2022-04-12 22:02:59 -03:00
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2022-04-12 21:54:18 -03:00
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<H2><A NAME="SECTION001021000000000000000">
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Pulse trains via waveshaping</A>
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</H2>
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<P>
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When we use waveshaping the shape of the formant is determined by
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a modulation term
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{m_a}[n] = f (a \cos(\omega n))
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\end{displaymath}
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-->
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<IMG
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WIDTH="147" HEIGHT="28" BORDER="0"
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SRC="img561.png"
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ALT="\begin{displaymath}
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{m_a}[n] = f (a \cos(\omega n))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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For small values of the index <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$">, the modulation term varies only slightly from
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the constant value <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img562.png"
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ALT="$f(0)$">, so most of the energy is concentrated at DC.
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As <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> increases, the energy spreads out among progressively higher harmonics
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of the fundamental <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. Depending on the function <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">, this spread
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may be orderly or disorderly. An orderly spread may be desirable and
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then again may not, depending on whether our goal is a predictable spectrum or
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a wide range of different (and perhaps hard-to-predict) spectra.
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<P>
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The waveshaping function <IMG
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WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img563.png"
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ALT="$f(x) = {e^x}$">, analyzed on
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Page <A HREF="node85.html#sect5.example.expon"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>,
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gives well-behaved, simple and predictable results. After normalizing suitably,
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we got the spectra shown in Figure <A HREF="node85.html#fig05.13">5.13</A>. A slight rewriting of the
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waveshaping modulator for this choice of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> (and taking the renormalization
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into account) gives:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{m_a}[n] = {e^{a \cdot (\cos(\omega n) - 1))}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="153" HEIGHT="28" BORDER="0"
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SRC="img564.png"
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ALT="\begin{displaymath}
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{m_a}[n] = {e^{a \cdot (\cos(\omega n) - 1))}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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= e ^ {
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{ -\left [
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b \sin {\omega \over 2}
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\right ] }
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^2
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="84" HEIGHT="24" BORDER="0"
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SRC="img565.png"
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ALT="\begin{displaymath}
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= e ^ {
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{ -\left [
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b \sin {\omega \over 2}
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\right ] }
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^2
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}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
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SRC="img566.png"
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ALT="${b^2}=2a$"> so that <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> is proportional to the bandwidth. This can
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be rewritten as
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{m_a}[n] = g ( b \sin {\omega \over 2} n )
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\end{displaymath}
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-->
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<IMG
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WIDTH="136" HEIGHT="35" BORDER="0"
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SRC="img567.png"
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ALT="\begin{displaymath}
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{m_a}[n] = g ( b \sin {\omega \over 2} n )
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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with
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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g(x) = e ^ {- x ^ 2}
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\end{displaymath}
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-->
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<IMG
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WIDTH="80" HEIGHT="28" BORDER="0"
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SRC="img568.png"
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ALT="\begin{displaymath}
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g(x) = e ^ {- x ^ 2}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Except for a missing normalization factor, this is a Gaussian distribution,
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sometimes called a "bell curve". The amplitudes of the harmonics are
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given by Bessel "I" type functions.
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<P>
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Another fine choice is the (again unnormalized) Cauchy distribution:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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h(x) = {1\over{1 + {x^2}}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="97" HEIGHT="40" BORDER="0"
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SRC="img569.png"
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ALT="\begin{displaymath}
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h(x) = {1\over{1 + {x^2}}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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which gives rise to a spectrum of exponentially falling harmonics:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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h(b \sin({\omega \over 2} n)) =
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G \cdot \left (
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{1\over 2} + H \cos(\omega n) + {H^2} \cos(2 \omega n)
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+ \cdots
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\right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="397" HEIGHT="45" BORDER="0"
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SRC="img570.png"
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ALT="\begin{displaymath}
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h(b \sin({\omega \over 2} n)) =
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G \cdot \left (
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{1\over 2...
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...H \cos(\omega n) + {H^2} \cos(2 \omega n)
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+ \cdots
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\right )
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img571.png"
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ALT="$G$"> and <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> are functions of the index <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">
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(explicit formulas are given in [<A
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HREF="node202.html#r-puckette95a">Puc95a</A>]).
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<P>
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In both this and the Gaussian case above, the bandwidth (counted in peaks,
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i.e., units of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">) is roughly proportional to the index <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">, and the
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amplitude of the DC term (the apex of the spectrum) is roughly proportional
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to <IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img572.png"
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ALT="$1/(1+b)$"> .
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For either waveshaping function (<IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$g$"> or <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img194.png"
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ALT="$h$">), if <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> is larger than about 2,
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the waveshape of <!-- MATH
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${m_a}(\omega n)$
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-->
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<IMG
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WIDTH="57" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img573.png"
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ALT="${m_a}(\omega n)$"> is
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approximately a (forward or backward) scan of the transfer function, so
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the resulting waveform looks
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like pulses whose widths decrease as the specified bandwidth increases.
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<P>
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<BR>
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<B> Next:</B> <A NAME="tex2html1928"
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<ADDRESS>
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