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571 lines
15 KiB
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<H1><A ID="SECTION001350000000000000000"></A>
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<A ID="sect9.phase"></A>
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<BR>
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Phase
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</H1>
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<P>
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So far we have operated on signals by altering the magnitudes of their
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windowed Fourier transforms, but leaving phases intact. The magnitudes
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encode the spectral envelope of the sound. The phases, on the other hand,
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encode frequency and time, in the sense that phase change from
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one window to a different one accumulates, over time, according to frequency.
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To make a transformation that allows independent control over frequency and
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time requires analyzing and reconstructing the phase.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.10"></A><A ID="12612"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.10:</STRONG>
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Phase in windowed Fourier analysis: (a) a complex sinusoid analyzed
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on three successive windows; (b) the result for a single channel (k=3), for
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the three windows.</CAPTION>
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<TR><TD><IMG
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WIDTH="447" HEIGHT="623" BORDER="0"
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SRC="img1176.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.10.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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In the analysis/synthesis examples of the previous section, the phase of the
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output is copied directly from the phase of the input. This is appropriate
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when the output signal corresponds in time with the input signal. Sometimes time
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modifications are desired, for instance to do time stretching or contraction.
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Alternatively the output phase might depend on more than one input, for instance
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to morph between one sound and another.
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<P>
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Figure <A HREF="#fig09.10">9.10</A> shows how the phase of the Fourier transform
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changes from window to window, given a complex sinusoid as input. The
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sinusoid's frequency is <!-- MATH
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$\alpha = 3\omega$
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-->
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<IMG
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WIDTH="53" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img1177.png"
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ALT="$\alpha = 3\omega$">, so that the peak in the Fourier transform
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is centered at <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1178.png"
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ALT="$k=3$">. If the initial phase is <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">, then the neighboring
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phases can be filled in as:
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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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\begin{array}{lll}
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{\angle S[0, 2] = \phi + \pi} &
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{\angle S[0, 3] = \phi} &
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{\angle S[0, 4] = \phi + \pi}\\
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{\angle S[1, 2] = \phi + H\alpha + \pi } &
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{\angle S[1, 3] = \phi + H\alpha} &
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{\angle S[1, 4] = \phi + H\alpha + \pi}\\
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{\angle S[2, 2] = \phi + 2H\alpha + \pi } &
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{\angle S[2, 3] = \phi + 2H\alpha} &
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{\angle S[2, 4] = \phi + 2H\alpha + \pi}\\
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\end{array}
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\end{displaymath}
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-->
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<IMG
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WIDTH="494" HEIGHT="64" BORDER="0"
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SRC="img1179.png"
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ALT="\begin{displaymath}
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\begin{array}{lll}
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{\angle S[0, 2] = \phi + \pi} &
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{\angl...
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...ha} &
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{\angle S[2, 4] = \phi + 2H\alpha + \pi}\\
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\end{array}\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 gives an excellent way of estimating the frequency <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $">: pick any
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channel whose amplitude is dominated by the sinusoid and subtract two
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successive phase to get <IMG
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WIDTH="28" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1180.png"
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ALT="$H\alpha$">:
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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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H \alpha = \angle S[1, 3] - \angle S[0, 3]
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\end{displaymath}
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-->
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<IMG
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WIDTH="169" HEIGHT="28" BORDER="0"
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SRC="img1181.png"
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ALT="\begin{displaymath}
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H \alpha = \angle S[1, 3] - \angle S[0, 3]
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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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<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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\alpha = {{\angle S[1, 3] - \angle S[0, 3] + 2 p \pi} \over H}
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\end{displaymath}
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-->
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<IMG
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WIDTH="203" HEIGHT="40" BORDER="0"
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SRC="img1182.png"
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ALT="\begin{displaymath}
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\alpha = {{\angle S[1, 3] - \angle S[0, 3] + 2 p \pi} \over H}
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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="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$"> is an integer. There are <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> possible frequencies, spaced by
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<IMG
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WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1183.png"
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ALT="$2\pi/H$">. If we are using an overlap of 4, that is, <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="$H=N/4$">, the frequencies
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are spaced by <!-- MATH
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$8\pi/N = 4 \omega$
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-->
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<IMG
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WIDTH="82" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1184.png"
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ALT="$8\pi/N = 4 \omega$">. Happily, this is the width of the main lobe
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for the Hann window, so no more than one possible value of <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $"> can explain
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any measured phase difference within the main lobe of a peak. The correct value
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of <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$"> to choose is that which gives a frequency closest to the nominal
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frequency of the channel, <IMG
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WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1145.png"
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ALT="$k\omega$">.
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<P>
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When computing phases for synthesizing a new or modified signal, we want to
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maintain the appropriate phase relationships between successive resynthesis
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windows, and also, simultaneously, between adjacent channels. These two sets of
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relationships are not always compatible, however. We will make it our first
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obligation to honor the relations between successive resynthesis windows, and
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worry about phase relationships between channels afterward.
