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HREF="node75.html">Modulation</A>
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HREF="node76.html">Taxonomy of spectra</A>
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<B> <A ID="tex2html1696"
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HREF="node201.html">Index</A></B>
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<H1><A ID="SECTION00920000000000000000"></A>
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<A ID="sect5.ringmod"></A>
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<BR>
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Multiplying audio signals
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</H1>
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<P>
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We have been routinely adding audio signals together, and multiplying them
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by slowly-varying signals (used, for example, as amplitude envelopes) since
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Chapter 1. For a full understanding of the algebra of audio
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signals we must also consider the situation where two audio signals,
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neither of which may be assumed to change slowly, are multiplied. The key to understanding
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what happens is the Cosine Product Formula:
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<A ID="eq-cosinemultiplication"></A>
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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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\cos(a) \cos (b) = {1 \over 2}
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{ \left [
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\parbox[t][0.1in]{0in}{\mbox{}}
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\cos (a+b) + \cos(a-b)
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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="287" HEIGHT="39" BORDER="0"
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SRC="img411.png"
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ALT="\begin{displaymath}
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\cos(a) \cos (b) = {1 \over 2}
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{ \left [
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\parbox[t][0.1in]{0in}{\mbox{}}
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\cos (a+b) + \cos(a-b)
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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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To see why this formula holds, we can use the formula for the cosine of a
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sum of two angles:
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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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\cos(a+b) = \cos(a)\cos(b) - \sin(a) \sin(b)
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\end{displaymath}
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-->
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<IMG
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WIDTH="274" HEIGHT="28" BORDER="0"
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SRC="img412.png"
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ALT="\begin{displaymath}
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\cos(a+b) = \cos(a)\cos(b) - \sin(a) \sin(b)
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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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to evaluate the right hand side of the cosine product formula; it then
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simplifies to the left hand side.
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<P>
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We can use this formula to see what happens when we multiply two sinusoids
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(Page <A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>):
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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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{\cos(\alpha n + \phi) \cos (\beta n + \xi)}
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=
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\end{displaymath}
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-->
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<IMG
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WIDTH="181" HEIGHT="28" BORDER="0"
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SRC="img413.png"
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ALT="\begin{displaymath}
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{\cos(\alpha n + \phi) \cos (\beta n + \xi)}
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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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<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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= {1 \over 2} { \left [ {
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\parbox[t][0.1in]{0in}{\mbox{}}
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{\cos \left ( (\alpha + \beta) n + (\phi + \xi) \right ) }
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+
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{\cos \left ( (\alpha - \beta) n + (\phi - \xi) \right ) }
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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="391" HEIGHT="39" BORDER="0"
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SRC="img414.png"
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ALT="\begin{displaymath}
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= {1 \over 2} { \left [ {
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\parbox[t][0.1in]{0in}{\mbox{}}...
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...t ( (\alpha - \beta) n + (\phi - \xi) \right ) }
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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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In words, multiply two sinusoids and you get a result with two partials,
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one at the sum of the two original frequencies, and one at their difference.
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(If the difference <IMG
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WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img415.png"
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ALT="$\alpha-\beta$"> happens to be negative, simply switch the
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original two sinusoids and the difference will then be positive.) These
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two new components are called
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<A ID="5635"></A><I>sidebands</I>.
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<P>
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This gives us a technique for shifting the component frequencies
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of a sound, called
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<A ID="5637"></A><A ID="5638"></A><I>ring modulation</I>, which is shown in its simplest form in Figure
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<A HREF="#fig05.02">5.2</A>. An oscillator provides a
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<A ID="5641"></A><I>carrier signal</I>, which
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is simply multiplied by the input. In this context the input is called the
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<A ID="5643"></A><I>modulating signal</I>.
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The term "ring modulation" is often used
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more generally to mean multiplying any two signals together, but here we'll
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just consider using a sinusoidal carrier signal. (The technique of ring
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modulation dates from the analog era [<A
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HREF="node202.html#r-strange72">Str95</A>]; digital
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multipliers now replace both the VCA (Section <A HREF="node12.html#sect1.synth">1.5</A>) and the
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ring modulator.)
