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<H1><A ID="SECTION001170000000000000000"></A>
<A ID="sect7-fractional"></A>
<BR>
Variable and fractional shifts
</H1>

<P>
Like any audio synthesis or processing technique, delay networks become much
more powerful and interesting if their characteristics can be made to change 
over time.  The gain parameters (such as <IMG
 WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.png"
 ALT="$g$"> in the recirculating comb filter)
may be controlled by envelope generators, varying them while avoiding clicks or
other artifacts.  The delay times (such as <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.png"
 ALT="$d$"> before) are not so easy to vary
smoothly for two reasons.

<P>
First, we have only defined time shifts for integer values of <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.png"
 ALT="$d$">, since for
fractional values of <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.png"
 ALT="$d$"> an expression such as <IMG
 WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img781.png"
 ALT="$x[n-d]$"> is not determined if
<IMG
 WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.png"
 ALT="$x[n]$"> is only defined for integer values of <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$">.  To make fractional delays
we will have to introduce some suitable interpolation scheme.  And if we
wish to vary <IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.png"
 ALT="$d$"> smoothly over time, it will not give good results simply
to hop from one integer to the next.

<P>
Second, even once we have achieved perfectly smoothly changing delay
times, the artifacts caused by varying delay time become noticeable even at very
small relative rates of change; while in most cases you may ramp an amplitude
control between any two values over 30 milliseconds without trouble, changing a
delay by only one sample out of every hundred makes a very noticeable shift
in pitch--indeed, one frequently will vary a delay deliberately
in order to hear the artifacts, only incidentally passing from one specific
delay time value to another one.

<P>
The first matter (fractional delays) can be dealt with using an
interpolation scheme, in exactly the same way as for wavetable lookup
(Section <A HREF="node31.html#sect2.interpolation">2.5</A>).  For example, suppose we want
a delay of <IMG
 WIDTH="53" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img782.png"
 ALT="$d=1.5$"> samples.  For each <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$"> we must estimate a value for
<IMG
 WIDTH="70" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img783.png"
 ALT="$x[n-1.5]$">.
We could do this using standard four-point interpolation, putting a cubic
polynomial through the four "known" points (0, x[n]), (1, x[n-1]), (2, x[n-2]),
(3, x[n-3]), and then evaluating the polynomial at the point 1.5.  Doing
this repeatedly for each value of <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$"> gives the delayed signal.

<P>
This four-point interpolation scheme can be used for any delay of at least one
sample.  Delays of less than one sample can't be calculated this way because we
need two input points at least as recent as the desired delay.  They were
available in the above example, but for a delay time of 0.5 samples, for
instance, we would need the value of <IMG
 WIDTH="58" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img784.png"
 ALT="$x[n+1]$">, which is in the future.

<P>
The accuracy of the estimate could be further improved by using higher-order
interpolation schemes.  However, there is a trade-off between quality and
computational efficiency.  Furthermore, if we move to higher-order
interpolation schemes, the minimum possible delay time will increase, causing
trouble in some situations.

<P>
The second matter to consider is the artifacts--whether wanted
or unwanted--that arise from changing delay lines.  In
general, a discontinuous change in delay time will give rise to a
discontinuous change in the output signal, since it is in effect
interrupted at one point and made to jump to another.  If the input is
a sinusoid, the result is a discontinuous phase change.

<P>
If it is desired to change the delay line occasionally between fixed delay
times (for instance, at the beginnings of musical notes), then we can use the
techniques for managing sporadic discontinuities that were introduced in
Section <A HREF="node61.html#sect4.declick">4.3</A>.  In effect these techniques all work by muting
the output in one way or another.  On the other hand, if it is
desired that the delay time change continuously--while we are listening
to the output--then we must take into account the artifacts that
result from the changes.

