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<h2>
<span class="refentrytitle">resonz</span>
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<p>resonz —
A bandpass filter with variable frequency response.
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<div class="refsect1" title="Description">
<a id="id3162882"></a>
<h2>Description</h2>
<p>
Implementations of a second-order, two-pole two-zero bandpass filter with variable frequency response.
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<div class="refsect1" title="Syntax">
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<h2>Syntax</h2>
<pre class="synopsis">ares <span class="command"><strong>resonz</strong></span> asig, kcf, kbw [, iscl] [, iskip]</pre>
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<div class="refsect1" title="Initialization">
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<h2>Initialization</h2>
<p>
The optional initialization variables for <span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> are identical to the i-time variables for <a class="link" href="reson.html" title="reson"><em class="citetitle">reson</em></a>.
</p>
<p>
<span class="emphasis"><em>iskip</em></span> -- initial disposition of internal data space. Since filtering incorporates a feedback loop of previous output, the initial status of the storage space used is significant. A zero value will clear the space; a non-zero value will allow previous information to remain. The default value is 0.
</p>
<p>
<span class="emphasis"><em>iscl</em></span> -- coded scaling factor for resonators. A value of 1 signifies a peak response factor of 1, i.e. all frequencies other than <span class="emphasis"><em>kcf</em></span> are attenuated in accordance with the (normalized) response curve. A value of 2 raises the response factor so that its overall RMS value equals 1. This intended equalization of input and output power assumes all frequencies are physically present; hence it is most applicable to white noise. A zero value signifies no scaling of the signal, leaving that to some later adjustment (see <a class="link" href="balance.html" title="balance"><em class="citetitle">balance</em></a>). The default value is 0.
</p>
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<div class="refsect1" title="Performance">
<a id="id3163793"></a>
<h2>Performance</h2>
<p>
<span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> are variations of the classic two-pole bandpass resonator (<a class="link" href="reson.html" title="reson"><em class="citetitle">reson</em></a>). Both filters have two zeroes in their transfer functions, in addition to the two poles. <span class="emphasis"><em>resonz</em></span> has its zeroes located at z = 1 and z = -1. <span class="emphasis"><em>resonr</em></span> has its zeroes located at +sqrt(<span class="emphasis"><em>R</em></span>) and -sqrt(<span class="emphasis"><em>R</em></span>), where <span class="emphasis"><em>R</em></span> is the radius of the poles in the complex z-plane. The addition of zeroes to <span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> results in the improved selectivity of the magnitude response of these filters at cutoff frequencies close to 0, at the expense of less selectivity of frequencies above the cutoff peak.
</p>
<p>
<span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> are very close to constant-gain as the center frequency is swept, resulting in a more efficient control of the magnitude response than with traditional two-pole resonators such as <span class="emphasis"><em>reson</em></span>.
</p>
<p>
<span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> produce a sound that is considerably different from <span class="emphasis"><em>reson</em></span>, especially for lower center frequencies; trial and error is the best way of determining which resonator is best suited for a particular application.
</p>
<p>
<span class="emphasis"><em>asig</em></span> -- input signal to be filtered
</p>
<p>
<span class="emphasis"><em>kcf</em></span> -- cutoff or resonant frequency of the filter, measured in Hz
</p>
<p>
<span class="emphasis"><em>kbw</em></span> -- bandwidth of the filter (the Hz difference between the upper and lower half-power points)
</p>
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<div class="refsect1" title="Technical History">
<a id="id3163909"></a>
<h2>Technical History</h2>
<p>
<span class="emphasis"><em>resonr</em></span> and <span class="emphasis"><em>resonz</em></span> were originally described in an article by Julius O. Smith and James B. Angell.<sup>1</sup> Smith and Angell recommended the <span class="emphasis"><em>resonz</em></span> form (zeros at +1 and -1) when computational efficiency was the main concern, as it has one less multiply per sample, while <span class="emphasis"><em>resonr</em></span> (zeroes at + and - the square root of the pole radius R) was recommended for situations when a perfectly constant-gain center peak was required.
</p>
<p>
Ken Steiglitz, in a later article <sup>2</sup>, demonstrated that <span class="emphasis"><em>resonz</em></span> had constant gain at the true peak of the filter, as opposed to <span class="emphasis"><em>resonr</em></span>, which displayed constant gain at the pole angle. Steiglitz also recommended <span class="emphasis"><em>resonz</em></span> for its sharper notches in the gain curve at zero and Nyquist frequency. Steiglitz's recent book <sup>3</sup> features a thorough technical discussion of <span class="emphasis"><em>reson</em></span> and <span class="emphasis"><em>resonz</em></span>, while Dodge and Jerse's textbook <sup>4</sup> illustrates the differences in the response curves of <span class="emphasis"><em>reson</em></span> and <span class="emphasis"><em>resonz</em></span>.
</p>
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<div class="refsect1" title="References">
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<h2>References</h2>
<p>
</p>
<div class="orderedlist">
<ol class="orderedlist" type="1">
<li class="listitem">
<p>
Smith, Julius O. and Angell, James B., "A Constant-Gain Resonator Tuned by a Single Coefficient," <span class="emphasis"><em>Computer Music Journal</em></span>, vol. 6, no. 4, pp. 36-39, Winter 1982.
</p>
</li>
<li class="listitem">
<p>
Steiglitz, Ken, "A Note on Constant-Gain Digital Resonators," <span class="emphasis"><em>Computer Music Journal</em></span>, vol. 18, no. 4, pp. 8-10, Winter 1994.
</p>
</li>
<li class="listitem">
<p>
Ken Steiglitz, <span class="emphasis"><em>A Digital Signal Processing Primer, with Applications to Digital Audio and Computer Music</em></span>. Addison-Wesley Publishing Company, Menlo Park, CA, 1996.
</p>
</li>
<li class="listitem">
<p>
Dodge, Charles and Jerse, Thomas A., <span class="emphasis"><em>Computer Music: Synthesis, Composition, and Performance</em></span>. New York: Schirmer Books, 1997, 2nd edition, pp. 211-214.
</p>
</li>
</ol>
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<p>
</p>
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<div class="refsect1" title="See Also">
<a id="id3164065"></a>
<h2>See Also</h2>
<p>
<a class="link" href="resonr.html" title="resonr"><em class="citetitle">resonr</em></a>
</p>
</div>
<div class="refsect1" title="Credits">
<a id="id3164082"></a>
<h2>Credits</h2>
<p>
</p>
<table border="0" summary="Simple list" class="simplelist">
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<td>Author: Sean Costello</td>
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<td>Seattle, Washington</td>
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<td>1999</td>
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<p>New in Csound version 3.55</p>
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