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<H2><A ID="SECTION001239000000000000000"></A>
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<A ID="sect8.timevarying"></A>
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<BR>
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Time-varying coefficients
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</H2>
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<P>
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In some recursive filter designs, changing the coefficients of the filter can
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inject energy into the system. A physical analogue is a child on a swing set.
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The child oscillates back and forth at the resonant frequency of the system,
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and pushing or pulling the child injects or extracts energy smoothly. However,
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if you decide to shorten the chain or move the swing set itself, you may inject
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an unpredictable amount of energy into the system. The same thing can happen
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when you change the coefficients in a resonant recirculating filter.
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<P>
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The simple one-zero and one-pole filters used here don't have this
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difficulty; if the feedback or feed-forward gain is changed smoothly (in the
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sense of an amplitude envelope) the output will behave smoothly as well.
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But one subtlety arises when trying to normalize a recursive filter's output
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when the feedback gain is close to one. For example, suppose we have a one-pole
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low-pass filter with gain 0.99 (for a cutoff frequency of 0.01 radians, or 70
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Hertz at the usual sample rate). To normalize this for unit DC gain we multiply
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by 0.01. Suppose now we wish to double the cutoff frequency by changing the
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gain slightly to 0.98. This is fine except that the normalizing factor
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suddenly doubles. If we multiply the filter's output by the normalizing
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factor, the output will suddenly, although perhaps only momentarily, jump by a
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factor of two.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.21"></A><A ID="10472"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.21:</STRONG>
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Normalizing a recirculating elementary filter:
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(a) correctly, by multiplying in the normalization factor at the input; (b)
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incorrectly, multiplying at the output.</CAPTION>
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<TR><TD><IMG
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WIDTH="387" HEIGHT="255" BORDER="0"
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SRC="img973.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.21.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 trick is to normalize at the <I>input</I> of the filter, not the output.
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Figure <A HREF="#fig08.21">8.21</A> (part a) shows a complex recirculating filter, with
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feedback gain <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$">, normalized at the input by <IMG
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WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img974.png"
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ALT="$1-\vert P\vert$"> so that the peak gain
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is one. Part (b) shows the wrong way to do it, multiplying at the output.
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<P>
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Things get more complicated when several elementary recirculating filters are
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put in series, since the correct normalizing factor is in general a function of
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all the coefficients. One possible approach, if such a filter is required to
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change rapidly, is to normalize each input separately as if it were acting
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alone, then multiplying the output, finally, by whatever further correction
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is needed.
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HREF="node139.html">Designing filters</A>
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<B> Previous:</B> <A ID="tex2html2775"
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HREF="node147.html">Butterworth band-pass filter</A>
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Miller Puckette
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2006-12-30
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