cq10x6</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">ctof_f64</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">fq</span> <span class="o">=</span> <span class="n">qtof</span><span class="p">(</span><span class="n">fq10x6</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="n">perr</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">f</span> <span class="o">-</span> <span class="n">fq</span><span class="p">)</span> <span class="o">/</span> <span class="n">f</span> <span class="o">*</span> <span class="mi">100</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">c</span><span class="si">:</span><span class="s2"> 3.1f</span><span class="si">}</span><span class="s2">° C </span><span class="se">\t</span><span class="si">{</span><span class="n">fq</span><span class="si">:</span><span class="s2"> 3.1f</span><span class="si">}</span><span class="s2">° F </span><span class="se">\t</span><span class="si">{</span><span class="n">f</span><span class="si">:</span><span class="s2"> 3.1f</span><span class="si">}</span><span class="s2">° F </span><span class="se">\t</span><span class="s2"> </span><span class="si">{</span><span class="n">perr</span><span class="si">:</span><span class="s2"> 2.2</span><span class="si">}</span><span class="s2"> %"</span><span class="p">)</span>
</pre></div>
<pre class="literal-block">
Celsius Q10.6 F64 Error
-100.0° C -147.7° F -148.0° F -0.2 %
-81.8° C -115.0° F -115.2° F -0.21 %
-63.6° C -82.2° F -82.4° F -0.22 %
-45.4° C -49.5° F -49.6° F -0.26 %
-27.1° C -16.8° F -16.9° F -0.45 %
-8.9° C 16.0° F 15.9° F 0.15 %
9.3° C 48.7° F 48.7° F 0.087 %
27.5° C 81.4° F 81.5° F 0.12 %
45.7° C 114.1° F 114.3° F 0.13 %
63.9° C 146.9° F 147.1° F 0.14 %
82.1° C 179.6° F 179.9° F 0.15 %
100.4° C 212.3° F 212.6° F 0.15 %
118.6° C 245.1° F 245.4° F 0.15 %
136.8° C 277.8° F 278.2° F 0.16 %
155.0° C 310.5° F 311.0° F 0.16 %
</pre>
<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pylab</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">linspace</span><span class="p">,</span> <span class="n">vectorize</span>
<span class="n">vctof_q10x6</span> <span class="o">=</span> <span class="n">vectorize</span><span class="p">(</span><span class="n">ctof_q10x6</span><span class="p">)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">155</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">512</span><span class="p">)</span>
<span class="n">cq</span> <span class="o">=</span> <span class="n">vftoq</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="n">fq</span> <span class="o">=</span> <span class="n">vctof_q10x6</span><span class="p">(</span><span class="n">cq</span><span class="p">)</span>
<span class="n">fq</span> <span class="o">=</span> <span class="n">vqtof</span><span class="p">(</span><span class="n">fq</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">)</span>
<span class="n">f</span> <span class="o">=</span> <span class="n">ctof_f64</span><span class="p">(</span><span class="n">c</span><span class="p">)</span>
<span class="n">perr</span> <span class="o">=</span> <span class="nb">abs</span><span class="p">(</span><span class="n">f</span> <span class="o">-</span> <span class="n">fq</span><span class="p">)</span><span class="o">/</span><span class="mf">100.0</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">c</span><span class="p">,</span> <span class="n">perr</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"° Celsius"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"</span><span class="si">% E</span><span class="s2">rror during conversion"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
<img alt="Percent error of fixed point to floating point" src="./images/fixed-point/output_21_0.png" />
</div>
</div>
<div class="section" id="fixed-point-signal-processing">
<h2>Fixed point signal processing</h2>
<ul class="simple">
<li>There are only two rules of fixed point signal processing:<ul>
<li>Thou shalt not overflow</li>
<li>Thou shalt not underflow</li>
</ul>
</li>
<li>Always design your filters so they are within range of operation<ul>
<li>Usually with FIR filters, it is not a problem</li>
<li>IIR filters become a challenge, can be avoided with low order
filters</li>
</ul>
</li>
</ul>
</div>
<div class="section" id="fir-filters-in-fixed-point">
<h2>FIR filters in fixed-point</h2>
<div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><stdint.h></span><span class="cp"></span>
<span class="k">typedef</span><span class="w"> </span><span class="kt">int16_t</span><span class="w"> </span><span class="n">q15_t</span><span class="p">;</span><span class="w"></span>
<span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">fir_instance_q15</span><span class="w"></span>
<span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">coeff</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">taps</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"> </span><span class="n">fir_q15_t</span><span class="p">;</span><span class="w"></span>
<span class="kt">void</span><span class="w"> </span><span class="nf">generic_fir_q15</span><span class="p">(</span><span class="n">fir_q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">self</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">dst</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">frames</span><span class="p">);</span><span class="w"></span>
</pre></div>
<ol class="arabic simple">
<li>CMSIS DSP documentation for <a class="reference external" href="https://arm-software.github.io/CMSIS_5/DSP/html/group__FIR.html#gad9f84fdadbb59292592c7fac81adb09c">fixed-point FIR
filter</a></li>
<li>CMSIS DSP implementation for FIR filter
