2022-04-12 22:02:59 -03:00
|
|
|
<!DOCTYPE html>
|
2022-04-12 21:54:18 -03:00
|
|
|
|
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
|
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
|
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
|
|
|
* with significant contributions from:
|
|
|
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
|
|
|
<HTML>
|
|
|
|
|
<HEAD>
|
2022-04-12 22:02:59 -03:00
|
|
|
|
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
|
|
|
|
|
2022-04-12 21:54:18 -03:00
|
|
|
<TITLE>Transition splicing</TITLE>
|
|
|
|
|
<META NAME="description" CONTENT="Transition splicing">
|
|
|
|
|
<META NAME="keywords" CONTENT="book">
|
|
|
|
|
<META NAME="resource-type" CONTENT="document">
|
|
|
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
|
|
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
|
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
|
|
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
|
|
|
|
|
|
<LINK REL="previous" HREF="node195.html">
|
|
|
|
|
<LINK REL="up" HREF="node193.html">
|
|
|
|
|
<LINK REL="next" HREF="node197.html">
|
|
|
|
|
</HEAD>
|
|
|
|
|
|
|
|
|
|
<BODY >
|
|
|
|
|
<!--Navigation Panel-->
|
|
|
|
|
<A NAME="tex2html3497"
|
|
|
|
|
HREF="node197.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3491"
|
|
|
|
|
HREF="node193.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3487"
|
|
|
|
|
HREF="node195.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3493"
|
|
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3495"
|
|
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html3498"
|
|
|
|
|
HREF="node197.html">Examples</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html3492"
|
|
|
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html3488"
|
|
|
|
|
HREF="node195.html">Sneaky triangle waves</A>
|
|
|
|
|
<B> <A NAME="tex2html3494"
|
|
|
|
|
HREF="node4.html">Contents</A></B>
|
|
|
|
|
<B> <A NAME="tex2html3496"
|
|
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
|
|
|
|
<H2><A NAME="SECTION001443000000000000000">
|
|
|
|
|
Transition splicing</A>
|
|
|
|
|
</H2>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
In the point of view developed in this chapter, the energy of the spectral
|
|
|
|
|
components of classical waves can be attributed entirely to their jumps and
|
|
|
|
|
corners. This is artificial, of course: the energy really emanates from the
|
|
|
|
|
entire waveform. Our derivation of the spectrum of the classical waveforms
|
|
|
|
|
uses the jumps and corners as a bookkeeping device, and this is possible
|
|
|
|
|
because the entire waveform is determined by their positions and magnitudes.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Taking this ruse even further, the problem of making band-limited versions of
|
|
|
|
|
classical waveforms can be attacked by making band-limited versions of the
|
|
|
|
|
jumps and corners. Since the jumps are the more serious foldover threat, we
|
|
|
|
|
will focus on them here, although the approach described here works perfectly
|
|
|
|
|
well for corners as well.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
|
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.09"></A><A NAME="14478"></A>
|
|
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.9:</STRONG>
|
|
|
|
|
A square wave, band-limited to partials 1, 3, 5, 7, 9, and 11. This
|
|
|
|
|
can be regarded approximately as a series of band-limited step functions arranged
|
|
|
|
|
end to end.</CAPTION>
|
|
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="474" HEIGHT="109" BORDER="0"
|
|
|
|
|
SRC="img1370.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.09.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
To construct a band-limited step function, all we have to do is add the Fourier
|
|
|
|
|
components of a square wave, as many as we like, and then harvest the step
|
|
|
|
|
function at any one of the jumps. Figure <A HREF="#fig10.09">10.9</A> shows the partial
|
|
|
|
|
Fourier sum corresponding to a square wave, using partials 1, 3, 5, 7, 9, and
|
|
|
|
|
11. The cutoff frequency can be taken as <IMG
|
|
|
|
|
WIDTH="29" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img1371.png"
|
|
|
|
|
ALT="$12\omega$"> (if <IMG
|
|
|
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img27.png"
|
|
|
|
|
ALT="$\omega $"> is the
|
|
|
|
|
fundamental frequency).
