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648 lines
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648 lines
19 KiB
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<H1><A ID="SECTION00650000000000000000"></A>
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<A ID="sect2.interpolation"></A>
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<BR>
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Interpolation
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</H1>
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<P>
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As mentioned before, interpolation schemes are often used to increase the
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accuracy of table lookup. Here we will give a somewhat simplified account of
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the effects of table sizes and interpolation schemes on the result of table
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lookup.
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<P>
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To speak of error in table lookup, we must view the wavetable as a sampled
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version of an underlying function. When we ask for a value of the
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underlying function which lies between the points of the wavetable, the error
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is the difference between the result of the wavetable lookup and the "ideal"
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value of the function at that point. The most revealing study of wavetable
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lookup error assumes that the underlying function is a sinusoid (Page
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<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). We can then understand what happens to other wavetables by
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considering them as superpositions (sums) of sinusoids.
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<P>
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The accuracy of lookup from a wavetable containing a sinusoid depends
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on two factors: the quality of the interpolation scheme, and the period
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of the sinusoid. In general, the longer the period of the sinusoid, the more
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accurate the result.
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<P>
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In the case of a synthetic wavetable, we might know its sinusoidal components
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from having specified them--in which case the issue becomes one of choosing a
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wavetable size appropriately, when calculating the wavetable, to match the
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interpolation algorithm and meet the desired standard of accuracy. In the case
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of recorded sounds, the accuracy analysis might lead us to adjust the sample
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rate of the recording, either at the outset or else by resampling later.
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<P>
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Interpolation error for a sinusoidal wavetable can have two components: first,
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the continuous signal (the theoretical result of reading the wavetable
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continuously in time, as if the output sample rate were infinite) might
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not be a pure sinusoid; and second, the amplitude might be wrong. (It
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is possible to get phase errors as well, but only through
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carelessness.)
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<P>
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In this treatment we'll only consider polynomial interpolation schemes such as
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rounding, linear interpolation, and cubic interpolation. These schemes amount
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to evaluating polynomials (of degree zero, one, and three, respectively) in the
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interstices between points of the wavetable. The idea is that, for any index
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<IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$">, we choose a nearby reference point <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img244.png"
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ALT="${x_0}$">, and let the output be calculated
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by some polynomial:
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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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{y_{\mathrm{INT}}}(x) =
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{a_0} + {a_1} (x - {x_0}) + {a_2} {{({x - x_0})}^ 2 } + \cdots +
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{a_n} {{({x - x_0})}^ n }
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\end{displaymath}
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-->
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<IMG
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WIDTH="421" HEIGHT="28" BORDER="0"
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SRC="img245.png"
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ALT="\begin{displaymath}
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{y_{\mathrm{INT}}}(x) =
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{a_0} + {a_1} (x - {x_0}) + {a_2} {{({x - x_0})}^ 2 } + \cdots +
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{a_n} {{({x - x_0})}^ 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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Usually we choose the polynomial which passes through the <IMG
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WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img246.png"
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ALT="$n+1$"> nearest
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points of the wavetable. For 1-point interpolation (a zero-degree polynomial)
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this means letting <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img96.png"
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ALT="$a_0$"> equal the nearest point of the wavetable. For
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two-point interpolation, we draw a line segment between the two points of the
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wavetable on either side of the desired point <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$">. We can let <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img247.png"
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ALT="$x_0$"> be the
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closest integer to the left of <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$"> (which we write as <!-- MATH
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$\lfloor x \rfloor$
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-->
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<IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img248.png"
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ALT="$\lfloor x \rfloor$">) and then the
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formula for linear interpolation is:
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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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{y_{\mathrm{INT}}}(x) =
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y[{x_0}] + (y[{x_0} + 1]- y[{x_0}]) \cdot (x - {x_0})
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\end{displaymath}
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-->
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<IMG
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WIDTH="321" HEIGHT="28" BORDER="0"
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SRC="img249.png"
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ALT="\begin{displaymath}
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{y_{\mathrm{INT}}}(x) =
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y[{x_0}] + (y[{x_0} + 1]- y[{x_0}]) \cdot (x - {x_0})
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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 is a polynomial, as in the previous formula, with
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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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{a_0} = y[{x_0}]
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\end{displaymath}
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-->
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<IMG
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WIDTH="67" HEIGHT="28" BORDER="0"
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SRC="img250.png"
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ALT="\begin{displaymath}
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{a_0} = y[{x_0}]
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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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{a_1} = y[{x_0} + 1]- y[{x_0}]
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\end{displaymath}
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-->
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<IMG
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WIDTH="148" HEIGHT="28" BORDER="0"
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SRC="img251.png"
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ALT="\begin{displaymath}
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{a_1} = y[{x_0} + 1]- y[{x_0}]
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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 general, you can fit exactly one polynomial of degree <IMG
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WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img252.png"
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ALT="$n-1$"> through any
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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="$n$"> points as long as their <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img243.png"
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ALT="$x$"> values are all different.