<a class="reference external" href="https://github.com/ARM-software/CMSIS/blob/f2cad4345783c948ed4a7f5cdb02cdc0856366f1/CMSIS/DSP_Lib/Source/FilteringFunctions/arm_fir_q15.c#L603">arm_fir_q15.c</a></li>
</ol>
<p>Generic FIR filter implementation in C:</p>
<div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf">"generic_fir_q15.h"</span><span class="cp"></span>
<span class="k">static</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="nf">q15_rsat</span><span class="p">(</span><span class="n">q63_t</span><span class="w"> </span><span class="n">x</span><span class="p">);</span><span class="w"></span>
<span class="kt">void</span><span class="w"> </span><span class="nf">generic_fir_q15</span><span class="p">(</span><span class="n">fir_q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">self</span><span class="p">,</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">dst</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="kt">size_t</span><span class="w"> </span><span class="n">frames</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="kt">size_t</span><span class="w"> </span><span class="n">taps</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">taps</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">size_t</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">index</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">state</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">coeff</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">coeff</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">size_t</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">frames</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Place input in circular buffer */</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="n">index</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">src</span><span class="p">[</span><span class="n">i</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Circularly increment index */</span><span class="w"></span>
<span class="w"> </span><span class="o">++</span><span class="n">index</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">index</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="n">taps</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/**</span>
<span class="cm"> * Use a large enough accumulator to avoid overflow. If coefficients</span>
<span class="cm"> * are Q1.15 and incoming sample data are also Q1.15, then each</span>
<span class="cm"> * multiplications will require at-least 32 bits (Q2.30), and the</span>
<span class="cm"> * worse case scenario of each addion will be an overflow which will</span>
<span class="cm"> * require an extra bit to capture. Without going into the assembly</span>
<span class="cm"> * level, it is more portable to get a larger data type (int64_t).</span>
<span class="cm"> */</span><span class="w"></span>
<span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Convolution Loop */</span><span class="w"></span>
<span class="w"> </span><span class="k">do</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">size_t</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="o">--</span><span class="n">k</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">((</span><span class="n">q31_t</span><span class="p">)</span><span class="n">coeff</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">((</span><span class="n">q31_t</span><span class="p">)</span><span class="n">state</span><span class="p">[</span><span class="n">k</span><span class="p">]);</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">size_t</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">taps</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">taps</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">,</span><span class="w"> </span><span class="o">--</span><span class="n">k</span><span class="p">)</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">((</span><span class="n">q31_t</span><span class="p">)</span><span class="n">coeff</span><span class="p">[</span><span class="n">j</span><span class="p">])</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">((</span><span class="n">q31_t</span><span class="p">)</span><span class="n">state</span><span class="p">[</span><span class="n">k</span><span class="p">]);</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">while</span><span class="w"> </span><span class="p">(</span><span class="mi">0</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">dst</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">q15_rsat</span><span class="p">(</span><span class="n">acc</span><span class="p">);</span><span class="w"> </span><span class="cm">/* Typically this is one machine instruction in DSPs */</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Store the updated circular buffer address */</span><span class="w"></span>
<span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">index</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="k">static</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="nf">q15_rsat</span><span class="p">(</span><span class="n">q63_t</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">vmax</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">32767</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">vmin</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-32768</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">rounding_value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">15</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Round to nearest Q1.15 */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">rounding_value</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">-</span><span class="n">rounding_value</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Rescale to Q1.15 */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">>></span><span class="w"> </span><span class="mi">15</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Saturate result */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="n">vmax</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">vmax</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">vmin</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">vmin</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">q15_t</span><span class="p">)</span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
