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<H1><A ID="SECTION00710000000000000000"></A>
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<A ID="sect3.sampling"></A>
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<BR>
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The sampling theorem
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</H1>
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<P>
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So far we have discussed digital audio signals as if they were capable of
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describing any function of time, in the sense that knowing the values the
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function takes on the integers should somehow determine the values it takes
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between them. This isn't really true. For instance, suppose some function
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<IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> (defined for real numbers) happens to attain the value 1 at all integers:
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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(n) = 1 \, , \, \, \, \, \, n = \, \, \ldots, -1, 0, 1, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="221" HEIGHT="28" BORDER="0"
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SRC="img298.png"
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ALT="\begin{displaymath}
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f(n) = 1 \, , \, \, \, \, \, n = \, \, \ldots, -1, 0, 1, \ldots
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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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We might guess that <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img299.png"
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ALT="$f(t)=1$"> for all real <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="$t$">. But perhaps <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> happens
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to be one for integers and zero everywhere else--that's a perfectly
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good function too, and nothing about the function's values at the integers
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distinguishes it from the simpler <IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img299.png"
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ALT="$f(t)=1$">. But intuition tells us that
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the constant function is in the <I>spirit</I> of digital audio signals,
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whereas the one that hides a secret between the samples isn't. A function
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that is "possible to sample" should be one for which we can use some reasonable
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interpolation scheme to deduce its values on non-integers from its values on
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integers.
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<P>
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It is customary at this point in discussions of computer music to invoke
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the famous
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<A ID="3554"></A><I>Nyquist theorem</I>.
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This states (roughly speaking) that if a function is a finite or even infinite
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combination of sinusoids, none of whose angular frequencies exceeds <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 $">,
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then, theoretically at least, it is fully determined by the function's values
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on the integers. One possible way of reconstructing the function would be
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as a limit of higher- and higher-order polynomial interpolation.
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<P>
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The angular frequency <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">, called the <I>Nyquist frequency</I>, corresponds
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to <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img300.png"
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ALT="$R/2$"> cycles per second if <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> is the sample rate. The corresponding period
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is two samples. The Nyquist frequency is the best we can do in the sense that
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any real sinusoid of higher frequency is equal, at the integers, to one whose
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frequency is lower than the Nyquist, and it is this lower frequency that will
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get reconstructed by the ideal interpolation process. For instance, a
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sinusoid with angular frequency between <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> and <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 $">, say <IMG
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WIDTH="43" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img301.png"
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ALT="$\pi + \omega$">,
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can be written as
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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((\pi + \omega)n + \phi) = \cos((\pi + \omega)n + \phi - 2\pi n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="314" HEIGHT="28" BORDER="0"
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SRC="img302.png"
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ALT="\begin{displaymath}
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\cos((\pi + \omega)n + \phi) = \cos((\pi + \omega)n + \phi - 2\pi 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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<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((\omega - \pi)n + \phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="139" HEIGHT="28" BORDER="0"
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SRC="img303.png"
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ALT="\begin{displaymath}
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= \cos((\omega - \pi)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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<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((\pi - \omega)n - \phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="139" HEIGHT="28" BORDER="0"
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SRC="img304.png"
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ALT="\begin{displaymath}
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= \cos((\pi - \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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for all integers <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$">. (If <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> weren't an integer the first step would fail.)
