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<H1><A NAME="SECTION00730000000000000000"></A>
<A NAME="sect3.controlstreams"></A>
<BR>
Control streams
</H1>
<P>
Control computations may come from a variety of sources, both internal and
external to the overall computation. Examples of internally engendered control
computations include sequencing (in which control computations must take place
at pre-determined times) or feature detection of the audio output (for
instance, watching for zero crossings in a signal). Externally engendered ones
may come from input devices such as MIDI controllers, the mouse and keyboard,
network packets, and so on. In any case, control computations may occur at
irregular intervals, unlike audio samples which correspond to a steadily
ticking sample clock.
<P>
<DIV ALIGN="CENTER"><A NAME="fig03.03"></A><A NAME="3592"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3.3:</STRONG>
Graphical representation of a control stream as a sequence
of points in time.</CAPTION>
<TR><TD><IMG
WIDTH="280" HEIGHT="51" BORDER="0"
SRC="img314.png"
ALT="\begin{figure}\psfig{file=figs/fig03.03.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
We will need a way of describing how information
flows between control and audio computations, which we will base on the
notion of a
<A NAME="3595"></A><I>control stream</I>.
This is simply a collection of numbers--possibly empty--that appear as a result of control
computations, whether
regularly or irregularly spaced in logical time.
The simplest possible control stream has no information other than a
<A NAME="3597"></A>
<I>time sequence</I>:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ldots , t[0], t[1], t[2], \ldots
\end{displaymath}
-->
<IMG
WIDTH="133" HEIGHT="28" BORDER="0"
SRC="img315.png"
ALT="\begin{displaymath}
\ldots , t[0], t[1], t[2], \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Although the time values are best given in units of samples,
their values aren't quantized; they may be arbitrary real numbers.
We do require them to be sorted in nondecreasing order:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\cdots \le t[0] \le t[1] \le t[2] \le \cdots
\end{displaymath}
-->
<IMG
WIDTH="186" HEIGHT="28" BORDER="0"
SRC="img316.png"
ALT="\begin{displaymath}
\cdots \le t[0] \le t[1] \le t[2] \le \cdots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Each item in the sequence
is called an
<A NAME="3599"></A><I>event</I>.
<P>
Control streams may be shown graphically as in Figure <A HREF="#fig03.03">3.3</A>. A number
line shows time and a sequence of arrows points to the times associated
with each event.
The control
stream shown has no data (it is a time sequence). If we want to show
data in the control stream we will write it at the base of each arrow.
<P>
A
<A NAME="3602"></A><I>numeric control stream</I>
is one that contains one number per time point, so that it
appears as a sequence of
ordered pairs:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ldots , \, (t[0], x[0]), \, (t[1], x[1]), \, \ldots
\end{displaymath}
-->
<IMG
WIDTH="202" HEIGHT="28" BORDER="0"
SRC="img317.png"
ALT="\begin{displaymath}
\ldots , \, (t[0], x[0]), \, (t[1], x[1]), \, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where the <IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img318.png"
ALT="$t[n]$"> are the time points
and the <IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$x[n]$"> are the signal's values
at those times.
<P>
A numeric control stream is roughly analogous to a "MIDI controller", whose
values change irregularly, for example when a physical control is moved by a
performer. Other control stream sources may have higher possible rates of
change and/or more precision. On the other hand, a time sequence might be a
sequence of pedal hits, which (MIDI implementation notwithstanding) shouldn't be
considered as having <I>values</I>, just <I>times</I>.
<P>
Numeric control streams are like audio signals in that
both are just time-varying numeric values. But
whereas the audio signal comes at a steady rate (and so the time values need
not be specified per sample), the control stream comes
unpredictably--perhaps evenly, perhaps unevenly, perhaps never.
<P>
Let us now look at what happens when we try to convert a numeric
control stream to
an audio signal. As before we'll choose a block size <IMG
WIDTH="45" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img312.png"
ALT="$B=4$">. We will consider
as a control stream a square wave of period 5.5:
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
(2, 1), (4.75, 0), (7.5, 1), (10.25, 0), (13, 1), \ldots
\end{displaymath}
-->
<IMG
WIDTH="296" HEIGHT="28" BORDER="0"
SRC="img319.png"
ALT="\begin{displaymath}
(2, 1), (4.75, 0), (7.5, 1), (10.25, 0), (13, 1), \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and demonstrate three ways it could be converted to an audio signal. Figure
<A HREF="#fig03.04">3.4</A> (part a) shows the simplest, fast-as-possible, conversion.
Each audio sample of output simply reflects the most recent value of the
control signal. So samples 0 through 3 (which are computed at logical time
4 because of the block size) are 1 in value because of the point (2, 1). The
next four samples are also one, because of the two points, (4.75, 0) and
(7.5, 1), the most recent still has the value 1.
<P>
Fast-as-possible conversion is most appropriate for control streams which do
not change frequently compared to the block size. Its main advantages are
simplicity of computation and the fastest possible response to changes. As the
figure shows, when the control stream's updates are too fast (on the order of
the block size), the audio signal may not be a good likeness of the sporadic
one. (If, as in this case, the control stream comes at regular intervals of
time, we can use the sampling theorem to analyze the result. Here the Nyquist
frequency associated with the block rate <IMG
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img320.png"
ALT="$R/B$"> is lower than the input square
wave's frequency, and so the output is aliased to a new frequency lower than
the Nyquist frequency.)