</pre></div>
<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">signal</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">zplane</span> <span class="kn">import</span> <span class="n">zplane</span>
<span class="k">def</span> <span class="nf">plot_response</span><span class="p">(</span><span class="n">fs</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="n">title</span><span class="p">,</span> <span class="n">sb</span><span class="o">=-</span><span class="mi">40</span><span class="p">):</span>
<span class="s2">"Utility function to plot response functions"</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="mf">0.5</span><span class="o">*</span><span class="n">fs</span><span class="o">*</span><span class="n">w</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">20</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">log10</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">h</span><span class="p">)))</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="n">sb</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.5</span><span class="o">*</span><span class="n">fs</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">grid</span><span class="p">(</span><span class="kc">True</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s1">'Frequency (Hz)'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s1">'Gain (dB)'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">sos2zpk</span><span class="p">(</span><span class="n">sos</span><span class="p">):</span>
<span class="n">b</span><span class="p">,</span> <span class="n">a</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">sos2tf</span><span class="p">(</span><span class="n">sos</span><span class="p">)</span>
<span class="n">z</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">tf2zpk</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="k">return</span> <span class="n">z</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">k</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="n">Fs</span> <span class="o">=</span> <span class="mf">16000.0</span> <span class="c1"># Sample Rate: Hz</span>
<span class="n">Fc</span> <span class="o">=</span> <span class="mf">3500.0</span> <span class="c1"># Cutoff frequency: Hz</span>
<span class="n">Tw</span> <span class="o">=</span> <span class="mf">500.0</span> <span class="c1"># Transition width: Hz</span>
<span class="n">ntaps</span> <span class="o">=</span> <span class="mi">45</span> <span class="c1"># Filter taps</span>
<span class="n">attenuation</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="c1"># Filter gain to avoid overflow</span>
<span class="c1"># Generate the FIR filter</span>
<span class="n">coeffs</span> <span class="o">=</span> <span class="n">attenuation</span> <span class="o">*</span> <span class="n">signal</span><span class="o">.</span><span class="n">remez</span><span class="p">(</span><span class="n">ntaps</span><span class="p">,</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="n">Fc</span><span class="p">,</span> <span class="n">Fc</span> <span class="o">+</span> <span class="n">Tw</span><span class="p">,</span> <span class="n">Fs</span><span class="o">/</span><span class="mf">2.0</span><span class="p">],</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Hz</span><span class="o">=</span><span class="n">Fs</span><span class="p">)</span>
<span class="n">qcoeffs</span> <span class="o">=</span> <span class="n">vquantize</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">15</span><span class="p">)</span>
<span class="c1"># Get filter response</span>
<span class="n">w</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">freqz</span><span class="p">(</span><span class="n">qcoeffs</span><span class="p">,</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">worN</span><span class="o">=</span><span class="mi">512</span><span class="p">)</span>
<span class="n">plot_response</span><span class="p">(</span><span class="n">Fs</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="s2">"Low-pass filter"</span><span class="p">,</span> <span class="n">sb</span><span class="o">=-</span><span class="mi">75</span><span class="p">)</span>
</pre></div>
<img alt="Frequency response of FIR filter" src="./images/fixed-point/output_25_0.png" />
<div class="highlight"><pre><span></span><span class="n">coeff_str</span> <span class="o">=</span> <span class="s2">","</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">d</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">vftoq</span><span class="p">(</span><span class="n">coeffs</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">15</span><span class="p">))</span>