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<P>
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Figure <A HREF="#fig02.11">2.11</A> shows the effect of using linear (two-point) interpolation
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to fill in a sinusoid of period 6. At the top are three traces: the original
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sinusoid, the linearly-interpolated result of using 6 points per period to
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represent the sinusoid, and finally, another sinusoid, of slightly smaller
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amplitude, which better matches the six-segment waveform. The error introduced
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by replacing the original sinusoid by the linearly interpolated version has
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two components: first, a (barely perceptible) change in amplitude, and second,
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a (very perceptible) distortion of the wave shape.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig02.11"></A><A ID="2374"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.11:</STRONG>
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Linear interpolation of a sinusoid: (upper graph) the original sinusoid, the
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interpolated sinusoid, and the best sinusoidal fit back to the interpolated
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version; (lower graph) the error, rescaled vertically.</CAPTION>
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<TR><TD><IMG
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WIDTH="488" HEIGHT="514" BORDER="0"
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SRC="img253.png"
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ALT="\begin{figure}\psfig{file=figs/fig02.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The bottom graph in the figure shows the difference between the interpolated
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waveform and the best-fitting sinusoid. This is a residual signal all of
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whose energy lies in overtones of the original sinusoid.
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As the number of points increases,
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the error decreases in magnitude. Since the error is the difference between
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a sinusoid and a sequence of approximating line segments, the magnitude of
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the error is
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roughly proportional to the square of the phase difference between each
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pair of points, or in other words, inversely proportional to the square
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of the number of points in the wavetable. Put another way, wavetable
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error decreases by 12 dB each time the table doubles in size. (This
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rule of thumb is only good for tables with 4 or more points.)
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<P>
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Four-point (cubic) interpolation works similarly. The interpolation formula
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is:
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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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{y_{\mathrm{INT}}}(x) =
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\end{displaymath}
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-->
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<IMG
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WIDTH="69" HEIGHT="28" BORDER="0"
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SRC="img254.png"
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ALT="\begin{displaymath}
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{y_{\mathrm{INT}}}(x) =
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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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-f (f-1)(f-2)/6 \cdot y[{x_0}-1]
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+ (f+1)(f-1)(f-2)/2 \cdot y[{x_0}]
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\end{displaymath}
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-->
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<IMG
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WIDTH="435" HEIGHT="28" BORDER="0"
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SRC="img255.png"
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ALT="\begin{displaymath}
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-f (f-1)(f-2)/6 \cdot y[{x_0}-1]
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+ (f+1)(f-1)(f-2)/2 \cdot y[{x_0}]
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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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- (f+1) f (f-2) / 2 \cdot y[{x_0}+1]
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+ (f+1) f (f-1) / 6 \cdot y[{x_0}+2]
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\end{displaymath}
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-->
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<IMG
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WIDTH="422" HEIGHT="28" BORDER="0"
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SRC="img256.png"
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ALT="\begin{displaymath}
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- (f+1) f (f-2) / 2 \cdot y[{x_0}+1]
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+ (f+1) f (f-1) / 6 \cdot y[{x_0}+2]
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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="79" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img257.png"
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ALT="$f = x - {x_0}$"> is the fractional part of the index. For tables
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with 4 or more points, doubling the number of points on the table tends
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to improve the RMS error by 24 dB. Table 2.1 shows the
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calculated RMS error for sinusoids at various periods for 1, 2, and
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4 point interpolation. (A slightly different quantity is measured in
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[<A
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HREF="node202.html#r-moore90">Moo90</A>, p.164]. There, the errors in amplitude and phase are also
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added in, yielding slightly more pessimistic results. See also
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[<A
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HREF="node202.html#r-hartmann87">Har87</A>].)
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<P>
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<BR><P></P>
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<DIV ALIGN="CENTER"><A ID="2392"></A>
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<TABLE>
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<CAPTION><STRONG>Table 2.1:</STRONG>
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RMS error for table lookup using 1, 2, and 4 point interpolation
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at various table sizes.</CAPTION>
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<TR><TD><TABLE CELLPADDING=3 BORDER="1">
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<TR><TD ALIGN="RIGHT">period</TD>
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<TD ALIGN="CENTER" COLSPAN=3>interpolation points</TD>
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</TR>
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<TR><TD ALIGN="RIGHT"> </TD>
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<TD ALIGN="RIGHT">1</TD>
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<TD ALIGN="RIGHT">2</TD>
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<TD ALIGN="RIGHT">4</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">2</TD>
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<TD ALIGN="RIGHT">-1.2</TD>
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<TD ALIGN="RIGHT">-17.1</TD>
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<TD ALIGN="RIGHT">-20.2</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">3</TD>
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<TD ALIGN="RIGHT">-2.0</TD>
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<TD ALIGN="RIGHT">-11.9</TD>
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<TD ALIGN="RIGHT">-15.5</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">4</TD>
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<TD ALIGN="RIGHT">-4.2</TD>
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<TD ALIGN="RIGHT">-17.1</TD>
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<TD ALIGN="RIGHT">-24.8</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">8</TD>
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<TD ALIGN="RIGHT">-10.0</TD>
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<TD ALIGN="RIGHT">-29.6</TD>
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<TD ALIGN="RIGHT">-48.4</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">16</TD>
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<TD ALIGN="RIGHT">-15.9</TD>
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<TD ALIGN="RIGHT">-41.8</TD>
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<TD ALIGN="RIGHT">-72.5</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">32</TD>
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<TD ALIGN="RIGHT">-21.9</TD>
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<TD ALIGN="RIGHT">-53.8</TD>
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<TD ALIGN="RIGHT">-96.5</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">64</TD>
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<TD ALIGN="RIGHT">-27.9</TD>
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<TD ALIGN="RIGHT">-65.9</TD>
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<TD ALIGN="RIGHT">-120.6</TD>
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</TR>
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<TR><TD ALIGN="RIGHT">128</TD>
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<TD ALIGN="RIGHT">-34.0</TD>
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<TD ALIGN="RIGHT">-77.9</TD>
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<TD ALIGN="RIGHT">-144.7</TD>
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</TR>
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</TABLE>
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<A ID="tab02.1"></A></TD></TR>
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</TABLE>
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</DIV><P></P>
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<BR>
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<P>
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The allowable input domain for table lookup depends on the number of points of
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interpolation. In general, when using <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="$k$">-point interpolation into a table
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with <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> points, the input may range over an interval of <IMG
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WIDTH="73" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img258.png"
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ALT="$N + 1 - k$"> points.