<P>
<DIV ALIGN="CENTER"><A NAME="fig03.04"></A><A NAME="3609"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3.4:</STRONG>
Three ways to change a control stream into an audio signal: (a)
as fast as possible; (b) delayed to the nearest sample; (c) with
two-point interpolation for higher delay accuracy.</CAPTION>
<TR><TD><IMG
WIDTH="574" HEIGHT="534" BORDER="0"
SRC="img321.png"
ALT="\begin{figure}\psfig{file=figs/fig03.04.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
Part (b) shows the result of nearest-sample conversion. Each new value of the
control stream at a time <IMG
WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img82.png"
ALT="$t$"> affects output samples starting from index
<!-- MATH
$\lfloor t \rfloor$
-->
<IMG
WIDTH="23" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img322.png"
ALT="$\lfloor t \rfloor$"> (the greatest integer not exceeding <IMG
WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img82.png"
ALT="$t$">).
This is equivalent to using fast-as-possible conversion at a block size of 1;
in other words, nearest-sample conversion hides the effect of the larger block
size. This is better than fast-as-possible conversion in cases where the
control stream might change quickly.
<P>
Part (c) shows sporadic-to-audio conversion, again at the nearest sample,
but now also using two-point interpolation to further increase the time
accuracy. Conceptually we can describe this as follows. Suppose the value
of the control stream was last equal to <IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$">, and that the next point is
<IMG
WIDTH="68" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img323.png"
ALT="$(n+f, y)$">, where <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$"> is an integer and <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$"> is the fractional part of the
time value (so <IMG
WIDTH="71" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img324.png"
ALT="$0 \le f < 1$">). The first point affected in the audio
output will be the sample at index <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">. But instead of setting the output
to <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img106.png"
ALT="$y$"> as before, we set it to
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
fx + (1-f)y,
\end{displaymath}
-->
<IMG
WIDTH="99" HEIGHT="28" BORDER="0"
SRC="img325.png"
ALT="\begin{displaymath}
fx + (1-f)y,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
in other words, to a weighted average of the previous and the new value, whose
weights favor the new value more if the time of the sporadic value is earlier,
closer to <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">. In the example shown, the transition from 0 to 1 at time 2
gives
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
0 \cdot x + 1 \cdot y = 1,
\end{displaymath}
-->
<IMG
WIDTH="108" HEIGHT="27" BORDER="0"
SRC="img326.png"
ALT="\begin{displaymath}
0 \cdot x + 1 \cdot y = 1,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
while the transition from 1 to 0 at time 4.75 gives:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
0.75 \cdot x + 0.25 \cdot y = 0.75.
\end{displaymath}
-->
<IMG
WIDTH="168" HEIGHT="27" BORDER="0"
SRC="img327.png"
ALT="\begin{displaymath}
0.75 \cdot x + 0.25 \cdot y = 0.75.
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This technique gives a still closer representation of the control signal
(at least, the portion of it that lies below the Nyquist frequency), at
the expense of more computation and slightly greater delay.
<P>
Numeric control streams may also be converted to audio signals using ramp
functions to smooth discontinuities. This is often used when a control stream
is used to control an amplitude, as described in Section <A HREF="node12.html#sect1.synth">1.5</A>. In
general there are three values to specify to set a ramp function in motion: a
start time and target value (specified by the control stream) and a target
time, often expressed as a delay after the start time.
<P>
In such situations it is almost always accurate enough to adjust the start and
ending times to match the first audio sample computed at a later logical time,
a choice which corresponds to the fast-as-possible scenario above. Figure
<A HREF="#fig03.05">3.5</A> (part a) shows the effect of ramping from 0, starting at
time 3, to a value of 1 at time 9, immediately starting back toward 0 at
time 15, with block size <IMG
WIDTH="45" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img312.png"
ALT="$B=4$">. The times 3, 9, and 15 are truncated to
0, 8, and 12, respectively.
<P>
<DIV ALIGN="CENTER"><A NAME="fig03.05"></A><A NAME="3616"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3.5:</STRONG>
Line segment smoothing of numeric control streams: (a) aligned to block
boundaries; (b) aligned to nearest sample.</CAPTION>
<TR><TD><IMG
WIDTH="342" HEIGHT="196" BORDER="0"
SRC="img328.png"
ALT="\begin{figure}\psfig{file=figs/fig03.05.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
In real situations the block size might be on the order of a millisecond, and
adjusting ramp endpoints to block boundaries works fine for controlling
amplitudes; reaching a target a fraction of a millisecond early or late rarely
makes an audible difference. However, other uses of ramps are more sensitive
to time quantization of endpoints. For example, if we wish to do something
repetitively every few milliseconds, the variation in segment lengths will make
for an audible aperiodicity.
<P>
For situations such as these, we can improve the ramp generation algorithm to
start and stop at arbitrary samples, as shown in Figure <A HREF="#fig03.05">3.5</A> (part b),
for example. Here the endpoints of the line segments line up exactly with
the requested samples 3, 9, and 15. We can go even further and adjust for
fractional samples, making the line segments touch the values 0 and 1 at
exactly specifiable points on a number line.
<P>
For example, suppose we want to repeat a recorded sound out of a wavetable 100
times per second, every 441 samples at the usual sample rate. Rounding errors
due to blocking at 64-sample boundaries could detune the playback by as
much as a whole tone in pitch; and even rounding to one-sample boundaries
could introduce variations up to about 0.2%, or three cents. This
situation would call for sub-sample accuracy in sporadic-to-audio conversion.
<P>
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2006-12-30
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