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<H1><A ID="SECTION001040000000000000000"></A>
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<A ID="sect6.paf"></A>
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<BR>
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Phase-aligned formant (PAF) generator
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</H1>
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<P>
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Combining the two-cosine carrier signal with the waveshaping pulse generator
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gives the
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<A ID="6911"></A><I>phase-aligned formant</I>
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generator, usually called by its acronym, PAF. (The PAF is the subject of a
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1994 patent owned by IRCAM.) The combined formula is,
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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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x[n] = {
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{\underbrace
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{g \left ( a \sin (\omega n/2) \right )}_{\mathrm{modulator}}
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}
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\;
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{\underbrace {\left [
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{p \cos( k \omega n) + q \cos( (k+1) \omega n)}
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\right ] } _\mathrm{carrier}}
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="369" HEIGHT="51" BORDER="0"
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SRC="img598.png"
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ALT="\begin{displaymath}
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x[n] = {
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{\underbrace
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{g \left ( a \sin (\omega n/2) \ri...
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...) + q \cos( (k+1) \omega n)}
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\right ] } _\mathrm{carrier}}
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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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Here the 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="$g$"> may be either the Gaussian or Cauchy waveshaping
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function, <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 $"> is the fundamental frequency, <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> is a modulation
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index controlling bandwidth, and <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> control the formant
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center frequency.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.08"></A><A ID="6919"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.8:</STRONG>
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The PAF generator as a block diagram.</CAPTION>
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<TR><TD><IMG
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WIDTH="478" HEIGHT="689" BORDER="0"
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SRC="img599.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.08.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig06.08">6.8</A> shows
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the PAF as a block diagram, separated into a phase
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generation step, a carrier, and a modulator. The phase generation step
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outputs a sawtooth signal at the fundamental frequency.
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The modulator is done by standard waveshaping, with a slight twist added.
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The formula for the modulator signals in the PAF call for an incoming
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sinusoid at half the fundamental frequency, i.e., <!-- MATH
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$\sin({\omega \over 2})$
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-->
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<IMG
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WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img600.png"
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ALT="$\sin({\omega \over 2})$">,
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and this nominally would require us to use a phasor tuned to half the
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fundamental frequency. However, since the waveshaping function is even,
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we may substitute the absolute value of the sinusoid:
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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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\left | \sin({\omega \over 2}) \right |
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\end{displaymath}
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-->
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<IMG
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WIDTH="53" HEIGHT="35" BORDER="0"
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SRC="img601.png"
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ALT="\begin{displaymath}
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\left \vert \sin({\omega \over 2}) \right \vert
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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 repeats at the frequency <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 $"> (the first half cycle is the same
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as the second one.) We can compute this simply by using a half-cycle
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sinusoid as a wavetable lookup function (with phase running from <IMG
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WIDTH="41" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img283.png"
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ALT="$-\pi/2$">
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to <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$">), and it is this rectified sinusoid that we pass to the
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waveshaping function.
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<P>
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Although the wavetable function is pictured over both negative and positive
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values (reaching from -10 to 10), in fact we're only using the positive
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side for lookup, ranging from 0 to <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">, the index of modulation.
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If the index of modulation exceeds the input range of the table (here set
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to stop at 10 as an example), the table lookup address should be
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clipped. The table should extend far enough into the tail of the waveshaping
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function so that the effect of clipping is inaudible.
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<P>
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The carrier signal is a weighted sum of two cosines, whose frequencies are
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increased by multiplication (by <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> and <IMG
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WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img602.png"
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ALT="$k+1$">, respectively) and wrapping. In
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this way all the lookup
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phases are controlled by the same sawtooth oscillator.
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<P>
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The quantities <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$">, and the wavetable index <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">
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are calculated as shown in Figure <A HREF="#fig06.09">6.9</A>. They are
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functions of the specified fundamental frequency, the formant center frequency,
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and the bandwidth, which are the original parameters of the algorithm. The
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quantity <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, not shown in the figure, is just <IMG
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WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img603.png"
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ALT="$1-q$">.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.09"></A><A ID="7044"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.9:</STRONG>
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Calculation of the time-varying parameters <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 waveshaping index),
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<I>k</I>, and <I>q</I> for use in the block diagram of Figure <A HREF="#fig06.08">6.8</A>.</CAPTION>
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<TR><TD><IMG
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WIDTH="322" HEIGHT="326" BORDER="0"
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SRC="img604.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.09.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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As described in the previous section, the quantities <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="$k$">, <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">, and <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> should
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only change at phase wraparound points, that is to say, at periods of
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<IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img138.png"
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ALT="$2 \pi / \omega$">. Since the calculation of <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="$k$">, etc., depends on the value
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of the parameter <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 $">, it follows that <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> itself should only be
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updated
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when the phase is a multiple of <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">; otherwise, a change in <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> could
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send the center frequency <IMG
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WIDTH="62" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img605.png"
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ALT="$(k+q)\omega$"> to an incorrect value for a
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(very audible)
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fraction of a period. In effect, all the parameter calculations should be
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synchronized to the phase of the original oscillator.