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.02"></A><A ID="5649"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.2:</STRONG>
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Block diagram for ring modulating an input signal with a sinusoid.</CAPTION>
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<TR><TD><IMG
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WIDTH="167" HEIGHT="121" BORDER="0"
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SRC="img416.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.02.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="#fig05.03">5.3</A> shows a variety of results that may be obtained by
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multiplying a (modulating) sinusoid of angular 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 $"> and
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peak amplitude
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<IMG
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WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img134.png"
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ALT="$2a$">, by a (carrier) sinusoid of angular frequency <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $"> and
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peak amplitude 1:
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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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\left [ 2 a \cos (\alpha n) \right ]
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\cdot
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\left [ \cos (\beta n) \right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="152" HEIGHT="28" BORDER="0"
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SRC="img417.png"
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ALT="\begin{displaymath}
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\left [ 2 a \cos (\alpha n) \right ]
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\cdot
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\left [ \cos (\beta n) \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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(For simplicity the phase terms are omitted.) Each part of the figure
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shows both the modulation signal and the result in the same spectrum.
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The modulating signal appears as a single 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 $">, at amplitude
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<IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$">.
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The product in general has two component frequencies,
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each at an amplitude of <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img409.png"
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ALT="$a/2$">.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.03"></A><A ID="5655"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.3:</STRONG>
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Sidebands arising from multiplying two sinusoids of frequency
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<IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $"> and <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $">: (a) with <!-- MATH
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$\alpha > \beta > 0$
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-->
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<IMG
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WIDTH="74" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img9.png"
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ALT="$\alpha > \beta > 0$">;
|
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(b) with <!-- MATH
|
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|
|
$\beta > \alpha$
|
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-->
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<IMG
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WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="$\beta > \alpha $"> so that
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the lower sideband is reflected about the <IMG
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WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="$f=0$"> axis; (c) with <IMG
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WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="$\alpha =\beta $">,
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for which the amplitude of the zero-frequency
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sideband depends on the phases of the two sinusoids; (d)
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with <IMG
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WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img13.png"
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ALT="$\alpha =0$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="390" HEIGHT="782" BORDER="0"
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SRC="img418.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.03.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
|
2022-04-12 23:17:03 -03:00
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Parts (a) and (b) of the figure show "general" cases where <IMG
|
2022-04-12 21:54:18 -03:00
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $"> and <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $"> are
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nonzero and different from each other. The component frequencies of the output
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are <!-- MATH
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$\alpha + \beta$
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-->
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<IMG
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WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img419.png"
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ALT="$\alpha + \beta$"> and <!-- MATH
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$\alpha - \beta$
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-->
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<IMG
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WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img415.png"
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ALT="$\alpha-\beta$">. In part (b), since <!-- MATH
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$\alpha-\beta<0$
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-->
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<IMG
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WIDTH="72" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img420.png"
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ALT="$\alpha-\beta<0$">,
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we get a negative frequency component. Since cosine is an even function, we
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have
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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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\cos((\alpha - \beta)n) = \cos((\beta - \alpha)n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="209" HEIGHT="28" BORDER="0"
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SRC="img421.png"
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ALT="\begin{displaymath}
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\cos((\alpha - \beta)n) = \cos((\beta - \alpha)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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so the negative component is exactly equivalent to one at the positive
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frequency <IMG
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WIDTH="43" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img422.png"
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ALT="$\beta-\alpha$">, at the same amplitude.
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<P>
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In the special case where <!-- MATH
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$\alpha = \beta$
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-->
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<IMG
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WIDTH="44" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="$\alpha =\beta $">, the second (difference) sideband
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has zero frequency. In this case phase will be significant so we rewrite
|
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the product with explicit phases, replacing <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="$\beta $"> by <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img7.png"
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ALT="$\alpha $">, to get:
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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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|
|
{2 a \cos(\alpha n + \phi) \cos (\alpha n + \xi)}
|
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|
=
|
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|
\end{displaymath}
|
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|
-->
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<IMG
|
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|
WIDTH="200" HEIGHT="28" BORDER="0"
|
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SRC="img423.png"
|
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|
|
ALT="\begin{displaymath}
|
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|
|
{2 a \cos(\alpha n + \phi) \cos (\alpha n + \xi)}
|
|
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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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|
|
=
|
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|
|
|
{a \cos \left ( 2 \alpha n + (\phi + \xi) \right ) }
|
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|
+
|
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|
|
{a \cos \left ( \phi - \xi \right ) }
|
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.