<span class="n">taps</span> <span class="o">=</span> <span class="n">coeffs</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"/*<autogen-fir>*/"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"q15_t fir_coeff[</span><span class="si">{</span><span class="n">taps</span><span class="si">}</span><span class="s2">] = </span><span class="se">{{</span><span class="si">{</span><span class="n">coeff_str</span><span class="si">}</span><span class="se">}}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"q15_t fir_state[</span><span class="si">{</span><span class="n">taps</span><span class="si">}</span><span class="s2">] = </span><span class="se">{{</span><span class="s2">0</span><span class="se">}}</span><span class="s2">;"</span><span class="p">);</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"uint32_t fir_taps = </span><span class="si">{</span><span class="n">taps</span><span class="si">}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"uint32_t fir_index = 0;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"fir_q15_t fir_filter = </span><span class="se">{{</span><span class="s2"> fir_coeff, fir_state, fir_index, fir_taps </span><span class="se">}}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"/*</autogen-fir>*/"</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="cm">/*<autogen-fir>*/</span><span class="w"></span>
<span class="n">q15_t</span><span class="w"> </span><span class="n">fir_coeff</span><span class="p">[</span><span class="mi">45</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">171</span><span class="p">,</span><span class="w"> </span><span class="mi">-201</span><span class="p">,</span><span class="w"> </span><span class="mi">-209</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">148</span><span class="p">,</span><span class="w"> </span><span class="mi">-33</span><span class="p">,</span><span class="w"> </span><span class="mi">-230</span><span class="p">,</span><span class="w"> </span><span class="mi">-29</span><span class="p">,</span><span class="w"> </span><span class="mi">271</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">90</span><span class="p">,</span><span class="w"> </span><span class="mi">-326</span><span class="p">,</span><span class="w"> </span><span class="mi">-188</span><span class="p">,</span><span class="w"> </span><span class="mi">375</span><span class="p">,</span><span class="w"> </span><span class="mi">328</span><span class="p">,</span><span class="w"> </span><span class="mi">-420</span><span class="p">,</span><span class="w"> </span><span class="mi">-537</span><span class="p">,</span><span class="w"> </span><span class="mi">458</span><span class="p">,</span><span class="w"> </span><span class="mi">888</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">-487</span><span class="p">,</span><span class="w"> </span><span class="mi">-1643</span><span class="p">,</span><span class="w"> </span><span class="mi">506</span><span class="p">,</span><span class="w"> </span><span class="mi">5183</span><span class="p">,</span><span class="w"> </span><span class="mi">7681</span><span class="p">,</span><span class="w"> </span><span class="mi">5183</span><span class="p">,</span><span class="w"> </span><span class="mi">506</span><span class="p">,</span><span class="w"> </span><span class="mi">-1643</span><span class="p">,</span><span class="w"> </span><span class="mi">-487</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">888</span><span class="p">,</span><span class="w"> </span><span class="mi">458</span><span class="p">,</span><span class="w"> </span><span class="mi">-537</span><span class="p">,</span><span class="w"> </span><span class="mi">-420</span><span class="p">,</span><span class="w"> </span><span class="mi">328</span><span class="p">,</span><span class="w"> </span><span class="mi">375</span><span class="p">,</span><span class="w"> </span><span class="mi">-188</span><span class="p">,</span><span class="w"> </span><span class="mi">-326</span><span class="p">,</span><span class="w"> </span><span class="mi">90</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">271</span><span class="p">,</span><span class="w"> </span><span class="mi">-29</span><span class="p">,</span><span class="w"> </span><span class="mi">-230</span><span class="p">,</span><span class="w"> </span><span class="mi">-33</span><span class="p">,</span><span class="w"> </span><span class="mi">148</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"> </span><span class="mi">-209</span><span class="p">,</span><span class="w"> </span><span class="mi">-201</span><span class="p">,</span><span class="w"> </span><span class="mi">171</span><span class="p">};</span><span class="w"></span>
<span class="n">q15_t</span><span class="w"> </span><span class="n">fir_state</span><span class="p">[</span><span class="mi">45</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">0</span><span class="p">};</span><span class="w"></span>
<span class="kt">uint32_t</span><span class="w"> </span><span class="n">fir_taps</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">45</span><span class="p">;</span><span class="w"></span>
<span class="kt">uint32_t</span><span class="w"> </span><span class="n">fir_index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"></span>
<span class="n">fir_q15_t</span><span class="w"> </span><span class="n">fir_filter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="n">fir_coeff</span><span class="p">,</span><span class="w"> </span><span class="n">fir_state</span><span class="p">,</span><span class="w"> </span><span class="n">fir_index</span><span class="p">,</span><span class="w"> </span><span class="n">fir_taps</span><span class="p">};</span><span class="w"></span>
<span class="cm">/*</autogen-fir>*/</span><span class="w"></span>
</pre></div>
</div>
<div class="section" id="iir-filters-in-fixed-point">
<h2>IIR filters in fixed-point</h2>
<div class="highlight"><pre><span></span><span class="cp">#include</span><span class="w"> </span><span class="cpf"><stdint.h></span><span class="cp"></span>
<span class="k">typedef</span><span class="w"> </span><span class="kt">int16_t</span><span class="w"> </span><span class="n">q15_t</span><span class="p">;</span><span class="w"></span>
