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HREF="node8.html">Measures of Amplitude</A>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION00500000000000000000">
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Sinusoids, amplitude and frequency</A>
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</H1>
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<P>
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Electronic music is usually made using a computer, by synthesizing or processing
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<A NAME="1011"></A><A NAME="1012"></A><A NAME="1013"></A><EM>digital audio signals</EM>. These are sequences of numbers,
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<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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..., x[n-1], x[n], x[n+1], ...
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ALT="\begin{displaymath}
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..., x[n-1], x[n], x[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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where the index <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="$n$">, called the
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<A NAME="1015"></A><I>sample number</I>, may range over some or all the integers. A single
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number in the sequence is called a <I>sample</I>.
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An example of a digital audio signal is the <I>Sinusoid</I>:
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<A NAME="eq-realsinusoid"></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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x[n] = a \cos (\omega n + \phi )
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\end{displaymath}
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-->
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<IMG
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WIDTH="140" HEIGHT="28" BORDER="0"
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SRC="img76.png"
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ALT="\begin{displaymath}
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x[n] = a \cos (\omega n + \phi )
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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 <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="$a$"> is the
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<A NAME="1020"></A><EM>amplitude</EM>, <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> is the
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<A NAME="1022"></A><EM>angular frequency</EM>, and
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<IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$"> is the initial
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<EM>phase</EM>.
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The phase is a function of the sample number <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, equal to
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<!-- MATH
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$\omega n + \phi$
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-->
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<IMG
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WIDTH="52" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img78.png"
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ALT="$\omega n + \phi$">. The initial phase is the phase at the zeroth sample (<IMG
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WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img79.png"
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ALT="$n=0$">).
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<P>
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Figure <A HREF="#fig01.01">1.1</A> (part a) shows a sinusoid graphically.
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The
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horizontal axis shows successive 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="$n$"> and the vertical axis shows the
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corresponding values of <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="$x[n]$">. The graph is drawn in such a way as to
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emphasize the sampled nature of the signal. Alternatively, we could draw it
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more simply as a continuous curve (part b). The upper drawing is the most
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faithful representation of the (digital audio) sinusoid, whereas the lower
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one can be considered an idealization of it.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig01.01"></A><A NAME="1028"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.1:</STRONG>
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A digital audio signal, showing its discrete-time nature (part a), and
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idealized as a continuous function (part b). This signal is
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a (real-valued) sinusoid, fifty points long, with amplitude 1, angular frequency
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0.24, and initial phase zero.</CAPTION>
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<TR><TD><IMG
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WIDTH="439" HEIGHT="336" BORDER="0"
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SRC="img81.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Sinusoids play a key role in audio processing because, if you shift one of them
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left or right by any number of samples, you get another one. This makes it
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easy to calculate the effect of all sorts of operations on sinusoids. Our ears
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use this same special property to help us parse incoming sounds, which is why
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sinusoids, and combinations of sinusoids, can be used to achieve many musical
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effects.
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<P>
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Digital audio signals do not have any intrinsic relationship with time, but
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to listen to them we must choose a
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<A NAME="1031"></A><I>sample rate</I>, usually given the variable name <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$">, which is the number
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of samples that fit into a second. The time <IMG
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WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$t$"> is related to the sample number
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<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="$n$"> by
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<IMG
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WIDTH="52" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img83.png"
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ALT="$Rt = n$">, or <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img84.png"
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ALT="$t = n/R$">. A sinusoidal signal with angular frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">
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has a real-time frequency equal to
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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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f = {{\omega R} \over {2 \pi}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="55" HEIGHT="39" BORDER="0"
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SRC="img85.png"
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ALT="\begin{displaymath}
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f = {{\omega R} \over {2 \pi}}
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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 Hertz (i.e., cycles per second), because a cycle is <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $"> radians and a
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second is <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> samples.
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<P>
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A real-world audio signal's amplitude might be expressed as a time-varying
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voltage or air pressure, but the samples of a digital audio signal are unitless
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numbers. We'll casually assume here that there is ample numerical accuracy
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so that we can ignore round-off errors, and that the numerical format is
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unlimited in range, so that samples may take any value we wish. However, most
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digital audio hardware works only over a fixed range of input and output
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values, most often between -1 and 1. Modern digital audio processing software
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usually uses a floating-point representation for signals. This allows us to
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use whatever units are most convenient for any given task, as long as the final
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audio output is within the hardware's range [<A
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HREF="node202.html#r-mathews69">Mat69</A>, pp. 4-10].
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html635"
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HREF="node8.html">Measures of Amplitude</A>
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<LI><A NAME="tex2html636"
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HREF="node9.html">Units of Amplitude</A>
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<LI><A NAME="tex2html637"
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HREF="node10.html">Controlling Amplitude</A>
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<LI><A NAME="tex2html638"
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HREF="node11.html">Frequency</A>
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<LI><A NAME="tex2html639"
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HREF="node12.html">Synthesizing a sinusoid</A>
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<LI><A NAME="tex2html640"
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HREF="node13.html">Superposing Signals</A>
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<LI><A NAME="tex2html641"
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HREF="node14.html">Periodic Signals</A>
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<LI><A NAME="tex2html642"
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HREF="node15.html">About the Software Examples</A>
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<UL>
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<LI><A NAME="tex2html643"
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HREF="node16.html">Quick Introduction to Pd</A>
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<LI><A NAME="tex2html644"
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HREF="node17.html">How to find and run the examples</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html645"
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HREF="node18.html">Examples</A>
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<UL>
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<LI><A NAME="tex2html646"
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HREF="node19.html">Constant amplitude scaler</A>
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<LI><A NAME="tex2html647"
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HREF="node20.html">Amplitude control in decibels</A>
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<LI><A NAME="tex2html648"
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HREF="node21.html">Smoothed amplitude control with an envelope generator</A>
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<LI><A NAME="tex2html649"
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HREF="node22.html">Major triad</A>
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<LI><A NAME="tex2html650"
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HREF="node23.html">Conversion between frequency and pitch</A>
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<LI><A NAME="tex2html651"
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HREF="node24.html">More additive synthesis</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html652"
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HREF="node25.html">Exercises</A>
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</UL>
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<!--End of Table of Child-Links-->
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<BR>
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<B> Next:</B> <A NAME="tex2html634"
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HREF="node8.html">Measures of Amplitude</A>
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<B> Up:</B> <A NAME="tex2html628"
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HREF="book.html">book</A>
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<B> Previous:</B> <A NAME="tex2html622"
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HREF="node6.html">Preface</A>
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<B> <A NAME="tex2html630"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html632"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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</BODY>
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</HTML>
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