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237 lines
7.3 KiB
237 lines
7.3 KiB
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<H2><A ID="SECTION0012311000000000000000">
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All-pass filters</A>
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</H2>
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<P>
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Sometimes a filter is applied to get a desired phase change, rather than to
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alter the amplitudes of the frequency components of a sound. To do this
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we would need a way to design a filter with a constant, unit frequency response
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but which changes the phase of an incoming sinusoid in a way that depends on its
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frequency. We have already seen in Chapter 7 that a delay of length <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">
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introduces a phase change of <IMG
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WIDTH="34" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img677.png"
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ALT="$- d \omega$">, at the angular 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 $">.
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Another class of filters, called
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<A ID="10502"></A><I>all-pass filters</I>,
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can make phase changes which are more interesting functions of <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">.
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<P>
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To design an all-pass filter, we start with two facts: first, an elementary
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recirculating filter and an elementary non-recirculating one cancel each other
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out perfectly if they have the same gain coefficient. In other words, if a
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signal has been put through a one-zero filter, either real or complex, the
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effect can be reversed by sequentially applying a one-pole filter, and vice
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versa.
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<P>
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The second fact is that the elementary non-recirculating filter of the second
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form has the same frequency response as that of the first form; they differ only
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in phase response. So if we combine an elementary recirculating filter with
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an elementary non-recirculating one of the second form, the frequency responses
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cancel out (to a flat gain independent of frequency) but the phase response
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is not constant.
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<P>
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To find the transfer function, we choose the same complex number <IMG
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WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img982.png"
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ALT="$P<1$"> as
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coefficient for both elementary filters and multiply their transfer functions:
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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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H(Z) = {{
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{\overline{P} - {Z^{-1}}}
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} \over {
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{1 - P{Z^{-1}}}
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="131" HEIGHT="44" BORDER="0"
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SRC="img983.png"
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ALT="\begin{displaymath}
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H(Z) = {{
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{\overline{P} - {Z^{-1}}}
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} \over {
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{1 - P{Z^{-1}}}
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}}
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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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The coefficient <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$"> controls both the location of the one pole (at <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$"> itself)
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and the zero (at <!-- MATH
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$1/\overline{P}$
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-->
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<IMG
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WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img984.png"
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ALT="$1/\overline{P}$">). Figure <A HREF="#fig08.23">8.23</A> shows the phase response of
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the all-pass filter for four real-valued choices <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$"> of the coefficient. At
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frequencies of <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="$0$">, <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 $">, the phase response is just that of a
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one-sample delay; but for frequencies in between, the phase response is bent
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upward or downward depending on the coefficient.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig08.23"></A><A ID="10511"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.23:</STRONG>
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Phase response of all-pass filters with different pole locations <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img57.png"
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ALT="$p$">.
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When the pole is located at zero, the filter reduces to a one-sample delay.</CAPTION>
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<TR><TD><IMG
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WIDTH="319" HEIGHT="218" BORDER="0"
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SRC="img985.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.23.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Complex coefficients give similar phase response curves, but the frequencies at
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which they cross the diagonal line in the figure are shifted according to the
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argument of the coefficient <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$">.
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<P>
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HREF="node151.html">Applications</A>
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<B> Up:</B> <A ID="tex2html2807"
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HREF="node139.html">Designing filters</A>
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<B> Previous:</B> <A ID="tex2html2803"
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HREF="node149.html">Impulse responses of recirculating</A>
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<B> <A ID="tex2html2809"
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<ADDRESS>
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Miller Puckette
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2006-12-30
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