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<div class="refentry" title="chebyshevpoly">
<a id="chebyshevpoly"></a>
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<a id="IndexChebyshevpoly" class="indexterm"></a>
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<h2>
<span class="refentrytitle">chebyshevpoly</span>
</h2>
<p>chebyshevpoly —
Efficiently evaluates the sum of Chebyshev polynomials of arbitrary order.
</p>
</div>
<div class="refsect1" title="Description">
<a id="idp15515752"></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" title="Syntax">
<a id="idp15554936"></a>
<h2>Syntax</h2>
<pre class="synopsis">aout <span class="command"><strong>chebyshevpoly</strong></span> ain, k0 [, k1 [, k2 [...]]]</pre>
</div>
<div class="refsect1" title="Performance">
<a id="idp15556080"></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" title="See Also">
<a id="idp15563512"></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" title="Examples">
<a id="idp15565480"></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="idp15566472"></a>
<p class="title">
<strong>Example 100. 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>
<pre class="programlisting">
<span class="csdtag"><CsoundSynthesizer></span>
<span class="csdtag"><CsOptions></span>
<span class="comment">; Select audio/midi flags here according to platform</span>
-odac <span class="comment">;;;RT audio out</span>
<span class="comment">;-iadc ;;;uncomment -iadc if RT audio input is needed too</span>
<span class="comment">; For Non-realtime ouput leave only the line below:</span>
<span class="comment">; -o chebyshevpoly.wav -W ;;; for file output any platform</span>
<span class="csdtag"></CsOptions></span>
<span class="csdtag"><CsInstruments></span>
<span class="ohdr">sr</span> <span class="op">=</span> 44100
<span class="ohdr">ksmps</span> <span class="op">=</span> 32
<span class="ohdr">nchnls</span> <span class="op">=</span> 2
<span class="comment">; time-varying mixture of first six harmonics</span>
<span class="oblock">instr</span> 1
<span class="comment">; According to the GEN13 manual entry,</span>
<span class="comment">; the pattern + - - + + - - for the signs of </span>
<span class="comment">; the chebyshev coefficients has nice properties.</span>
<span class="comment">; these six lines control the relative powers of the harmonics</span>
k1 <span class="opc">line</span> 1.0, p3, 0.0
k2 <span class="opc">line</span> <span class="op">-</span>0.5, p3, 0.0
k3 <span class="opc">line</span> <span class="op">-</span>0.333, p3, <span class="op">-</span>1.0
k4 <span class="opc">line</span> 0.0, p3, 0.5
k5 <span class="opc">line</span> 0.0, p3, 0.7
k6 <span class="opc">line</span> 0.0, p3, <span class="op">-</span>1.0
<span class="comment">; play the sine wave at a frequency of 256 Hz with amplitude = 1.0</span>
ax <span class="opc">oscili</span> 1, 256, 1
<span class="comment">; waveshape it</span>
ay <span class="opc">chebyshevpoly</span> ax, 0, k1, k2, k3, k4, k5, k6
<span class="comment">; avoid clicks, scale final amplitude, and output</span>
adeclick <span class="opc">linseg</span> 0.0, 0.05, 1.0, p3 <span class="op">-</span> 0.1, 1.0, 0.05, 0.0
<span class="opc">outs</span> ay <span class="op">*</span> adeclick <span class="op">*</span> 10000, ay <span class="op">*</span> adeclick <span class="op">*</span> 10000
<span class="oblock">endin</span>
<span class="csdtag"></CsInstruments></span>
<span class="csdtag"><CsScore></span>
<span class="stamnt">f</span>1 0 32768 10 1 <span class="comment">; a sine wave</span>
<span class="stamnt">i</span>1 0 5
<span class="stamnt">e</span>
<span class="csdtag"></CsScore></span>
<span class="csdtag"></CsoundSynthesizer></span>
</pre>
</div>
</div>
<p><br class="example-break" />
</p>
</div>
<div class="refsect1" title="Credits">
<a id="idp15568760"></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>
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