<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<title>chebyshevpoly</title>
<link rel="stylesheet" type="text/css" href="csound.css" />
<link rel="stylesheet" type="text/css" href="syntax-highlighting.css" />
<meta name="generator" content="DocBook XSL Stylesheets Vsnapshot" />
<link rel="home" href="index.html" title="The Canonical Csound Reference Manual" />
<link rel="up" href="OpcodesTop.html" title="Orchestra Opcodes and Operators" />
<link rel="prev" href="cbrt.html" title="cbrt" />
<link rel="next" href="checkbox.html" title="checkbox" />
</head>
<body>
<div class="navheader">
<table width="100%" summary="Navigation header">
<tr>
<th colspan="3" align="center">chebyshevpoly</th>
</tr>
<tr>
<td width="20%" align="left"><a accesskey="p" href="cbrt.html">Prev</a> </td>
<th width="60%" align="center">Orchestra Opcodes and Operators</th>
<td width="20%" align="right"> <a accesskey="n" href="checkbox.html">Next</a></td>
</tr>
</table>
<hr />
</div>
<div class="refentry">
<a id="chebyshevpoly"></a>
<div class="titlepage"></div>
<a id="IndexChebyshevpoly" class="indexterm"></a>
<div class="refnamediv">
<h2>
<span class="refentrytitle">chebyshevpoly</span>
</h2>
<p>chebyshevpoly —
Efficiently evaluates the sum of Chebyshev polynomials of arbitrary order.
</p>
</div>
<div class="refsect1">
<a id="idm281472947391656"></a>
<h2>Description</h2>
<p>
The <span class="emphasis"><em>chebyshevpoly</em></span> opcode calculates the value of a polynomial expression with a single a-rate input variable that is made up of a linear combination of the first N Chebyshev polynomials of the first kind. Each Chebyshev polynomial, Tn(x), is weighted by a k-rate coefficient, <span class="emphasis"><em>kn</em></span>, so that the opcode is calculating a sum of any number of terms in the form <span class="emphasis"><em>kn*Tn(x)</em></span>. Thus, the <span class="emphasis"><em>chebyshevpoly</em></span> opcode allows for the waveshaping of an audio signal with a <span class="emphasis"><em>dynamic</em></span> transfer function that gives precise control over the harmonic content of the output.
</p>
</div>
<div class="refsect1">
<a id="idm281472947406040"></a>
<h2>Syntax</h2>
<pre class="synopsis">aout <span class="command"><strong>chebyshevpoly</strong></span> ain, k0 [, k1 [, k2 [...]]]</pre>
</div>
<div class="refsect1">
<a id="idm281472947334152"></a>
<h2>Performance</h2>
<p>
<span class="emphasis"><em>ain</em></span> -- the input signal used as the independent variable of the Chebyshev polynomials ("x").
</p>
<p>
<span class="emphasis"><em>aout</em></span> -- the output signal ("y").
</p>
<p>
<span class="emphasis"><em>k0, k1, k2, ...</em></span> -- k-rate multipliers for each Chebyshev polynomial.
</p>
<p>
This opcode is very useful for dynamic waveshaping of an audio signal. Traditional waveshaping techniques utilize a lookup table for the transfer function -- usually a sum of Chebyshev polynomials. When a sine wave at full-scale amplitude is used as an index to read the table, the precise harmonic spectrum as defined by the weights of the Chebyshev polynomials is produced. A dynamic spectrum is acheived by varying the amplitude of the input sine wave, but this produces a non-linear change in the spectrum.
</p>
<p>
By directly calculating the Chebyshev polynomials, the <span class="emphasis"><em>chebyshevpoly</em></span> opcode allows more control over the spectrum and the number of harmonic partials added to the input can be varied with time. The value of each <span class="emphasis"><em>kn</em></span> coefficient directly controls the amplitude of the nth harmonic partial if the input <span class="emphasis"><em>ain</em></span> is a sine wave with amplitude = 1.0. This makes <span class="emphasis"><em>chebyshevpoly</em></span> an efficient additive synthesis engine for N partials that requires only one oscillator instead of N oscillators. The amplitude or waveform of the input signal can also be changed for different waveshaping effects.