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If <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img259.png"
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ALT="$k=1$"> (i.e., no interpolation at all), the domain is 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="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">
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(including the endpoint at <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="$0$"> but excluding the one at <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">) assuming input
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values are truncated (as is done for non-interpolated table lookup in Pd). The
|
|
domain is from -<IMG
|
|
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img98.png"
|
|
ALT="$1/2$"> to <IMG
|
|
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img260.png"
|
|
ALT="$N-1/2$"> if, instead, we round the input to the nearest
|
|
integer instead of interpolating. In either case, the domain stretches over a
|
|
length of <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> points.
|
|
|
|
<P>
|
|
For two-point interpolation, the input must lie between the first and last
|
|
points, that is, between <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> and <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img171.png"
|
|
ALT="$N-1$">. So the <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> points suffice to define
|
|
the function over a domain of length <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img171.png"
|
|
ALT="$N-1$">. For four-point interpolation,
|
|
we cannot get values for inputs between 0 and 1 (not having the required
|
|
two points to the left of the input) and neither can we for the space between
|
|
the last two points (<IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img261.png"
|
|
ALT="$N-2$"> and <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img171.png"
|
|
ALT="$N-1$">). So in this case the domain reaches from
|
|
<IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img262.png"
|
|
ALT="$1$"> to <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img261.png"
|
|
ALT="$N-2$"> and has length <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img263.png"
|
|
ALT="$N-3$">.
|
|
|
|
<P>
|
|
Periodic waveforms stored in wavetables require special treatment at the
|
|
ends of the table. For example, suppose we wish to store a pure sinusoid of
|
|
length <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">. For non-interpolating table lookup, it suffices to set, for
|
|
example,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N-1
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="255" HEIGHT="28" BORDER="0"
|
|
SRC="img264.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N-1
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
For two-point interpolation, we need <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img265.png"
|
|
ALT="$N+1$"> points:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="228" HEIGHT="28" BORDER="0"
|
|
SRC="img266.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
In other words, we must repeat the first (<IMG
|
|
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img79.png"
|
|
ALT="$n=0$">) point at the end, so that the
|
|
last segment from <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img171.png"
|
|
ALT="$N-1$"> to <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> reaches back to the beginning value.
|
|
|
|
<P>
|
|
For four-point interpolation, the cycle must be adjusted to start at the point
|
|
<IMG
|
|
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img267.png"
|
|
ALT="$n=1$">, since we can't get properly interpolated values out for inputs less than
|
|
one. If, then, one cycle of the wavetable is arranged from <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img262.png"
|
|
ALT="$1$"> to <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">, we
|
|
must supply extra points for <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> (copied from <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">), and also <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img265.png"
|
|
ALT="$N+1$"> and <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img268.png"
|
|
ALT="$N+2$">,
|
|
copied from <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img262.png"
|
|
ALT="$1$"> and <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img269.png"
|
|
ALT="$2$">, to make a table of length <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img270.png"
|
|
ALT="$N+3$">. For the same sinusoid
|
|
as above, the table should contain:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = \cos(2 \pi (n-1) / N) ,\, n = 0, \ldots, N+2
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="294" HEIGHT="28" BORDER="0"
|
|
SRC="img271.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = \cos(2 \pi (n-1) / N) ,\, n = 0, \ldots, N+2
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
|
|
<P>
|
|
<HR>
|
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<!--Navigation Panel-->
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<B> Next:</B> <A ID="tex2html1003"
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HREF="node32.html">Examples</A>
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<B> Up:</B> <A ID="tex2html997"
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HREF="node26.html">Wavetables and samplers</A>
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<B> Previous:</B> <A ID="tex2html991"
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HREF="node30.html">Timbre stretching</A>
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
|
|
2006-12-30
|
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</ADDRESS>
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</HTML>
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