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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<H1><A ID="SECTION00570000000000000000"></A>
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<A ID="sect1.fourier"></A>
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<BR>
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Periodic Signals
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</H1>
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A signal <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="$x[n]$"> is said to repeat at a period <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img135.png"
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ALT="$\tau$"> if
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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 + \tau] = x[n]
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\end{displaymath}
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-->
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<IMG
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WIDTH="102" HEIGHT="28" BORDER="0"
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SRC="img136.png"
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ALT="\begin{displaymath}
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x[n + \tau] = x[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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for all <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="$n$">. Such a signal would also repeat at periods <IMG
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WIDTH="20" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img137.png"
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ALT="$2 \tau$"> and so on;
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the smallest <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img135.png"
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ALT="$\tau$"> if any at which a signal repeats is called the signal's
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<A ID="1175"></A><I>period</I>.
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In discussing periods of digital audio signals, we quickly run into the
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difficulty of describing signals whose "period" isn't an integer, so that the
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equation above doesn't make sense. For now we'll effectively
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ignore this difficulty by supposing that the signal <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="$x[n]$"> may somehow be
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interpolated between the samples so that it's well defined whether <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="$n$"> is an
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integer or not.
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<P>
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A sinusoid has a period (in samples) of <!-- MATH
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$2 \pi / \omega$
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-->
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<IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img138.png"
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ALT="$2 \pi / \omega$"> where
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<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 angular frequency. More generally, any sum of sinusoids
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with frequencies <!-- MATH
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$2 \pi k/ \omega$
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-->
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<IMG
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WIDTH="48" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img139.png"
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ALT="$2 \pi k/ \omega$">, for integers <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, will repeat after
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<!-- MATH
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$2 \pi / \omega$
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-->
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<IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img138.png"
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ALT="$2 \pi / \omega$"> samples.
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Such a sum is called a <I>Fourier Series</I>:
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<A ID="eq-fourierseries"></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_0} +
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{a_1} \cos \left ( \omega n + {\phi_1} \right ) +
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{a_2} \cos \left ( 2 \omega n + {\phi_2} \right ) + \cdots +
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{a_p} \cos \left ( p \omega n + {\phi_p} \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="507" HEIGHT="29" BORDER="0"
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SRC="img140.png"
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ALT="\begin{displaymath}
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x[n] = {a_0} +
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{a_1} \cos \left ( \omega n + {\phi_1} \rig...
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... + \cdots +
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{a_p} \cos \left ( p \omega n + {\phi_p} \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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Moreover, if we make certain technical assumptions (in effect that signals
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only contain frequencies up to a finite bound), we can represent any periodic
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signal as such a sum. This is the discrete-time variant of Fourier analysis
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which will reappear in Chapter <A HREF="node163.html#chapter-fourier">9</A>.
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<P>
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The angular frequencies of the sinusoids above are all integer
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multiples of <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 $">. They are called the
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<A ID="1187"></A><I>harmonics</I>
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of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, which in turn is called the
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<A ID="1189"></A><I>fundamental</I>. In terms of pitch, the harmonics
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<!-- MATH
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$\omega, 2 \omega, \ldots$
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-->
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<IMG
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WIDTH="65" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img141.png"
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ALT="$\omega, 2 \omega, \ldots$"> are at intervals of 0, 1200, 1902, 2400, 2786,
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3102, 3369, 3600, ..., cents above the fundamental; this sequence of pitches
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is sometimes called the <I>harmonic series</I>. The first six of these are
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all quite close to multiples of 100; in other words, the first six harmonics
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of a pitch in the Western scale land close to (but not always exactly on) other
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pitches of the same scale; the third and sixth miss only by 2 cents and the
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fifth misses by 14.
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<P>
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<A ID="r-intervals"></A>Put another way, the frequency ratio 3:2 (a perfect fifth in Western
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terminology) is almost exactly
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seven half-steps, 4:3 (a perfect fourth) is just as near to five half-steps,
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and the ratios 5:4 and 6:5 (perfect major and minor thirds) are fairly close to
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intervals of four and three half-steps, respectively.
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<P>
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A Fourier series (with only three terms) is shown in Figure <A HREF="#fig01.08">1.8</A>.
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The first three graphs are of sinusoids, whose frequencies are in a 1:2:3
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ratio. The common period is marked on the horizontal axis. Each sinusoid
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has a different amplitude and initial phase. The sum of the three, at
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bottom, is not a sinusoid, but it still maintains the periodicity shared
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by the three component sinusoids.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig01.08"></A><A ID="1196"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.8:</STRONG>
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A Fourier series, showing three sinusoids and their sum. The
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three component sinusoids have frequencies in the ratio 1:2:3.</CAPTION>
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<TR><TD><IMG
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WIDTH="438" HEIGHT="373" BORDER="0"
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SRC="img142.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.08.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Leaving questions of phase aside, we can use a bank of sinusoidal oscillators
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to synthesize periodic tones, or even to morph smoothly through a succession
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of periodic tones, by specifying the fundamental frequency and the (possibly
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time-varying) amplitudes of the partials. Figure <A HREF="#fig01.09">1.9</A> shows a
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block diagram for doing this.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig01.09"></A><A ID="1202"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.9:</STRONG>
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Using many oscillators to synthesize a waveform with desired harmonic
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amplitudes.</CAPTION>
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<TR><TD><IMG
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WIDTH="482" HEIGHT="391" BORDER="0"
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SRC="img143.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.09.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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This is an example of
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<A ID="1205"></A><I>additive synthesis</I>; more generally the term can be applied to networks
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in which the frequencies of the oscillators are independently controllable.
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The early days of computer music rang with the sound of additive synthesis.
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<ADDRESS>
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Miller Puckette
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2006-12-30
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