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
If we double the period of the square wave, to arrive at the same cutoff
|
|
|
|
|
frequency, we would add twice as many Fourier partials, up to number 23, for
|
|
|
|
|
instance. Extending this process forever, we would eventually see the ideal
|
|
|
|
|
band-limited step function, twice per (arbitrarily long) period.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
In practice we can do quite well using only the first two partials (one and
|
|
|
|
|
three times the fundamental). Figure <A HREF="#fig10.10">10.10</A> (part a) shows
|
|
|
|
|
a two-partial approximation of a square wave. The cutoff frequency is
|
|
|
|
|
four times the fundamental; so if the period of the waveform is eight
|
|
|
|
|
samples, the cutoff is at the Nyquist frequency. Part (b) of the
|
|
|
|
|
figure shows how we could use this step function to synthesize, approximately,
|
|
|
|
|
a square wave of twice the period. If the cutoff frequency is the Nyquist
|
|
|
|
|
frequency, the period of the waveform of part (b) is 16 samples. Each
|
|
|
|
|
transition lasts 4 samples, because the band-limited square wave has a period
|
|
|
|
|
of eight samples.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
|
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.10"></A><A NAME="14485"></A>
|
|
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.10:</STRONG>
|
|
|
|
|
Stretching a band-limited square wave: (a) the original waveform;
|
|
|
|
|
(b) after splicing in horizontal segments; (c) using the same step transition
|
|
|
|
|
for a sawtooth wave.</CAPTION>
|
|
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="474" HEIGHT="376" BORDER="0"
|
|
|
|
|
SRC="img1372.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.10.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
We can make a band-limited sawtooth wave by adding the four-sample-long
|
|
|
|
|
transition to a ramp function so that the end of the resulting function meets
|
|
|
|
|
smoothly with itself end to end, as shown in part (c) of the
|
|
|
|
|
figure.
|
|
|
|
|
There is one transition per period, so the period must be at least four
|
|
|
|
|
samples; the highest fundamental frequency we can synthesize this way is half
|
|
|
|
|
the Nyquist frequency. For this or lower fundamental frequency, the foldover
|
|
|
|
|
products all turn out to be at least 60 dB quieter than the fundamental.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
|
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.11"></A><A NAME="14490"></A>
|
|
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.11:</STRONG>
|
|
|
|
|
Block diagram for making a sawtooth wave with a spliced transition.</CAPTION>
|
|
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="199" HEIGHT="389" BORDER="0"
|
|
|
|
|
SRC="img1373.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.11.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Figure <A HREF="#fig10.11">10.11</A> shows how to generate a sawtooth wave with a spliced
|
|
|
|
|
transition. The two parameters are <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img112.png"
|
|
|
|
|
ALT="$f$">, the fundamental frequency, and <IMG
|
|
|
|
|
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img21.png"
|
|
|
|
|
ALT="$b$">,
|
|
|
|
|
the band limit, assumed to be at least as large as <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img112.png"
|
|
|
|
|
ALT="$f$">. We start with a
|
|
|
|
|
digital sawtooth wave (a phasor) ranging from -0.5 to 0.5 in value. The
|
|
|
|
|
transition will take place at the middle of the cycle, when the phasor crosses
|
|
|
|
|
0. The wavetable is traversed in a constant amount of time, <IMG
|
|
|
|
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img1374.png"
|
|
|
|
|
ALT="$1/b$">, regardless
|
|
|
|
|
of <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img112.png"
|
|
|
|
|
ALT="$f$">. The table lookup is taken to be non-wraparound, so that inputs out of
|
|
|
|
|
range output either -0.5 or 0.5.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
At the end of the cycle the phasor discontinuously jumps from -0.5 to 0.5, but
|
|
|
|
|
the output of the transition table jumps an equal and opposite amount, so the
|
|
|
|
|
result is continuous. During the portion of the waveform in which the
|
|
|
|
|
transition table is read at one or the other end-point, the output describes
|
|
|
|
|
a straight line segment.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
<HR>
|
|
|
|
|
<!--Navigation Panel-->
|
|
|
|
|
<A NAME="tex2html3497"
|
|
|
|
|
HREF="node197.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3491"
|
|
|
|
|
HREF="node193.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3487"
|
|
|
|
|
HREF="node195.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3493"
|
|
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<A NAME="tex2html3495"
|
|
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html3498"
|
|
|
|
|
HREF="node197.html">Examples</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html3492"
|
|
|
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html3488"
|
|
|
|
|
HREF="node195.html">Sneaky triangle waves</A>
|
|
|
|
|
<B> <A NAME="tex2html3494"
|
|
|
|
|
HREF="node4.html">Contents</A></B>
|
|
|
|
|
<B> <A NAME="tex2html3496"
|
|
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
<ADDRESS>
|
|
|
|
|
Miller Puckette
|
|
|
|
|
2006-12-30
|
|
|
|
|
</ADDRESS>
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
