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<H1><A ID="SECTION001170000000000000000"></A>
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<A ID="sect7-fractional"></A>
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<BR>
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Variable and fractional shifts
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</H1>
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<P>
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Like any audio synthesis or processing technique, delay networks become much
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more powerful and interesting if their characteristics can be made to change
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over time. The gain parameters (such as <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="$g$"> in the recirculating comb filter)
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may be controlled by envelope generators, varying them while avoiding clicks or
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other artifacts. The delay times (such as <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> before) are not so easy to vary
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smoothly for two reasons.
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<P>
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First, we have only defined time shifts for integer values of <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">, since for
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fractional values of <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> an expression such as <IMG
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WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img781.png"
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ALT="$x[n-d]$"> is not determined if
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<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="$x[n]$"> is only defined for integer values of <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="$n$">. To make fractional delays
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we will have to introduce some suitable interpolation scheme. And if we
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wish to vary <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> smoothly over time, it will not give good results simply
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to hop from one integer to the next.
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<P>
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Second, even once we have achieved perfectly smoothly changing delay
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times, the artifacts caused by varying delay time become noticeable even at very
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small relative rates of change; while in most cases you may ramp an amplitude
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control between any two values over 30 milliseconds without trouble, changing a
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delay by only one sample out of every hundred makes a very noticeable shift
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in pitch--indeed, one frequently will vary a delay deliberately
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in order to hear the artifacts, only incidentally passing from one specific
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delay time value to another one.
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<P>
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The first matter (fractional delays) can be dealt with using an
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interpolation scheme, in exactly the same way as for wavetable lookup
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(Section <A HREF="node31.html#sect2.interpolation">2.5</A>). For example, suppose we want
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a delay of <IMG
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WIDTH="53" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img782.png"
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ALT="$d=1.5$"> samples. For each <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="$n$"> we must estimate a value for
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<IMG
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WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img783.png"
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ALT="$x[n-1.5]$">.
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We could do this using standard four-point interpolation, putting a cubic
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polynomial through the four "known" points (0, x[n]), (1, x[n-1]), (2, x[n-2]),
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(3, x[n-3]), and then evaluating the polynomial at the point 1.5. Doing
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this repeatedly for each value of <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="$n$"> gives the delayed signal.
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<P>
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This four-point interpolation scheme can be used for any delay of at least one
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sample. Delays of less than one sample can't be calculated this way because we
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need two input points at least as recent as the desired delay. They were
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available in the above example, but for a delay time of 0.5 samples, for
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instance, we would need the value of <IMG
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WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img784.png"
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ALT="$x[n+1]$">, which is in the future.
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<P>
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The accuracy of the estimate could be further improved by using higher-order
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interpolation schemes. However, there is a trade-off between quality and
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computational efficiency. Furthermore, if we move to higher-order
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interpolation schemes, the minimum possible delay time will increase, causing
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trouble in some situations.
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<P>
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The second matter to consider is the artifacts--whether wanted
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or unwanted--that arise from changing delay lines. In
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general, a discontinuous change in delay time will give rise to a
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discontinuous change in the output signal, since it is in effect
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interrupted at one point and made to jump to another. If the input is
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a sinusoid, the result is a discontinuous phase change.
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<P>
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If it is desired to change the delay line occasionally between fixed delay
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times (for instance, at the beginnings of musical notes), then we can use the
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techniques for managing sporadic discontinuities that were introduced in
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Section <A HREF="node61.html#sect4.declick">4.3</A>. In effect these techniques all work by muting
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the output in one way or another. On the other hand, if it is
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desired that the delay time change continuously--while we are listening
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to the output--then we must take into account the artifacts that
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result from the changes.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig07.17"></A><A ID="8124"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.17:</STRONG>
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A variable length delay line, whose output is the input from some
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previous time. The output samples can't be newer than the input samples, nor
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older than the length <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D$"> of the delay line. The slope of the input/output
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curve controls the momentary transposition of the output.</CAPTION>
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<TR><TD><IMG
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WIDTH="447" HEIGHT="405" BORDER="0"
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SRC="img785.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.17.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="#fig07.17">7.17</A> shows the relationship between input and output time in a
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variable delay line. The delay line is assumed to have a fixed maximum length
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<IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D$">. At each sample of output (corresponding to a point on the horizontal
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axis), we output one (possibly interpolated) sample of the delay line's input.
