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<H1><A ID="SECTION001180000000000000000">
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Fidelity of interpolating delay lines</A>
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</H1>
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<P>
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Since they are in effect doing wavetable lookup, variable delay lines
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introduce distortion to the signals they operate on. Moreover, a
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subtler problem can come up even when the delay line is not changing in length:
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the frequency response, in real situations, is never perfectly flat for a
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delay line whose length is not an integer.
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<P>
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If the delay time is changing from sample to sample, the distortion results of
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Section <A HREF="node31.html#sect2.interpolation">2.5</A> apply. To use them, we suppose that the
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delay line input can be broken down into sinusoids and consider separately what
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happens to each individual sinusoid. We can use Table 2.1
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(Page <A HREF="node31.html#tab02.1"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>) to predict
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the RMS level of the combined distortion products for an interpolated
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variable delay line.
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<P>
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We'll assume here that we want to use four-point interpolation. For sinusoids
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with periods longer than 32 samples (that is, for frequencies below 1/16 of the
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Nyquist frequency) the distortion is 96 dB or better--unlikely ever to be
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noticeable. At a 44 kHz. sample rate, these periods would correspond to
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frequencies up to about 1400 Hertz. At higher frequencies the quality degrades,
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and above 1/4 the Nyquist frequency the distortion products, which are only
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down about 50 dB, will probably be audible.
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<P>
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The situation for a complex tone depends primarily on the amplitudes and
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frequencies of its higher partials. Suppose, for instance, that a tone's
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partials above 5000 Hertz are at least 20 dB less than its strongest partial, and
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that above 10000 Hertz they are down 60 dB. Then as a rough estimate, the
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distortion products from the range 5000-10000 will each be limited to
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about -68 dB and those from above 10000 Hertz will be limited to about -75 dB
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(because the worst figure in the table is about -15 dB and this must be
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added to the strength of the partial involved.)
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<P>
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If the high-frequency content of the input signal does turn out to give
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unacceptable distortion products, in general it is more effective to increase
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the sample rate than the number of points of interpolation. For periods
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greater than 4 samples, doubling the period (by doubling the sample
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rate, for example) decreases distortion by about 24 dB.
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<P>
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The 4-point interpolating delay line's frequency response is nearly
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flat up to half the Nyquist frequency, but thereafter it dives quickly.
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Suppose (to pick the worst case) that the delay is set halfway between two
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integers, say 1.5. Cubic interpolation gives:
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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[1.5] = {{-x[0] + 9x[1] + 9 x[2] - x[3]} \over 8}
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\end{displaymath}
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-->
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<IMG
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WIDTH="250" HEIGHT="40" BORDER="0"
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SRC="img792.png"
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ALT="\begin{displaymath}
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x[1.5] = {{-x[0] + 9x[1] + 9 x[2] - x[3]} \over 8}
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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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Now let <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="$x[n]$"> be a (real-valued) unit-amplitude sinusoid with angular
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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 $">, whose phase is zero at <IMG
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WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img793.png"
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ALT="$1.5$">:
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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] = \cos(\omega \cdot (n - 1.5))
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\end{displaymath}
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-->
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<IMG
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WIDTH="164" HEIGHT="28" BORDER="0"
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SRC="img794.png"
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ALT="\begin{displaymath}
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x[n] = \cos(\omega \cdot (n - 1.5))
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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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and compute <IMG
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WIDTH="41" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img795.png"
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ALT="$x[1.5]$"> using the above formula:
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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[1.5] = {{9 \cos(\omega/2) - \cos(3 \omega / 2)}\over4}
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\end{displaymath}
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-->
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<IMG
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WIDTH="218" HEIGHT="40" BORDER="0"
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SRC="img796.png"
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ALT="\begin{displaymath}
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x[1.5] = {{9 \cos(\omega/2) - \cos(3 \omega / 2)}\over4}
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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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This is the peak value of the sinusoid that comes back out of the delay
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line, and since the peak amplitude going in was one, this shows the
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frequency response of the delay line. This is graphed in Figure
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<A HREF="#fig07.18">7.18</A>. At half the Nyquist frequency (<!-- MATH
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$\omega = \pi / 2$
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-->
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<IMG
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WIDTH="60" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img797.png"
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ALT="$\omega = \pi / 2$">) the gain
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is about
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-1 dB, which is a barely perceptible drop in amplitude. At the Nyquist
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frequency
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itself, however, the gain is zero.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig07.18"></A><A ID="8143"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.18:</STRONG>
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Gain of a four-point interpolating delay line with a delay halfway
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between two integers. The DC gain is one.</CAPTION>
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<TR><TD><IMG
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WIDTH="447" HEIGHT="172" BORDER="0"
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SRC="img798.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.18.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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As with the results for distortion, the frequency response improves radically
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with a doubling of sample rate. If we run our delay at a sample rate of 88200
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Hertz instead of the standard 44100, we will get only about 1 dB of roll-off all
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the way up to 20000 Hertz.
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HREF="node113.html">Variable and fractional shifts</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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