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<H2><A ID="SECTION001374000000000000000">
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Phase vocoder time bender</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.18"></A><A ID="12750"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.18:</STRONG>
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Phase vocoder for time stretching and contraction.</CAPTION>
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<TR><TD><IMG
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WIDTH="589" HEIGHT="851" BORDER="0"
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SRC="img1240.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.18.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 phase vocoder usually refers to the general technique of passing from
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(complex-valued) channel amplitudes to pairs consisting of (real-valued)
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magnitudes and phase precession rates ("frequencies"), and back, as
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described in Figure <A HREF="node175.html#fig09.11">9.11</A> (Section <A HREF="node175.html#sect9.phase">9.5</A>). In
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Example I07.phase.vocoder.pd (Figure <A HREF="#fig09.18">9.18</A>), we use this technique with the
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specific aim of time-stretching and/or time-contracting a recorded sound
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under real-time control. That is, we control, at any moment in real time,
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the location in the recorded sound we hear. Two new objects are used:
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{lrshift\~}}$
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-->
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<IMG
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WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1241.png"
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ALT="\fbox{\texttt{lrshift\~}}">:
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<A ID="12905"></A>shift a block left or right (according to its creation argument). If the
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argument is positive, each block of the output is the input shifted that number
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of spaces to the right, filling zeros in as needed on the left. A negative
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argument shifts to the left, filling zeros in at the right.
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{q8\_rsqrt\~}}$
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-->
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<IMG
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WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1243.png"
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ALT="\fbox{\texttt{q8\_rsqrt\~}}">:
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<A ID="12906"></A>quick and approximate reciprocal square root. Outputs the reciprocal of the
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square root of its input, good to about a part in 256, using much less
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computation than a full-precision square root and reciprocal would.
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<P>
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The process starts with a sub-patch, <TT>pd read-windows</TT>, that outputs
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two Hann-windowed blocks of the recorded sound, a "back" one and a
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"front" one 1/4 window further forward in the recording. The window
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shown uses the two outputs of the sub-patch to guide the amplitude
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and phase change of each channel of its own output.
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<P>
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The top two <TT>tabreceive~</TT> objects recall the previous block of complex
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amplitudes sent to the <TT>rifft~</TT> object at bottom, corresponding to
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<IMG
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WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1198.png"
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ALT="$S[m-1,k]$"> in
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the discussion of Section <A HREF="node175.html#sect9.phase">9.5</A>. The patch as a whole computes
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<IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> and then its Hann windowed inverse FT for output.
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<P>
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After normalizing <IMG
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WIDTH="80" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1198.png"
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ALT="$S[m-1,k]$">, its complex conjugate (the normalized inverse)
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is multiplied by the windowed Fourier
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transform of the "back" window <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1199.png"
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ALT="$T[k]$">, giving the product <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$"> of
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Page <A HREF="node176.html#sect9.phaserelationship"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>.
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Next, depending on the value of the parameter "lock", the computed value of
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<IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$"> is conditionally replaced with the phase-locking version <IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1245.png"
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ALT="$R'[k]$">. This
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is done using <TT>lrshift~</TT> objects, whose outputs are added into <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1206.png"
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ALT="$R[k]$"> if
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"lock" is set to one, or otherwise not if it is zero.
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The result is then normalized and multiplied by the Hann-windowed Fourier transform
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of the "front" window (<IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1196.png"
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ALT="$T'[k]$">) to give <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$">.
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<P>
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Three other applications of Fourier analysis/resynthesis, not pictured here,
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are provided in the Pd examples. First, Example I08.pvoc.reverb.pd shows how to make a
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phase vocoder whose output recirculates as in a reverberator, except that
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individual channels are replaced by the input when it is more powerful than
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what is already recirculating. The result is a more coherent-sounding reverberation
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effect than can be made in the classical way using delay lines.
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<P>
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Example I09.sheep.from.goats.pd demonstrates the (imperfect) technique of separating pitched
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signals from noisy ones, channel by channel, based on the phase coherence we
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should expect from a Hann-windowed sinusoid. If three adjacent channels are
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approximately <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> radians out of phase from each other, they are judged to
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belong to a sinusoidal peak. Channels belonging to sinusoidal peaks are
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replaced with zero to extract the noisy portion of the signal, or all others
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are replaced with zero to give the sinusoidal portion.
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<P>
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Example I10.phase.bash.pd returns to the wavetable looping sampler of Figure
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<A HREF="node29.html#fig02.07">2.7</A>, and shows how to align the phases of the sample so that all
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components of the signal have zero phase at points 0, <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">, <IMG
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WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1200.png"
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ALT="$2N$">, and so on. In
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this way, two copies of a looping sampler placed <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples apart can be
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coherently cross-faded. A synthetic, pitched version of the original soundfile
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can be made using daisy-chained cross-fades.
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<P>
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HREF="node181.html">Timbre stamp ("vocoder")</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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