<span class="k">typedef</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">iir_instance_q15</span><span class="w"></span>
<span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">coeff</span><span class="p">;</span><span class="w"> </span><span class="cm">/* b0, b1, b2, a1, a2... */</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="p">;</span><span class="w"> </span><span class="cm">/* x1, x2, y1, y2, ... */</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">nstage</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="kt">int32_t</span><span class="w"> </span><span class="n">shift</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"> </span><span class="n">iir_q15_t</span><span class="p">;</span><span class="w"></span>
<span class="kt">void</span><span class="w"> </span><span class="nf">generic_iir_q15</span><span class="p">(</span><span class="n">iir_q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">self</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">dst</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">frames</span><span class="p">);</span><span class="w"></span>
</pre></div>
<ol class="arabic simple">
<li>CCRMA’s info on <a class="reference external" href="https://ccrma.stanford.edu/~jos/filters/Direct_Form_I.html">Direct form
I</a></li>
<li>CMSIS DSP documentation on <a class="reference external" href="https://arm-software.github.io/CMSIS_5/DSP/html/group__BiquadCascadeDF1.html#ga27af5bfa40dad7c877e48eedc8d67558">fixed-point IIR
filters</a></li>
<li>CMSIS DSP implementation of IIR filter
<a class="reference external" href="https://github.com/ARM-software/CMSIS/blob/f2cad4345783c948ed4a7f5cdb02cdc0856366f1/CMSIS/DSP_Lib/Source/FilteringFunctions/arm_biquad_cascade_df1_q15.c#L75">arm_biquad_cascade_df1_q15.c</a></li>
</ol>
<p>Generic IIR filter implementation in C</p>
<div class="highlight"><pre><span></span><span class="cm">/* IIR Filter Implementation */</span><span class="w"></span>
<span class="k">static</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="nf">q15_rsat</span><span class="p">(</span><span class="n">q63_t</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int32_t</span><span class="w"> </span><span class="n">shift</span><span class="p">);</span><span class="w"></span>
<span class="kt">void</span><span class="w"> </span><span class="nf">generic_iir_q15</span><span class="p">(</span><span class="n">iir_q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">self</span><span class="p">,</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">dst</span><span class="p">,</span><span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">src</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">frames</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">nstage</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">nstage</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="kt">int32_t</span><span class="w"> </span><span class="n">shift</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">shift</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">coeff</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">coeff</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">state</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">self</span><span class="o">-></span><span class="n">state</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">nstage</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">i</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">b0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coeff</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">b1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coeff</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">b2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coeff</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">a1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coeff</span><span class="p">[</span><span class="mi">3</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">coeff</span><span class="p">[</span><span class="mi">4</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">xptr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">dst</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">src</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="o">*</span><span class="n">yptr</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dst</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">q31_t</span><span class="w"> </span><span class="n">x0</span><span class="p">,</span><span class="w"> </span><span class="n">x1</span><span class="p">,</span><span class="w"> </span><span class="n">x2</span><span class="p">,</span><span class="w"> </span><span class="n">y0</span><span class="p">,</span><span class="w"> </span><span class="n">y1</span><span class="p">,</span><span class="w"> </span><span class="n">y2</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">x1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">0</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="n">x2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">1</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="n">y1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">2</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="n">y2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">3</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="kt">uint32_t</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">frames</span><span class="p">;</span><span class="w"> </span><span class="o">++</span><span class="n">j</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">acc</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">x0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">xptr</span><span