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<refentry id="resonz">
  <refmeta>
    <refentrytitle>resonz</refentrytitle>
  </refmeta>
 
  <refnamediv>
    <refname>resonz</refname>
    <refpurpose>
      A bandpass filter with variable frequency response.
      <indexterm id="IndexResonz"><primary>resonz</primary></indexterm>
    </refpurpose>
  </refnamediv>
 
  <refsect1>
    <title>Description</title>
    <para>
      Implementations of a second-order, two-pole two-zero bandpass filter with variable frequency response.
    </para>
  </refsect1>
 
  <refsect1>
    <title>Syntax</title>
    <synopsis>ar <command>resonz</command> asig, kcf, kbw [, iscl] [, iskip]</synopsis>
  </refsect1>
 
  <refsect1>
    <title>Initialization</title>
    <para>
      The optional initialization variables for <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> are identical to the i-time variables for <link linkend="reson"><citetitle>reson</citetitle></link>.
    </para>

    <para>
      <emphasis>iskip</emphasis> -- 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.
    </para>

    <para>
      <emphasis>iscl</emphasis> -- coded scaling factor for resonators. A value of 1 signifies a peak response factor of 1, i.e.  all frequencies other than <emphasis>kcf</emphasis> 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 <link linkend="balance"><citetitle>balance</citetitle></link>). The default value is 0.
    </para>
  </refsect1>
 
  <refsect1>
    <title>Performance</title>
    <para>
      <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> are variations of the classic two-pole bandpass resonator (<link linkend="reson"><citetitle>reson</citetitle></link>). Both filters have two zeroes in their transfer functions, in addition to the two poles. <emphasis>resonz</emphasis> has its zeroes located at z = 1 and z = -1. <emphasis>resonr</emphasis> has its zeroes located at +sqrt(<emphasis>R</emphasis>) and -sqrt(<emphasis>R</emphasis>), where <emphasis>R</emphasis> is the radius of the poles in the complex z-plane. The addition of zeroes to <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> 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.
    </para>

    <para>
      <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> 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 <emphasis>reson</emphasis>.
    </para>

    <para>
      <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> produce a sound that is considerably different from <emphasis>reson</emphasis>, especially for lower center frequencies; trial and error is the best way of determining which resonator is best suited for a particular application.
    </para>

    <para>
      <emphasis>asig</emphasis> -- input signal to be filtered
    </para>

    <para>
      <emphasis>kcf</emphasis> -- cutoff or resonant frequency of the filter, measured in Hz
    </para>

    <para>
      <emphasis>kbw</emphasis> -- bandwidth of the filter (the Hz difference between the upper and lower half-power points)
    </para>
 </refsect1>
 
  <refsect1>
    <title>Technical History</title>
    <para>
      <emphasis>resonr</emphasis> and <emphasis>resonz</emphasis> were originally described in an article by Julius O. Smith and James B. Angell.<superscript>1</superscript> Smith and Angell recommended the <emphasis>resonz</emphasis> form (zeros at +1 and -1) when computational efficiency was the main concern, as it has one less multiply per sample, while <emphasis>resonr</emphasis> (zeroes at + and - the square root of the pole radius R) was recommended for situations when a perfectly constant-gain center peak was required.
    </para>

    <para>
      Ken Steiglitz, in a later article <superscript>2</superscript>, demonstrated that <emphasis>resonz</emphasis> had constant gain at the true peak of the filter, as opposed to <emphasis>resonr</emphasis>, which displayed constant gain at the pole angle. Steiglitz also recommended <emphasis>resonz</emphasis> for its sharper notches in the gain curve at zero and Nyquist frequency. Steiglitz's recent book <superscript>3</superscript> features a thorough technical discussion of <emphasis>reson</emphasis> and <emphasis>resonz</emphasis>, while Dodge and Jerse's textbook <superscript>4</superscript> illustrates the differences in the response curves of <emphasis>reson</emphasis> and <emphasis>resonz</emphasis>.
    </para>
  </refsect1>
 
  <refsect1>
    <title>References</title>
    <para>
      <orderedlist>
        <listitem>
          <para>
            Smith, Julius O. and Angell, James B., &quot;A Constant-Gain Resonator Tuned by a Single Coefficient,&quot; <emphasis>Computer Music Journal</emphasis>, vol. 6, no. 4, pp. 36-39, Winter 1982.
          </para>
        </listitem>

        <listitem>
          <para>
            Steiglitz, Ken, &quot;A Note on Constant-Gain Digital Resonators,&quot; <emphasis>Computer Music Journal</emphasis>, vol. 18, no. 4, pp. 8-10, Winter 1994.
          </para>
        </listitem>

        <listitem>
          <para>
            Ken Steiglitz, <emphasis>A Digital Signal Processing Primer, with Applications to Digital Audio and Computer Music</emphasis>. Addison-Wesley Publishing Company, Menlo Park, CA, 1996.
          </para>
        </listitem>

        <listitem>
          <para>
            Dodge, Charles and Jerse, Thomas A., <emphasis>Computer Music: Synthesis, Composition, and Performance</emphasis>. New York: Schirmer Books, 1997, 2nd edition, pp. 211-214.
          </para>
        </listitem>
      </orderedlist>
    </para>
  </refsect1>

  <refsect1>
    <title>See Also</title>
    <para>
      <link linkend="resonr"><citetitle>resonr</citetitle></link>
    </para>
  </refsect1>
 
  <refsect1>
    <title>Credits</title>
    <para>
      <simplelist>
        <member>Author: &namesean;</member>
        <member>Seattle, Washington</member>
        <member>1999</member>
      </simplelist>
    </para>

    <para>New in Csound version 3.55</para>
  </refsect1>
</refentry>