</p>
<p>
If we consider the input parameter <span class="emphasis"><em>ain</em></span> to be "x" and the output <span class="emphasis"><em>aout</em></span> to be "y", then the <span class="emphasis"><em>chebyshevpoly</em></span> opcode calculates the following equation:
</p>
<div class="literallayout">
<p><br />
y = k0*T0(x) + k1*T1(x) + k2*T2(x) + k3*T3(x) + ...<br />
</p>
</div>
<p>
</p>
<p>
where the Tn(x) are defined by the recurrence relation
</p>
<div class="literallayout">
<p><br />
T0(x) = 1,<br />
T1(x) = x, <br />
Tn(x) = 2x*T[n-1](x) - T[n-2](x)<br />
</p>
</div>
<p>
</p>
<p>
More information about Chebyshev polynomials can be found on Wikipedia at
<a class="ulink" href="http://en.wikipedia.org/wiki/Chebyshev_polynomial" target="_top">
<em class="citetitle">http://en.wikipedia.org/wiki/Chebyshev_polynomial</em>
</a>
</p>
</div>
<div class="refsect1">
<a id="idm281472947321224"></a>
<h2>See Also</h2>
<p>
<a class="link" href="polynomial.html" title="polynomial"><em class="citetitle">polynomial</em></a>,
<a class="link" href="mac.html" title="mac"><em class="citetitle">mac</em></a>
<a class="link" href="maca.html" title="maca"><em class="citetitle">maca</em></a>
</p>
</div>
<div class="refsect1">
<a id="idm281472947317288"></a>
<h2>Examples</h2>
<p>
Here is an example of the chebyshevpoly opcode. It uses the file <a class="ulink" href="examples/chebyshevpoly.csd" target="_top"><em class="citetitle">chebyshevpoly.csd</em></a>.
</p>
<div class="example">
<a id="idm281472947315400"></a>
<p class="title">
<strong>Example 120. Example of the chebyshevpoly opcode.</strong>
</p>
<div class="example-contents">
<p>See the sections <a class="link" href="UsingRealTime.html" title="Real-Time Audio"><em class="citetitle">Real-time Audio</em></a> and <a class="link" href="CommandFlags.html" title="Csound command line"><em class="citetitle">Command Line Flags</em></a> for more information on using command line flags.</p>
<div class="refsect1">
<a id="idm281472781054504"></a>
<pre class="programlisting">
<span class="nt"><CsoundSynthesizer></span>
<span class="nt"><CsOptions></span>
<span class="c1">; Select audio/midi flags here according to platform</span>
-odac <span class="c1">;;;RT audio out</span>
<span class="c1">;-iadc ;;;uncomment -iadc if RT audio input is needed too</span>
<span class="c1">; For Non-realtime ouput leave only the line below:</span>
<span class="c1">; -o chebyshevpoly.wav -W ;;; for file output any platform</span>
<span class="nt"></CsOptions></span>
<span class="nt"><CsInstruments></span>
<span class="vg">sr</span> <span class="o">=</span> <span class="mi">44100</span>
<span class="vg">ksmps</span> <span class="o">=</span> <span class="mi">32</span>
<span class="vg">nchnls</span> <span class="o">=</span> <span class="mi">2</span>
<span class="c1">; time-varying mixture of first six harmonics</span>
<span class="kd">instr</span> <span class="nf">1</span>
<span class="c1">; According to the GEN13 manual entry,</span>
<span class="c1">; the pattern + - - + + - - for the signs of </span>
<span class="c1">; the chebyshev coefficients has nice properties.</span>
<span class="c1">; these six lines control the relative powers of the harmonics</span>
k<span class="n">1</span> <span class="nb">line</span> <span class="mf">1.0</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="mf">0.0</span>
k<span class="n">2</span> <span class="nb">line</span> <span class="o">-</span><span class="mf">0.5</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="mf">0.0</span>