class="p">[</span><span class="n">j</span><span class="p">];</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">b0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">x0</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">b1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">x1</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">b2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">x2</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="p">(</span><span class="n">a1</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">y1</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">acc</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="p">(</span><span class="n">a2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">y2</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">y0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">q15_rsat</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span><span class="w"> </span><span class="n">shift</span><span class="p">);</span><span class="w"></span>
<span class="w"> </span><span class="n">x2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x1</span><span class="p">;</span><span class="w"> </span><span class="cm">/* new x[n-1] */</span><span class="w"></span>
<span class="w"> </span><span class="n">x1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x0</span><span class="p">;</span><span class="w"> </span><span class="cm">/* new x[n-2] */</span><span class="w"></span>
<span class="w"> </span><span class="n">y2</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y1</span><span class="p">;</span><span class="w"> </span><span class="cm">/* new y[n-1] */</span><span class="w"></span>
<span class="w"> </span><span class="n">y1</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y0</span><span class="p">;</span><span class="w"> </span><span class="cm">/* new y[n-2] */</span><span class="w"></span>
<span class="w"> </span><span class="n">yptr</span><span class="p">[</span><span class="n">j</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y0</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x1</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x2</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y1</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="p">[</span><span class="mi">3</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">y2</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">state</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">4</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">coeff</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="mi">5</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="p">}</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
<span class="k">static</span><span class="w"> </span><span class="n">q15_t</span><span class="w"> </span><span class="nf">q15_rsat</span><span class="p">(</span><span class="n">q63_t</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="kt">int32_t</span><span class="w"> </span><span class="n">shift</span><span class="p">)</span><span class="w"> </span><span class="p">{</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">vmax</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">32767</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">vmin</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">-32768</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">const</span><span class="w"> </span><span class="n">q63_t</span><span class="w"> </span><span class="n">rounding_value</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o"><<</span><span class="w"> </span><span class="mi">15</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Round to nearest Q1.15 */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">+=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">></span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">rounding_value</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="o">-</span><span class="n">rounding_value</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Rescale to Q by caller */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">>>=</span><span class="w"> </span><span class="n">shift</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="cm">/* Saturate result */</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">>=</span><span class="w"> </span><span class="n">vmax</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">vmax</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o"><=</span><span class="w"> </span><span class="n">vmin</span><span class="p">)</span><span class="w"> </span><span class="o">?</span><span class="w"> </span><span class="n">vmin</span><span class="w"> </span><span class="o">:</span><span class="w"> </span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="p">(</span><span class="n">q15_t</span><span class="p">)</span><span class="n">x</span><span class="p">;</span><span class="w"></span>
<span class="p">}</span><span class="w"></span>
</pre></div>
<div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="n">linalg</span> <span class="k">as</span> <span class="n">LA</span>
<span class="n">Fs</span> <span class="o">=</span> <span class="mf">16000.0</span> <span class="c1"># Sample Rate: Hz</span>
<span class="n">Fc</span> <span class="o">=</span> <span class="mf">4000.0</span> <span class="c1"># Cutoff frequency: Hz</span>
<span class="c1"># Create the prototype filter</span>