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The vertical axis shows which sample (integer or fractional) to use from the
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input signal. Letting <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="$n$"> denote the output sample number, the vertical axis
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shows the quantity <IMG
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WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img786.png"
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ALT="$n - d[n]$">, where <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img787.png"
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ALT="$d[n]$"> is the (time-varying) delay in
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samples. If we denote the input sample location by:
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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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y[n] = n - d[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="102" HEIGHT="28" BORDER="0"
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SRC="img788.png"
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ALT="\begin{displaymath}
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y[n] = n - d[n]
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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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then the output of the delay line 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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z[n] = x[y[n]]
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\end{displaymath}
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-->
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<IMG
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WIDTH="91" HEIGHT="28" BORDER="0"
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SRC="img168.png"
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ALT="\begin{displaymath}
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z[n] = x[y[n]]
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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 the signal <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$"> is evaluated at the point <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$">, interpolating
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appropriately in case <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> is not an integer. This is exactly the formula
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for wavetable lookup (Page <A HREF="node26.html#chapter-wavetable"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). We can use all the
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properties of wavetable lookup of recorded sounds to predict the behavior
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of variable delay lines.
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<P>
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There remains one difference between delay lines and wavetables:
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the material in the delay line is constantly being refreshed. Not only
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can we not read into the future, but, if the
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the delay line is <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D$"> samples in length, we can't read further than <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img40.png"
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ALT="$D$"> samples
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into the past either:
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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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0 < d[n] < D
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\end{displaymath}
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-->
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<IMG
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WIDTH="89" HEIGHT="28" BORDER="0"
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SRC="img789.png"
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ALT="\begin{displaymath}
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0 < d[n] < D
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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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or, negating this and adding <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="$n$"> to each side,
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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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n > y[n] > n - D.
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\end{displaymath}
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-->
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<IMG
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WIDTH="124" HEIGHT="28" BORDER="0"
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SRC="img790.png"
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ALT="\begin{displaymath}
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n > y[n] > n - D.
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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 last relationship appears as the region between the two diagonal lines
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in Figure <A HREF="#fig07.17">7.17</A>; the function <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> must stay within this strip.
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<P>
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Returning to Section <A HREF="node28.html#sect2.sampling">2.2</A>, we can use the Momentary
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Transposition Formulas for wavetables to calculate the transposition <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img318.png"
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ALT="$t[n]$"> of
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the output. This gives the Momentary Transposition Formula for delay lines:
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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[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])
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\end{displaymath}
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-->
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<IMG
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WIDTH="306" HEIGHT="28" BORDER="0"
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SRC="img791.png"
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ALT="\begin{displaymath}
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t[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])
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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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<A ID="eq.momentarydel"></A>If <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img787.png"
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ALT="$d[n]$"> does not change with <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="$n$">, the transposition factor is <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img262.png"
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ALT="$1$"> and the
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sound emerges from the delay line at the same speed as it went in. But if the
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delay time is increasing as a function of <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="$n$">, the resulting sound is transposed
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downward, and if <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img787.png"
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ALT="$d[n]$"> decreases, upward.
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<P>
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This is called the
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<A ID="8132"></A><I>Doppler effect</I>, and it occurs in nature as well.
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The air that sound travels through can sometimes be thought of as a delay
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line. Changing the length of the delay line corresponds to moving the
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listener toward or away from a stationary sound source; the Doppler effect
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from the changing path length works precisely the same in the delay line as it
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would be in the physical air.
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<P>
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Returning to Figure <A HREF="#fig07.17">7.17</A>, we can predict that there is no pitch
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shift at the beginning, but then when the slope of the path decreases the
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pitch will drop for an interval of time before going back to the original
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pitch (when the slope returns to one). The delay time can be manipulated
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to give any desired transposition, but the greater the transposition, the
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less long we can maintain it before we run off the bottom or the top of the
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diagonal region.
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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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