|
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|
|
\end{displaymath}
|
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|
-->
|
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|
<IMG
|
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|
WIDTH="266" HEIGHT="28" BORDER="0"
|
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|
|
SRC="img424.png"
|
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|
|
ALT="\begin{displaymath}
|
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|
=
|
|
|
|
|
{a \cos \left ( 2 \alpha n + (\phi + \xi) \right ) }
|
|
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|
+
|
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|
|
{a \cos \left ( \phi - \xi \right ) }
|
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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 second term has zero frequency; its amplitude depends on the relative phase
|
|
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|
|
of the two sinusoids and
|
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|
|
ranges from <IMG
|
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|
WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img425.png"
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ALT="$+a$"> to <IMG
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WIDTH="24" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
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|
SRC="img426.png"
|
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ALT="$-a$">
|
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|
as the phase difference <IMG
|
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|
|
WIDTH="40" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
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|
|
SRC="img427.png"
|
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|
|
ALT="$\phi - \xi$"> varies from <IMG
|
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|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
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|
SRC="img179.png"
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ALT="$0$"> to <IMG
|
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|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
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|
|
SRC="img41.png"
|
|
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|
|
ALT="$\pi $"> radians. This
|
|
|
|
|
situation is shown in part (c) of Figure <A HREF="#fig05.03">5.3</A>.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Finally, part (d) shows a carrier signal whose frequency is zero. Its value is
|
|
|
|
|
the constant <IMG
|
|
|
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
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|
|
SRC="img4.png"
|
|
|
|
|
ALT="$a$"> (not <IMG
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|
WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
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|
|
SRC="img134.png"
|
|
|
|
|
ALT="$2a$">; zero frequency is a special case). Here we
|
|
|
|
|
get only one sideband, of amplitude <IMG
|
|
|
|
|
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
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|
|
SRC="img409.png"
|
|
|
|
|
ALT="$a/2$"> as usual.
|
|
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|
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|
|
|
|
|
<P>
|
|
|
|
|
We can use the distributive rule for multiplication to find out what
|
|
|
|
|
happens when we multiply signals together which consist of more than one
|
|
|
|
|
partial each. For example, in the situation above we can replace the
|
|
|
|
|
signal of frequency <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img7.png"
|
|
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|
|
ALT="$\alpha $"> with a sum of several sinusoids, such as:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
{a_1} \cos({\alpha _1} n ) + \cdots + {a_k} \cos({\alpha _k} n )
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="214" HEIGHT="28" BORDER="0"
|
|
|
|
|
SRC="img428.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
{a_1} \cos({\alpha _1} n ) + \cdots + {a_k} \cos({\alpha _k} n )
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
|
|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
Multiplying by the signal of frequency <IMG
|
|
|
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img8.png"
|
|
|
|
|
ALT="$\beta $"> gives partials at frequencies
|
|
|
|
|
equal to:
|
|
|
|
|
<BR><P></P>
|
|
|
|
|
<DIV ALIGN="CENTER">
|
|
|
|
|
<!-- MATH
|
|
|
|
|
\begin{displaymath}
|
|
|
|
|
\alpha_1 + \beta, \alpha_1 - \beta, \ldots,
|
|
|
|
|
\alpha_k + \beta, \alpha_k - \beta
|
|
|
|
|
\end{displaymath}
|
|
|
|
|
-->
|
|
|
|
|
|
|
|
|
|
<IMG
|
|
|
|
|
WIDTH="236" HEIGHT="27" BORDER="0"
|
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|
|
SRC="img429.png"
|
|
|
|
|
ALT="\begin{displaymath}
|
|
|
|
|
\alpha_1 + \beta, \alpha_1 - \beta, \ldots,
|
|
|
|
|
\alpha_k + \beta, \alpha_k - \beta
|
|
|
|
|
\end{displaymath}">
|
|
|
|
|
</DIV>
|
|
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|
|
<BR CLEAR="ALL">
|
|
|
|
|
<P></P>
|
|
|
|
|
As before if any frequency is negative we take its absolute value.
