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<H1><A NAME="SECTION001340000000000000000">
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Fourier analysis and reconstruction of audio signals</A>
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</H1>
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<P>
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Fourier analysis can sometimes be used to resolve the component sinusoids in an
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audio signal. Even when it can't go that far, it can separate a
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signal into frequency regions, in the sense that for each <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="$k$">, the <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="$k$">th point
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of the Fourier transform would be affected only by components close to
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the nominal frequency <IMG
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WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1145.png"
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ALT="$k\omega$">. This suggests many interesting operations
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we could perform on a signal by taking its Fourier transform, transforming
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the result, and then reconstructing a new, transformed, signal from the
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modified transform.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig09.07"></A><A NAME="12565"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.7:</STRONG>
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Sliding-window analysis and resynthesis of an audio signal using
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Fourier transforms. In this example the signal is filtered by multiplying the
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Fourier transform with a desired frequency response.</CAPTION>
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<TR><TD><IMG
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WIDTH="444" HEIGHT="680" BORDER="0"
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SRC="img1146.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.07.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="#fig09.07">9.7</A> shows how to carry out a Fourier analysis, modification,
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and reconstruction of an audio signal. The first step is to divide the
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signal into
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<A NAME="12569"></A><I>windows</I>,
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which are segments of the signal, 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="$N$"> samples each, usually with some
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overlap. Each window is then shaped by multiplying it by a windowing
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function (Hann, for example). Then the Fourier transform is calculated for
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the <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="$N$"> points <!-- MATH
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$k = 0, 1, \ldots, N-1$
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-->
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<IMG
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WIDTH="133" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1147.png"
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ALT="$k = 0, 1, \ldots, N-1$">. (Sometimes it is desirable to
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calculate
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the Fourier transform for more points than this, but these <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="$N$"> points will
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suffice here.)
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<P>
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The Fourier analysis gives us a two-dimensional array of complex numbers.
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Let <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="$H$"> denote the
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<A NAME="12571"></A><I>hop size</I>,
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the number of samples each window is advanced past the
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previous window. Then for each <!-- MATH
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$m = \ldots, 0, 1, \ldots$
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-->
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<IMG
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WIDTH="115" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1148.png"
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ALT="$m = \ldots, 0, 1, \ldots$">, the <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">th window
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consists of the <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="$N$"> points starting at the point <IMG
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WIDTH="32" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1149.png"
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ALT="$mH$">. The <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 point
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of the <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">th window is <IMG
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WIDTH="61" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1150.png"
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ALT="$mH+n$">. The windowed Fourier transform is thus
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equal to:
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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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S[m, k] = {\cal FT}\{w(n)X[n-mH]\} (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="247" HEIGHT="28" BORDER="0"
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SRC="img1151.png"
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ALT="\begin{displaymath}
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S[m, k] = {\cal FT}\{w(n)X[n-mH]\} (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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This is both a function of time (<IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">, in units of <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="$H$"> samples) and of
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frequency (<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="$k$">, as a multiple of the fundamental 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 $">). Fixing
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the frame number <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$"> and looking
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at the windowed Fourier transform as a function of <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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S[k] = S[m, k]
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\end{displaymath}
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-->
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<IMG
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WIDTH="98" HEIGHT="28" BORDER="0"
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SRC="img1152.png"
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ALT="\begin{displaymath}
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S[k] = S[m, k]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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gives us a measure of the momentary spectrum of the signal <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$">. On the other
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hand, fixing a frequency <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> we can look at it as the <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">th channel of an
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<IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">-channel signal:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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C[m] = S[m, k]
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\end{displaymath}
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-->
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<IMG
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WIDTH="104" HEIGHT="28" BORDER="0"
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SRC="img1153.png"
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ALT="\begin{displaymath}
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C[m] = S[m, k]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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From this point of view, the windowed Fourier transform separates the original
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signal <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> into <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> narrow frequency regions, called <I>bands</I>.
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<P>
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Having computed the windowed Fourier transform, we next apply any desired
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modification. In the figure, the modification is simply to replace the upper
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half of the spectrum by zero, which gives a highly selective low-pass filter.
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(Two other possible modifications, narrow-band companding and vocoding, are
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described in the following sections.)
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<P>
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Finally we reconstruct an output signal. To do this we apply the inverse of
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the Fourier transform (labeled "iFT" in the figure). As shown in
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Section <A HREF="node166.html#sect9-IFT">9.1.2</A> this can be done by taking another Fourier transform,
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normalizing, and flipping the result backwards. In case the reconstructed
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window does not go smoothly to zero at its two ends, we apply the Hann
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windowing function a second time. Doing this to each successive window of
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the input, we then add the outputs, using the same overlap as for the analysis.
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<P>
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If we use the Hann window and an overlap of four (that is, choose <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$"> a multiple
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of four and space each window <IMG
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WIDTH="68" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1154.png"
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ALT="$H=N/4$"> samples past the previous one), we can
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reconstruct the original signal faithfully by omitting the "modification"
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step. This is because the iFT undoes the work of the <IMG
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WIDTH="27" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1155.png"
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ALT="$FT$">, and so we
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are multiplying each window by the Hann function squared. The output is
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thus the input, times the Hann window function squared, overlap-added by four.
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An easy check shows that this comes to the constant <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1156.png"
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ALT="$3/2$">, so the output
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equals the input times a constant factor.
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<P>
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The ability to reconstruct the input signal exactly is useful because some
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types of modification may be done by degrees, and so the output can be made
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to vary smoothly between the input and some transformed version of it.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html3146"
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HREF="node173.html">Narrow-band companding</A>
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<LI><A NAME="tex2html3147"
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HREF="node174.html">Timbre stamping (classical vocoder)</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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