<span class="n">z</span><span class="p">,</span><span class="n">p</span><span class="p">,</span><span class="n">k</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">iirfilter</span><span class="p">(</span><span class="mi">7</span><span class="p">,</span> <span class="n">Fc</span><span class="p">,</span> <span class="n">rs</span><span class="o">=</span><span class="mi">70</span><span class="p">,</span> <span class="n">rp</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">btype</span><span class="o">=</span><span class="s1">'low'</span><span class="p">,</span> <span class="n">analog</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">ftype</span><span class="o">=</span><span class="s1">'cheby2'</span><span class="p">,</span> <span class="n">fs</span><span class="o">=</span><span class="n">Fs</span><span class="p">,</span> <span class="n">output</span><span class="o">=</span><span class="s1">'zpk'</span><span class="p">)</span>
<span class="c1"># Convert it to 2nd order stages (biquad) with a gain of 1</span>
<span class="c1"># Note keep the pairing odd for greater use of precision</span>
<span class="n">sos</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">zpk2sos</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">pairing</span><span class="o">=</span><span class="s1">'keep_odd'</span><span class="p">)</span>
<span class="c1"># Modify feed-forward gain until it's under -6 dB of attenuation</span>
<span class="n">gain</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">/</span> <span class="n">sos</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">sos</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">=</span> <span class="n">sos</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">:</span><span class="mi">3</span><span class="p">]</span> <span class="o">*</span> <span class="n">gain</span>
<span class="nb">print</span><span class="p">(</span><span class="n">sos</span><span class="p">)</span>
<span class="c1"># Quantize</span>
<span class="n">qsos</span> <span class="o">=</span> <span class="n">vquantize</span><span class="p">(</span><span class="n">sos</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
<span class="n">w</span><span class="p">,</span> <span class="n">h</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">sosfreqz</span><span class="p">(</span><span class="n">qsos</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Max gain: </span><span class="si">{</span><span class="nb">max</span><span class="p">(</span><span class="nb">abs</span><span class="p">(</span><span class="n">h</span><span class="p">))</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">zplane</span><span class="p">(</span><span class="n">z</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
<span class="n">plot_response</span><span class="p">(</span><span class="n">Fs</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">h</span><span class="p">,</span> <span class="s2">"Low-pass filter"</span><span class="p">,</span> <span class="n">sb</span><span class="o">=-</span><span class="mi">96</span><span class="p">)</span>
</pre></div>
<pre class="literal-block">
[[ 0.125 0.125 0. 1. -0.2315115 0. ]
[ 0.125 0.17078507 0.125 1. -0.52869293 0.13074684]
[ 0.125 0.06031606 0.125 1. -0.70554983 0.35305028]
[ 0.125 0.00634657 0.125 1. -0.96279273 0.72736088]]
Max gain: 0.036533575929996835
</pre>
<img alt="Frequency response of IIR filter" src="./images/fixed-point/output_28_1.png" />
<img alt="Pole-zero diagram of IIR filter" src="./images/fixed-point/output_28_2.png" />
<div class="highlight"><pre><span></span><span class="c1"># Check the step response to detect any overflow over a short period of time</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">sosfilt</span><span class="p">(</span><span class="n">vquantize</span><span class="p">(</span><span class="n">sos</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">14</span><span class="p">),</span> <span class="n">signal</span><span class="o">.</span><span class="n">unit_impulse</span><span class="p">(</span><span class="mi">32</span><span class="p">))</span>
<span class="n">ys</span> <span class="o">=</span> <span class="n">signal</span><span class="o">.</span><span class="n">sosfilt</span><span class="p">(</span><span class="n">vquantize</span><span class="p">(</span><span class="n">sos</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">14</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="mi">32</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Impulse response:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{</span><span class="n">y</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Step response: </span><span class="se">\n</span><span class="si">{</span><span class="n">ys</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
<pre class="literal-block">
Impulse response:
[ 0.00244969 0.01305221 0.03750753 0.07332111 0.10464949 0.10941449
0.07655378 0.01950579 -0.03047698 -0.04673595 -0.02733165 0.00550324
0.02516314 0.02096245 0.00240548 -0.01281883 -0.01419637 -0.00445748
0.00599136 0.00902022 0.00434855 -0.00236218 -0.00543642 -0.00351961
0.00056275 0.00310108 0.00257681 0.0002259 -0.00165651 -0.00175919
-0.00048893 0.00080876]
Step response:
[0.00244969 0.0155019 0.05300943 0.12633054 0.23098003 0.34039452
0.4169483 0.4364541 0.40597712 0.35924117 0.33190953 0.33741276
0.36257591 0.38353836 0.38594384 0.37312501 0.35892864 0.35447116
0.36046253 0.36948275 0.3738313 0.37146912 0.3660327 0.36251309
0.36307584 0.36617691 0.36875372 0.36897963 0.36732312 0.36556393
0.365075 0.36588375]
</pre>
<div class="highlight"><pre><span></span><span class="n">M</span><span class="p">,</span><span class="n">N</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">14</span>