<P>

<DIV ALIGN="CENTER"><A ID="fig07.17"></A><A ID="8124"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.17:</STRONG>
A variable length delay line, whose output is the input from some
previous time.  The output samples can't be newer than the input samples, nor
older than the length <IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$D$"> of the delay line.  The slope of the input/output
curve controls the momentary transposition of the output.</CAPTION>
<TR><TD><IMG
 WIDTH="447" HEIGHT="405" BORDER="0"
 SRC="img785.png"
 ALT="\begin{figure}\psfig{file=figs/fig07.17.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>

<P>
Figure <A HREF="#fig07.17">7.17</A> shows the relationship between input and output time in a
variable delay line.  The delay line is assumed to have a fixed maximum length
<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$D$">.  At each sample of output (corresponding to a point on the horizontal
axis), we output one (possibly interpolated) sample of the delay line's input. 
The vertical axis shows which sample (integer or fractional) to use from the
input signal. Letting <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$"> denote the output sample number, the vertical axis
shows the quantity <IMG
 WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img786.png"
 ALT="$n - d[n]$">, where <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img787.png"
 ALT="$d[n]$"> is the (time-varying) delay in
samples. If we denote the input sample location by:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
y[n] = n - d[n]
\end{displaymath}
 -->

<IMG
 WIDTH="102" HEIGHT="28" BORDER="0"
 SRC="img788.png"
 ALT="\begin{displaymath}
y[n] = n - d[n]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then the output of the delay line is:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
z[n] = x[y[n]]
\end{displaymath}
 -->

<IMG
 WIDTH="91" HEIGHT="28" BORDER="0"
 SRC="img168.png"
 ALT="\begin{displaymath}
z[n] = x[y[n]]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where the signal <IMG
 WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img243.png"
 ALT="$x$"> is evaluated at the point <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.png"
 ALT="$y[n]$">, interpolating
appropriately in case <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.png"
 ALT="$y[n]$"> is not an integer.  This is exactly the formula
for wavetable lookup (Page <A HREF="node26.html#chapter-wavetable"><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]"
 SRC="crossref.png"></A>).  We can use all the
properties of wavetable lookup of recorded sounds to predict the behavior
of variable delay lines.

<P>
There remains one difference between delay lines and wavetables:
the material in the delay line is constantly being refreshed.  Not only
can we not read into the future, but, if the
the delay line is <IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$D$"> samples in length, we can't read further than <IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img40.png"
 ALT="$D$"> samples
into the past either:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
0 < d[n] < D
\end{displaymath}
 -->

<IMG
 WIDTH="89" HEIGHT="28" BORDER="0"
 SRC="img789.png"
 ALT="\begin{displaymath}
0 &lt; d[n] &lt; D
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
or, negating this and adding <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$"> to each side,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
n > y[n] > n - D.
\end{displaymath}
 -->

<IMG
 WIDTH="124" HEIGHT="28" BORDER="0"
 SRC="img790.png"
 ALT="\begin{displaymath}
n &gt; y[n] &gt; n - D.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This last relationship appears as the region between the two diagonal lines
in Figure <A HREF="#fig07.17">7.17</A>; the function <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img2.png"
 ALT="$y[n]$"> must stay within this strip.

<P>
Returning to Section <A HREF="node28.html#sect2.sampling">2.2</A>, we can use the Momentary
Transposition Formulas for wavetables to calculate the transposition <IMG
 WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img318.png"
 ALT="$t[n]$"> of
the output.  This gives the Momentary Transposition Formula for delay lines:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
t[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])
\end{displaymath}
 -->

<IMG
 WIDTH="306" HEIGHT="28" BORDER="0"
 SRC="img791.png"
 ALT="\begin{displaymath}
t[n] = y[n] - y[n-1] = 1 - (d[n] - d[n-1])
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<A ID="eq.momentarydel"></A>If <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img787.png"
 ALT="$d[n]$"> does not change with <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$">, the transposition factor is <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img262.png"
 ALT="$1$"> and the
sound emerges from the delay line at the same speed as it went in.  But if the
delay time is increasing as a function of <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$">, the resulting sound is transposed
downward, and if <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img787.png"
 ALT="$d[n]$"> decreases, upward.

<P>
This is called the
<A ID="8132"></A><I>Doppler effect</I>, and it occurs in nature as well.
The air that sound travels through can sometimes be thought of as a delay
line.   Changing the length of the delay line corresponds to moving the
listener toward or away from a stationary sound source; the Doppler effect
from the changing path length works precisely the same in the delay line as it
would be in the physical air.

<P>
Returning to Figure <A HREF="#fig07.17">7.17</A>, we can predict that there is no pitch
shift at the beginning, but then when the slope of the path decreases the
pitch will drop for an interval of time before going back to the original
pitch (when the slope returns to one).  The delay time can be manipulated
to give any desired transposition, but the greater the transposition, the
less long we can maintain it before we run off the bottom or the top of the
diagonal region.

<P>
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Miller Puckette
2006-12-30
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