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<P>
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Suppose we want to construct the <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$">th spectrum <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> for resynthesis
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(having already constructed the previous one, number <IMG
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WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1185.png"
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ALT="$m-1$">). Suppose
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we wish the phase relationships between windows <IMG
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WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1185.png"
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ALT="$m-1$"> and <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$"> to be those of
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a signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">, but that the phases of window number <IMG
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WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1185.png"
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ALT="$m-1$"> might have come
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from somewhere else and can't be assumed to be in line with our wishes.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.11"></A><A ID="12631"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.11:</STRONG>
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Propagating phases in resynthesis. Each phase, such as that of
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<IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$"> here, depends on the previous output phase and the difference of the
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input phases.</CAPTION>
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<TR><TD><IMG
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WIDTH="459" HEIGHT="405" BORDER="0"
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SRC="img1186.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.12"></A><A ID="12636"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.12:</STRONG>
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Phases of one channel of the analysis windows and two successive
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resynthesis windows.</CAPTION>
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<TR><TD><IMG
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WIDTH="355" HEIGHT="244" BORDER="0"
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SRC="img1187.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.12.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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To find out how much the phase of each channel should differ from the previous
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one, we do two analyses of the signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">, separated by the same hop size
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<IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img25.png"
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ALT="$H$"> that we're using for resynthesis:
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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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T[k] = {\cal FT}(W(n)X[n]) (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="180" HEIGHT="28" BORDER="0"
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SRC="img1188.png"
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ALT="\begin{displaymath}
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T[k] = {\cal FT}(W(n)X[n]) (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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<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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T'[k] = {\cal FT}(W(n)X[n+H]) (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="219" HEIGHT="28" BORDER="0"
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SRC="img1189.png"
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ALT="\begin{displaymath}
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T'[k] = {\cal FT}(W(n)X[n+H]) (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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Figure <A HREF="#fig09.11">9.11</A> shows the process of phase accumulation, in which the
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output phases each depend on the previous output phase and the phase difference
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for two windowed analyses of the input. Figure <A HREF="#fig09.12">9.12</A> illustrates the
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phase relationship in the complex plane.
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The phase of the new output <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]$"> should be that of the previous one plus the
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difference between the phases of the two analyses:
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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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\angle S[m, k] = \angle S[m-1, k] +
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\left ( \angle T'[k] - \angle T[k] \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="299" HEIGHT="28" BORDER="0"
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SRC="img1190.png"
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ALT="\begin{displaymath}
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\angle S[m, k] = \angle S[m-1, k] +
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\left ( \angle T'[k] - \angle T[k] \right )
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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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<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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= \angle \left (
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{{S[m-1, k] T'[k]}
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\over
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{T[k]}}
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\right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="163" HEIGHT="45" BORDER="0"
|
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SRC="img1191.png"
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ALT="\begin{displaymath}
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= \angle \left (
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{{S[m-1, k] T'[k]}
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\over
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{T[k]}}
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\right )
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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 we used the fact that multiplying or dividing two complex numbers gives
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the sum or difference of their arguments.
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<P>
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If the desired magnitude is a real number <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$">, then we should set <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img64.png"
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ALT="$S[m, k]$">
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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] \; = \;
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a
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\; \cdot \;
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{
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{ \left |
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{{S[m-1, k] T'[k]}
|
|
\over
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{T[k]}}
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\right |}
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^
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{-1}
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}
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\; \cdot \;
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|
{
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{{S[m-1, k] T'[k]}
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\over
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{T[k]}}
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}
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\end{displaymath}
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-->
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<IMG
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WIDTH="383" HEIGHT="48" BORDER="0"
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SRC="img1192.png"
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ALT="\begin{displaymath}
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S[m, k] \; = \;
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a
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\; \cdot \;
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|
{
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{ \left \vert
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{{S[m-...
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...{-1}
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}
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\; \cdot \;
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|
{
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{{S[m-1, k] T'[k]}
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\over
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{T[k]}}
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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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The magnitudes of the second and third terms cancel out, so that the magnitude
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of <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]$"> reduces to <IMG
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|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
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ALT="$a$">; the first two terms are real numbers so the
|
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argument is controlled by the last term.
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|
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<P>
|
|
If we want to end up with the magnitude from the spectrum <IMG
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img557.png"
|
|
ALT="$T$"> as well, we can
|
|
set <IMG
|
|
WIDTH="75" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1193.png"
|
|
ALT="$a = \vert T'[k]\vert$"> and simplify:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
S[m, k] \; = \;
|
|
{
|
|
{ \left |
|
|
{{S[m-1, k]}
|
|
\over
|
|
{T[k]}}
|
|
\right |}
|
|
^
|
|
{-1}
|
|
}
|
|
\; \cdot \;
|
|
{
|
|
{{S[m-1, k] T'[k]}
|
|
\over
|
|
{T[k]}}
|
|
}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="321" HEIGHT="48" BORDER="0"
|
|
SRC="img1194.png"
|
|
ALT="\begin{displaymath}
|
|
S[m, k] \; = \;
|
|
{
|
|
{ \left \vert
|
|
{{S[m-1, k]}
|
|
\over
|
|
{...
|
|
...{-1}
|
|
}
|
|
\; \cdot \;
|
|
{
|
|
{{S[m-1, k] T'[k]}
|
|
\over
|
|
{T[k]}}
|
|
}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<!--Table of Child-Links-->
|
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
|
|
|
<UL>
|
|
<LI><A ID="tex2html3188"
|
|
HREF="node176.html">Phase relationships between channels</A>
|
|
</UL>
|
|
<!--End of Table of Child-Links-->
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html3186"
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|
HREF="node176.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
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<A ID="tex2html3180"
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HREF="node163.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
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<A ID="tex2html3174"
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HREF="node174.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
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<A ID="tex2html3182"
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HREF="node4.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
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<A ID="tex2html3184"
|
|
HREF="node201.html">
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<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html3187"
|
|
HREF="node176.html">Phase relationships between channels</A>
|
|
<B> Up:</B> <A ID="tex2html3181"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> Previous:</B> <A ID="tex2html3175"
|
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HREF="node174.html">Timbre stamping (classical vocoder)</A>
|
|
<B> <A ID="tex2html3183"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html3185"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
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<ADDRESS>
|
|
Miller Puckette
|
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2006-12-30
|
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</ADDRESS>
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</BODY>
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</HTML>
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