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So a sinusoid with frequency between <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> and <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 $"> is equal, on the
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integers at least, to one with frequency between 0 and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">; you simply can't
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tell the two apart. And since any conversion hardware should do the "right"
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thing and reconstruct the lower-frequency sinusoid, any higher-frequency one
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you try to synthesize will come out your speakers at the wrong
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frequency--specifically, you will hear the unique frequency between 0 and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">
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that the higher frequency lands on when reduced in the above way. This
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phenomenon is called
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<I>foldover</I>,
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<A ID="3558"></A>
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because the half-line of frequencies from 0
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to <IMG
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WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img305.png"
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ALT="$\infty$"> is folded back and forth, in lengths of <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">, onto the interval
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from 0 to <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">. The word
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<I>aliasing</I>
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<A ID="3560"></A>means the same thing. Figure <A HREF="#fig03.01">3.1</A> shows that sinusoids of angular
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frequencies <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$"> and <IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$3\pi /2$">, for instance, can't be distinguished
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as digital audio signals.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig03.01"></A><A ID="3564"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 3.1:</STRONG>
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Two real sinusoids, with angular frequencies <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\pi /2$"> and <IMG
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WIDTH="37" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="$3\pi /2$">,
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showing that they coincide at integers. A digital audio signal can't
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distinguish between the two.</CAPTION>
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<TR><TD><IMG
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WIDTH="311" HEIGHT="110" BORDER="0"
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SRC="img306.png"
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ALT="\begin{figure}\psfig{file=figs/fig03.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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We conclude that when, for instance, we're computing values of a
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Fourier series (Page <A HREF="node14.html#eq-fourierseries"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>),
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either as a wavetable or as a real-time signal, we had better leave out any
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sinusoid
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in the sum whose frequency exceeds <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 $">. But the picture in general is not
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this simple, since most techniques other than additive synthesis don't lead to
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neat, band-limited signals (ones whose components stop at some limited
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frequency). For example, a sawtooth wave of frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, of the form
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put out by Pd's <TT>phasor~</TT> object but considered as a continuous
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function <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img307.png"
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ALT="$f(t)$">, expands 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(t) = {1 \over 2} - {1 \over \pi}
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{ \left (
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\sin(\omega t) + {{\sin(2 \omega t)} \over 2} +
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{{\sin(3 \omega t)} \over 3} + \cdots
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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="357" HEIGHT="45" BORDER="0"
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SRC="img308.png"
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ALT="\begin{displaymath}
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|
f(t) = {1 \over 2} - {1 \over \pi}
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{ \left (
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|
\sin(\omega ...
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|
...\over 2} +
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{{\sin(3 \omega t)} \over 3} + \cdots
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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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|
which enjoys arbitrarily high frequencies; and moreover the hundredth partial
|
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|
is only 40 dB weaker than the first one. At any but very low values of
|
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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 $">, the partials above <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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|
ALT="$\pi $"> will be audibly present--and, because of
|
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|
|
foldover, they will be heard at incorrect frequencies. (This does not mean
|
|
|
|
|
that one shouldn't use sawtooth waves as phase generators--the wavetable
|
|
|
|
|
lookup step magically corrects the sawtooth's foldover--but one should think
|
|
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|
|
twice before using a sawtooth wave itself as a digital sound source.)
|
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|
<P>
|
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|
Many synthesis techniques, even if not strictly band-limited, give partials
|
|
|
|
|
which may be made to drop off more rapidly than <IMG
|
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|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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|
SRC="img309.png"
|
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|
ALT="$1/n$"> as in the sawtooth
|
|
|
|
|
example, and are thus more forgiving to work with digitally. In any case,
|
|
|
|
|
it is always a good idea to keep the possibility of foldover in mind, and
|
|
|
|
|
to train your ears to recognize it.
|
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|
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|
|
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|
|
<P>
|
|
|
|
|
The first line of defense against foldover is simply to use high sample rates;
|
|
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|
|
it is a good practice to systematically use the highest sample rate that your
|
|
|
|
|
computer can easily handle. The highest practical rate will vary according to
|
|
|
|
|
whether you are working in real time or not, CPU time and memory constraints,
|
|
|
|
|
and/or input and output hardware, and sometimes even software-imposed
|
|
|
|
|
limitations.
|
|
|
|
|
|
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|
|
<P>
|
|
|
|
|
A very non-technical treatment of sampling theory is given in
|
|
|
|
|
[<A
|
|
|
|
|
HREF="node202.html#r-ballora03">Bal03</A>]. More detail can be found in [<A
|
|
|
|
|
HREF="node202.html#r-mathews69">Mat69</A>, pp. 1-30].
|
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<P>
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<HR>
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