|
|
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|
|
|
|
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|
|
<P>
|
|
|
|
|
Figure <A HREF="#fig05.04">5.4</A> shows the result of multiplying a complex periodic signal
|
|
|
|
|
(with several components tuned in the ratio 0:1:2:<IMG
|
|
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|
|
WIDTH="22" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
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|
|
SRC="img430.png"
|
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|
|
ALT="$\cdots$">) by a
|
|
|
|
|
sinusoid. Both the spectral envelope and the component frequencies of the
|
|
|
|
|
result are changed according to relatively simple rules.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<DIV ALIGN="CENTER"><A ID="fig05.04"></A><A ID="5669"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.4:</STRONG>
|
|
|
|
|
Result of ring modulation of a complex signal by a pure sinusoid:
|
|
|
|
|
(a) the original signal's spectrum and spectral envelope; (b) modulated
|
|
|
|
|
by a relatively low modulating frequency (1/3 of the fundamental); (c)
|
|
|
|
|
modulated by a higher frequency, 10/3 of the fundamental.</CAPTION>
|
|
|
|
|
<TR><TD><IMG
|
|
|
|
|
WIDTH="431" HEIGHT="413" BORDER="0"
|
|
|
|
|
SRC="img431.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig05.04.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
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|
|
|
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|
|
|
<P>
|
|
|
|
|
The resulting spectrum is essentially the original spectrum combined with its
|
|
|
|
|
reflection about the vertical axis. This combined spectrum is then shifted to
|
|
|
|
|
the right by the carrier frequency. Finally, if any components of the
|
|
|
|
|
shifted spectrum are still left of the vertical axis, they are reflected about
|
|
|
|
|
it to make positive frequencies again.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
In part (b) of the figure, the carrier frequency (the frequency of the
|
|
|
|
|
sinusoid) is below the fundamental frequency of the complex signal. In this
|
|
|
|
|
case the shifting is by a relatively small distance, so that re-folding the
|
|
|
|
|
spectrum at the end almost places the two halves on top of each other. The
|
|
|
|
|
result is a spectral envelope roughly the same as the original (although half
|
|
|
|
|
as high) and a spectrum twice as dense.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
A special case, not shown, is to use a carrier frequency half the
|
|
|
|
|
fundamental. In this case, pairs of partials will fall on top of each other,
|
|
|
|
|
and will have the ratios 1/2 : 3/2 : 5/2 :<IMG
|
|
|
|
|
WIDTH="22" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img430.png"
|
|
|
|
|
ALT="$\cdots$"> to give an odd-partial-only
|
|
|
|
|
signal an octave below the original. This is a very simple and effective
|
|
|
|
|
octave divider for a harmonic signal, assuming you know or can find its
|
|
|
|
|
fundamental frequency. If you want even partials as well as odd ones (for the
|
|
|
|
|
octave-down signal), simply mix the original signal with the modulated one.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
Part (c) of the figure shows the effect of using a modulating frequency
|
|
|
|
|
much higher than the fundamental frequency of the complex signal. Here the
|
|
|
|
|
unfolding effect is much more clearly visible (only one partial, the leftmost
|
|
|
|
|
one, had to be reflected to make its frequency positive). The spectral envelope
|
|
|
|
|
is now widely displaced from the original; this displacement is often a more
|
|
|
|
|
strongly audible effect than the relocation of partials.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
As another special case, the carrier frequency may be a multiple of the
|
|
|
|
|
fundamental of the complex periodic signal; then the partials
|
|
|
|
|
all land back on other partials of the same fundamental, and the only effect
|
|
|
|
|
is the shift in spectral envelope.
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
<HR>
|
|
|
|
|
<!--Navigation Panel-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html1697"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node78.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
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