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HREF="node86.html">Sinusoidal waveshaping: evenness and</A>
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<H2><A NAME="SECTION00955000000000000000">
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Waveshaping using an exponential function</A>
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</H2>
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<A NAME="sect5.example.expon"></A>
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<P>
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We return again to the spectra computed on Page <A HREF="node78.html#eq-waveshaping"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>,
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corresponding to waveshaping functions of the form <IMG
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WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img504.png"
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ALT="$f(x) = x^k$">. We note
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with pleasure that not only are they all in phase (so that they can
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be superposed with easily predictable results) but also that the spectra
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spread out as <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> increases. Also, in a series
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of the form,
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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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f(x) = {f_0} + {f_1} x + {f_2} {x^2} + \cdots,
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\end{displaymath}
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-->
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<IMG
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WIDTH="205" HEIGHT="28" BORDER="0"
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SRC="img505.png"
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ALT="\begin{displaymath}
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f(x) = {f_0} + {f_1} x + {f_2} {x^2} + \cdots,
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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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a higher index of modulation will lend more relative weight to the higher
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power terms in the expansion; as we saw seen earlier, if the index of
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modulation is <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 various <IMG
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WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img506.png"
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ALT="$x^k$"> terms are multiplied by <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img507.png"
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ALT="$f_0$">,
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<IMG
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WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img508.png"
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ALT="$af_1$">, <IMG
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WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img509.png"
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ALT="${a^2}{f_2}$">, and so on.
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<P>
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Now suppose we wish to arrange for different terms in the above expansion
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to dominate the result in a predictable way as a function 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="$a$">.
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To choose the simplest possible example, suppose we wish <IMG
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WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img507.png"
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ALT="$f_0$"> to be the largest
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term for <IMG
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WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img510.png"
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ALT="$0<a<1$">, then for it to be overtaken by the more quickly growing
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<IMG
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WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img508.png"
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ALT="$af_1$"> term for <IMG
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WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img511.png"
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ALT="$1<a<2$">, which is then overtaken by the <IMG
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WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img509.png"
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ALT="${a^2}{f_2}$"> term for
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<IMG
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WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img512.png"
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ALT="$2<a<3$"> and so on, so that each <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">th term takes over at index <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">.
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To make this happen we just require that
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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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{f_1} = {f_0} , 2 {f_2} = {f_1}, 3 {f_3} = {f_2} , \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="207" HEIGHT="27" BORDER="0"
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SRC="img513.png"
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ALT="\begin{displaymath}
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{f_1} = {f_0} , 2 {f_2} = {f_1}, 3 {f_3} = {f_2} , \ldots
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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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and so choosing <IMG
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WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img514.png"
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ALT="${f_0}=0$">, we get <IMG
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WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img515.png"
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ALT="${f_1}=1$">, <IMG
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WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img516.png"
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ALT="${f_2}=1/2$">, <IMG
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WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img517.png"
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ALT="${f_3}=1/6$">, and in
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general,
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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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{f_k} = {1 \over {1 \cdot 2 \cdot 3 \cdot ... \cdot k}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="129" HEIGHT="38" BORDER="0"
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SRC="img518.png"
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ALT="\begin{displaymath}
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{f_k} = {1 \over {1 \cdot 2 \cdot 3 \cdot ... \cdot k}}
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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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These happen to be the coefficients of the power series for the function
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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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f(x) = {e ^ x}
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\end{displaymath}
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-->
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<IMG
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WIDTH="65" HEIGHT="28" BORDER="0"
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SRC="img519.png"
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ALT="\begin{displaymath}
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f(x) = {e ^ x}
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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="52" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img520.png"
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ALT="$e \approx 2.7$"> is Euler's constant.
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<P>
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Before plugging in <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img521.png"
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ALT="$e^x$"> as a transfer function it's wise to plan how we
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will deal with signal amplitude, since <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img521.png"
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ALT="$e^x$"> grows quickly as
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<IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$"> increases. If we're going to plug in a sinusoid of amplitude <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 maximum output
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will be <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img522.png"
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ALT="$e^a$">, occurring whenever the phase is zero. A simple and natural
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choice is simply to divide by <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img522.png"
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ALT="$e^a$"> to reduce the peak to one, giving:
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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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f(a \cos(\omega n)) =
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{{{e^{a \cos(\omega n)}}} \over {e^a}} = {e^{a (\cos(\omega n) - 1)}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="275" HEIGHT="42" BORDER="0"
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SRC="img523.png"
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ALT="\begin{displaymath}
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f(a \cos(\omega n)) =
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{{{e^{a \cos(\omega n)}}} \over {e^a}} = {e^{a (\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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This is realized in Example E06.exponential.pd. Resulting spectra for
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<IMG
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WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img524.png"
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ALT="$a=$"> 0, 4, and 16 are shown in Figure <A HREF="#fig05.13">5.13</A>. As the waveshaping index
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rises, progressively less energy is present in the fundamental; the energy
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is increasingly spread over the partials.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig05.13"></A><A NAME="5876"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.13:</STRONG>
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Spectra of waveshaping output using an exponential transfer function.
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Indices of modulation of 0, 4, and 16 are shown; note the different vertical
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scales.</CAPTION>
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