k<span class="n">3</span> <span class="nb">line</span> <span class="o">-</span><span class="mf">0.333</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span>
k<span class="n">4</span> <span class="nb">line</span> <span class="mf">0.0</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="mf">0.5</span>
k<span class="n">5</span> <span class="nb">line</span> <span class="mf">0.0</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="mf">0.7</span>
k<span class="n">6</span> <span class="nb">line</span> <span class="mf">0.0</span><span class="p">,</span> <span class="nb">p3</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.0</span>
<span class="c1">; play the sine wave at a frequency of 256 Hz with amplitude = 1.0</span>
a<span class="n">x</span> <span class="nb">oscili</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">256</span><span class="p">,</span> <span class="mi">1</span>
<span class="c1">; waveshape it</span>
a<span class="n">y</span> <span class="nb">chebyshevpoly</span> a<span class="n">x</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> k<span class="n">1</span><span class="p">,</span> k<span class="n">2</span><span class="p">,</span> k<span class="n">3</span><span class="p">,</span> k<span class="n">4</span><span class="p">,</span> k<span class="n">5</span><span class="p">,</span> k<span class="n">6</span>
<span class="c1">; avoid clicks, scale final amplitude, and output</span>
a<span class="n">declick</span> <span class="nb">linseg</span> <span class="mf">0.0</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="nb">p3</span> <span class="o">-</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">0.05</span><span class="p">,</span> <span class="mf">0.0</span>
<span class="nb">outs</span> a<span class="n">y</span> <span class="o">*</span> a<span class="n">declick</span> <span class="o">*</span> <span class="mi">10000</span><span class="p">,</span> a<span class="n">y</span> <span class="o">*</span> a<span class="n">declick</span> <span class="o">*</span> <span class="mi">10000</span>
<span class="kd">endin</span>
<span class="nt"></CsInstruments></span>
<span class="nt"><CsScore></span>
<span class="nb">f</span><span class="mi">1</span> <span class="mi">0</span> <span class="mi">32768</span> <span class="mi">10</span> <span class="mi">1</span> <span class="c1">; a sine wave</span>
<span class="nb">i</span><span class="mi">1</span> <span class="mi">0</span> <span class="mi">5</span>
<span class="nb">e</span>
<span class="nt"></CsScore></span>
<span class="nt"></CsoundSynthesizer></span>
</pre>
</div>
</div>
</div>
<p><br class="example-break" />
</p>
</div>
<div class="refsect1">
<a id="idm281472947311112"></a>
<h2>Credits</h2>
<p>
</p>
<table border="0" summary="Simple list" class="simplelist">
<tr>
<td>Author: Anthony Kozar</td>
</tr>
<tr>
<td>January 2008</td>
</tr>
</table>
<p>
</p>
<p>New in Csound version 5.08</p>
</div>
</div>
<div class="navfooter">
<hr />
<table width="100%" summary="Navigation footer">
<tr>
<td width="40%" align="left"><a accesskey="p" href="cbrt.html">Prev</a> </td>
<td width="20%" align="center">
<a accesskey="u" href="OpcodesTop.html">Up</a>
</td>
<td width="40%" align="right"> <a accesskey="n" href="checkbox.html">Next</a></td>
</tr>
<tr>
<td width="40%" align="left" valign="top">cbrt </td>
<td width="20%" align="center">
<a accesskey="h" href="index.html">Home</a>
</td>
<td width="40%" align="right" valign="top"> checkbox</td>
</tr>
</table>
</div>
</body>
</html>
distrib
>
Mageia
>
7
>
armv7hl
>
media
>
core-release
>
by-pkgid
>
5fcfcb7517038d1f44ab4e478e6e61fa
>
files
>
439