<span class="n">sos2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">delete</span><span class="p">(</span><span class="n">sos</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">flatten</span><span class="p">()</span> <span class="c1"># Remove a0 column</span>
<span class="n">coeff_str</span> <span class="o">=</span> <span class="s2">","</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">d</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">vftoq</span><span class="p">(</span><span class="n">sos2</span><span class="p">,</span> <span class="n">M</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span> <span class="c1"># "flatten" array so it's one dimention</span>
<span class="n">iir_stage</span> <span class="o">=</span> <span class="n">sos</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="c1"># Get number of stages</span>
<span class="n">iir_shift</span> <span class="o">=</span> <span class="n">N</span> <span class="c1"># Get the shift amount</span>
<span class="c1"># Write out header file</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"/*<autogen-iir>*/"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"#define NSTAGE (</span><span class="si">{</span><span class="n">iir_stage</span><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"q15_t iir_coeff[5 * NSTAGE] = </span><span class="se">{{</span><span class="si">{</span><span class="n">coeff_str</span><span class="si">}</span><span class="se">}}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"q15_t iir_state[4 * NSTAGE] = </span><span class="se">{{</span><span class="s2">0</span><span class="se">}}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"uint32_t iir_stage = NSTAGE;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"int32_t iir_shift = </span><span class="si">{</span><span class="n">iir_shift</span><span class="si">}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"iir_q15_t iir_filter = </span><span class="se">{{</span><span class="s2">iir_coeff, iir_state, iir_stage, iir_shift</span><span class="se">}}</span><span class="s2">;"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"/*</autogen-iir>*/</span><span class="se">\n</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span><span class="cm">/*<autogen-iir>*/</span><span class="w"></span>
<span class="cp">#define NSTAGE (4)</span>
<span class="n">q15_t</span><span class="w"> </span><span class="n">iir_coeff</span><span class="p">[</span><span class="mi">5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">NSTAGE</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">-3793</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">4980</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">-8662</span><span class="p">,</span><span class="w"> </span><span class="mi">2142</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">1759</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">-11560</span><span class="p">,</span><span class="w"></span>
<span class="w"> </span><span class="mi">5784</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">185</span><span class="p">,</span><span class="w"> </span><span class="mi">3645</span><span class="p">,</span><span class="w"> </span><span class="mi">-15774</span><span class="p">,</span><span class="w"> </span><span class="mi">11917</span><span class="p">};</span><span class="w"></span>
<span class="n">q15_t</span><span class="w"> </span><span class="n">iir_state</span><span class="p">[</span><span class="mi">4</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">NSTAGE</span><span class="p">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="mi">0</span><span class="p">};</span><span class="w"></span>
<span class="kt">uint32_t</span><span class="w"> </span><span class="n">iir_stage</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">NSTAGE</span><span class="p">;</span><span class="w"></span>
<span class="kt">int32_t</span><span class="w"> </span><span class="n">iir_shift</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">14</span><span class="p">;</span><span class="w"></span>
<span class="n">iir_q15_t</span><span class="w"> </span><span class="n">iir_filter</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">{</span><span class="n">iir_coeff</span><span class="p">,</span><span class="w"> </span><span class="n">iir_state</span><span class="p">,</span><span class="w"> </span><span class="n">iir_stage</span><span class="p">,</span><span class="w"> </span><span class="n">iir_shift</span><span class="p">};</span><span class="w"></span>
<span class="cm">/*</autogen-iir>*/</span><span class="w"></span>
</pre></div>
</div>
<div class="section" id="other-functions">
<h2>Other functions</h2>
<ol class="arabic simple">
<li>Trigonometry:
<a class="reference external" href="https://en.wikipedia.org/wiki/Libfixmath">libfixmath</a> offers a
great set of math functions for your fixed point needs.</li>
<li><a class="reference external" href="https://www.arm.com/why-arm/technologies/cmsis">CMSIS</a> and
<a class="reference external" href="https://github.com/ARM-software/CMSIS_5">CMSIS-DSP</a>, A great
library for programming for embedded ARM environment. Supported by
ARM!</li>
<li>Espressif, a new player in slilcon industry from China, has a great
set of DSP libraries
<a class="reference external" href="https://github.com/espressif/esp-dsp">esp-dsp</a></li>
<li><a class="reference external" href="https://github.com/pulp-platform/pulp-dsp">PLUP DSP</a> is an open
source FPGA DSP that is based on RISC-V architecture.</li>
<li>I have also dabbled in Trigonometry functions in fixed point with
HiFi architecture, which can be found in my
<a class="reference external" href="https://gist.github.com/ashafq/0578c16cf8c93ba63d9b993b496f9067">Github Gist [0578c]</a></li>
</ol>
</div>
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