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<P>
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Having the oscillator's phase control the updating of its own frequency
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is an example of
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<A ID="6933"></A><I>feedback</I>,
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which in general means using any process's output as one of its inputs.
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When processing digital audio signals at a fixed sample rate (as we're doing),
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it is never possible to have the process's <I>current</I> output as an input,
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since at the time we would need it we haven't yet calculated it. The best
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we can hope for is to use the previous sample of output--in effect, adding
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one sample of delay. In block environments (such as Max, Pd, and Csound) the
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situation becomes more complicated, but we will delay discussing that until
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the next chapter (and simply wish away the problem in the examples at the
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end of this one).
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<P>
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The amplitude of the central peak in the spectrum of the PAF generator is
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roughly <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="$1/(1+b)$">; in other words, close to unity when 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="$b$"> is
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smaller than one, and falling off inversely with larger values of <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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For values of <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="$b$"> less than about ten, the loudness of the output does
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not vary greatly, since the introduction of other partials, even at lower
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amplitudes, offsets the decrease of the center partial's amplitude. However,
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if using the PAF to generate formants with specified peak amplitudes, the
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output should be multiplied by <IMG
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WIDTH="37" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img606.png"
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ALT="$1+b$"> (or even, if necessary, a better
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approximation of the correction factor, whose exact value depends on the
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waveshaping function).
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This amplitude correction should be ramped, not
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sampled-and-held.
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<P>
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Since the expansion of the waveshaping (modulator) signal consists of all
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cosine terms (i.e., since they all have initial phase zero), as do the two
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components of the carrier, it follows from the cosine product formula that the
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components of the result are all cosines as well. This means that any number of
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PAF generators, if they are made to share the same oscillator for phase
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generation, will all be in phase and combining them gives the sum of the
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individual spectra. So we can make a multiple-formant version as shown in
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Figure <A HREF="#fig06.10">6.10</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.10"></A><A ID="6939"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.10:</STRONG>
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Block diagram for making a spectrum with two formants using the PAF
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generator.</CAPTION>
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<TR><TD><IMG
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WIDTH="493" HEIGHT="466" BORDER="0"
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SRC="img607.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.10.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig06.11">6.12</A> shows a possible output of a pair of formants generated
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this way; the first formant is centered halfway between partials 3 and 4, and
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the second at partial 12, with lower amplitude and bandwidth. The Cauchy
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waveshaping function was used, which makes linearly sloped spectra (viewed
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in dB). The two superpose additively, so that the spectral envelope curves
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smoothly from one formant to the other. The lower formant also adds to its
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own reflection about the vertical axis, so that it appears slightly curved
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upward there.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.11"></A><A ID="6945"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.11:</STRONG>
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Spectrum from a two-formant PAF generator.</CAPTION>
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<TR><TD><IMG
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WIDTH="310" HEIGHT="230" BORDER="0"
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SRC="img608.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The PAF generator can be altered if desired to make inharmonic spectra by
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sliding the partials upward or downward in frequency. To do this, add a second
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oscillator to the phase of both carrier cosines, but not to the phase of the
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modulation portion of the diagram, nor to the controlling phase of the
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sample-and-hold units. It turns out that the sample-and-hold strategy for
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smooth parameter updates still works; and furthermore, multiple PAF generators
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sharing the same phase generation portion will still be in phase with each
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other.
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<P>
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This technique for superposing spectra does not work as predictably for phase
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modulation as it does for the PAF generator; the partials of the phase
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modulation output have complicated phase relationships and they seem difficult
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to combine coherently. In general, phase modulation will give more complicated
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patterns of spectral evolution, whereas the PAF is easier to predict and turn to
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specific desired effects.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.11"></A><A ID="6952"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.12:</STRONG>
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Spectrum from a two-formant PAF generator.</CAPTION>
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<TR><TD><IMG
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WIDTH="318" HEIGHT="235" BORDER="0"
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SRC="img609.png"
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ALT="\begin{figure}\vspace*{-15pt}
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\centerline{\psfig{file=figs/fig06.11.ps,width=3.8in}}
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\vspace*{-13pt}\vspace*{13pt}
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\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<HR>
|
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Miller Puckette
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2006-12-30
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