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<H2><A ID="SECTION001371000000000000000">
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Fourier analysis and resynthesis in Pd</A>
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</H2>
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<P>
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Example I01.Fourier.analysis.pd (Figure <A HREF="#fig09.14">9.14</A>, part a) demonstrates computing the
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Fourier transform of an audio signal using the <TT>fft~</TT> object:
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.14"></A><A ID="12858"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
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Fourier analysis in Pd: (a) the <!-- MATH
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$\mathrm{fft}\sim$
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-->
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<IMG
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WIDTH="35" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1214.png"
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ALT="$\mathrm{fft}\sim$"> object; (b)
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using a subwindow to control block size of the Fourier transform; (c)
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the subwindow, using a real Fourier transform (the <TT>fft~</TT>object) and the
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Hann windowing function.</CAPTION>
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<TR><TD><IMG
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WIDTH="530" HEIGHT="293" BORDER="0"
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SRC="img1215.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.14.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{fft\~}}$
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-->
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<IMG
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WIDTH="47" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1216.png"
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ALT="\fbox{\texttt{fft\~}}">:
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<A ID="12898"></A><A ID="12696"></A>Fast Fourier transform. The two inlets take audio signals representing the real
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and imaginary parts of a complex-valued signal. The window size <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="$N$"> is given
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by Pd's block size. One Fourier transform is done on each block.
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<P>
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The Fast Fourier transform [<A
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HREF="node202.html#r-smith03">SI03</A>] reduces the computational cost of
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Fourier analysis in Pd to only that of between 5 and 15 <TT>osc~</TT> objects in
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typical configurations. The FFT algorithm in its simplest form takes <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="$N$"> to
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be a power of two, which is also (normally) a constraint on block sizes in Pd.
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<P>
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Example I02.Hann.window.pd (Figure <A HREF="#fig09.14">9.14</A>, parts b and c) shows how to control the
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block size using a <TT>block~</TT> object, how to apply a Hann window, and a
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different version of the Fourier transform. Part (b) shows the invocation of a
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subwindow which in turn is shown in part (c). New objects are:
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{rfft\~}}$
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-->
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<IMG
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WIDTH="56" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1218.png"
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ALT="\fbox{\texttt{rfft\~}}">:
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<A ID="12899"></A>real Fast Fourier transform. The imaginary part of the input is assumed to
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be zero. Only the first <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1220.png"
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ALT="$N/2+1$"> channels of output are filled in (the others
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are determined by symmetry). This takes half the computation time of the
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(more general) <TT>fft~</TT>object.
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{tabreceive\~}}$
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-->
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<IMG
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WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1221.png"
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ALT="\fbox{\texttt{tabreceive\~}}">:
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<A ID="12900"></A>repeatedly outputs the contents of a wavetable. Each
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block of computation outputs the same first <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 of the table.
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<P>
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In this example, the table "$0-hann" holds a Hann window function
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of length 512, in agreement with the specified block size. The signal
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to be analyzed appears (from the parent patch) via the <TT>inlet~</TT> object.
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The channel amplitudes (the output of the <TT>rfft~</TT> object) are reduced
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to real-valued magnitudes: the real and imaginary parts are squared separately,
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the two squares are added, and the result passed to the <TT>sqrt~</TT> object.
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Finally the magnitude is written (controlled by a connection not shown in
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the figure) via <TT>tabwrite~</TT> to another table, "$0-magnitude", for
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graphing.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.15"></A><A ID="12707"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.15:</STRONG>
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Fourier analysis and resynthesis, using <TT>block~</TT> to specify an
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overlap of 4, and <TT>rifft~</TT> to reconstruct the signal after modification.</CAPTION>
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<TR><TD><IMG
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WIDTH="646" HEIGHT="413" BORDER="0"
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SRC="img1223.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.15.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Example I03.resynthesis.pd (Figure <A HREF="#fig09.15">9.15</A>) shows how to analyze and resynthesize
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an audio signal following the strategy of Figure <A HREF="node172.html#fig09.07">9.7</A>.
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As before there is a sub-window to do the work at a block size appropriate to
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the task; the figure shows only the sub-window.
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We need one new object for the inverse Fourier transform:
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<P>
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<BR><!-- MATH
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$\fbox{\texttt{rifft\~}}$
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-->
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<IMG
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WIDTH="64" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
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SRC="img1224.png"
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ALT="\fbox{\texttt{rifft\~}}">:
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<A ID="12901"></A>real inverse Fast Fourier transform. Using the first <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1220.png"
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ALT="$N/2+1$"> points of its
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inputs (taken to be a real/imaginary pair), and assuming the appropriate values
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for the other channels by symmetry, reconstructs a real-valued output. No
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normalization is done, so that a <TT>rfft~</TT>/<TT>rifft~</TT> pair together
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result in a gain of <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="$N$">. The <TT>ifft~</TT> object is also available
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which computes an unnormalized inverse for the <TT>fft~</TT> object,
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reconstructing a complex-valued output.
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<P>
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The <TT>block~</TT> object, in the subwindow, is invoked with a second argument
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which specifies an overlap factor of 4. This dictates that the sub-window will
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run four times every <IMG
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WIDTH="63" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1226.png"
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ALT="$N=512$"> samples, at regular intervals of 128 samples. The
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<TT>inlet~</TT> object does the necessary buffering and rearranging of samples
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so that its output always gives the 512 latest samples of input in order. In
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the other direction, the <TT>outlet~</TT> object adds segments of
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its previous four inputs to carry out the overlap-add scheme shown in Figure
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<A HREF="node172.html#fig09.07">9.7</A>.
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<P>
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The 512-sample blocks are multiplied by the Hann window both at
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the input and the output. If the <TT>rfft~</TT> and <TT>rifft~</TT> objects
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were connected without any modifications in between, the output would
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faithfully reconstruct the input.
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<P>
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A modification is applied, however: each channel is multiplied by a
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(positive real-valued) gain. The complex-valued amplitude for each channel is
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scaled by separately multiplying the real and imaginary parts by the gain. The
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gain (which depends on the channel) comes from another table, named
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"$0-gain". The result is a graphical equalization filter; by mousing in the
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graphical window for this table, you can design gain-frequency curves.
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<P>
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There is an inherent delay introduced by using <TT>block~</TT> to increase
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the block size (but none if it is used, as shown in Chapter 7, to reduce
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block size relative to the parent window.) The delay can be measured from
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the inlet to the outlet of the sub-patch, and is equal to the difference of the
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two block sizes. In this example the buffering delay is 512-64=448 samples.
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Blocking delay does not depend on overlap, only on block sizes.
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<P>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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