187 lines
6.4 KiB
HTML
187 lines
6.4 KiB
HTML
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||
|
|
|
||
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
||
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
||
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
||
|
|
* with significant contributions from:
|
||
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
||
|
|
<HTML>
|
||
|
|
<HEAD>
|
||
|
|
<TITLE>One-pole, one-zero high-pass filter</TITLE>
|
||
|
|
<META NAME="description" CONTENT="One-pole, one-zero high-pass filter">
|
||
|
|
<META NAME="keywords" CONTENT="book">
|
||
|
|
<META NAME="resource-type" CONTENT="document">
|
||
|
|
<META NAME="distribution" CONTENT="global">
|
||
|
|
|
||
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
||
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
||
|
|
|
||
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
||
|
|
|
||
|
|
<LINK REL="next" HREF="node142.html">
|
||
|
|
<LINK REL="previous" HREF="node140.html">
|
||
|
|
<LINK REL="up" HREF="node139.html">
|
||
|
|
<LINK REL="next" HREF="node142.html">
|
||
|
|
</HEAD>
|
||
|
|
|
||
|
|
<BODY >
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2688"
|
||
|
|
HREF="node142.html">
|
||
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||
|
|
<A NAME="tex2html2682"
|
||
|
|
HREF="node139.html">
|
||
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||
|
|
<A NAME="tex2html2676"
|
||
|
|
HREF="node140.html">
|
||
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||
|
|
<A NAME="tex2html2684"
|
||
|
|
HREF="node4.html">
|
||
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||
|
|
<A NAME="tex2html2686"
|
||
|
|
HREF="node201.html">
|
||
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||
|
|
<BR>
|
||
|
|
<B> Next:</B> <A NAME="tex2html2689"
|
||
|
|
HREF="node142.html">Shelving filter</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2683"
|
||
|
|
HREF="node139.html">Designing filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2677"
|
||
|
|
HREF="node140.html">One-pole low-pass filter</A>
|
||
|
|
<B> <A NAME="tex2html2685"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2687"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<BR>
|
||
|
|
<BR>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
|
||
|
|
<H2><A NAME="SECTION001232000000000000000"></A>
|
||
|
|
<A NAME="sect08.highpass"></A>
|
||
|
|
<BR>
|
||
|
|
One-pole, one-zero high-pass filter
|
||
|
|
</H2>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
Sometimes an audio signal carries an unwanted constant offset, or in other
|
||
|
|
words, a zero-frequency component. For example, the waveshaping spectra of
|
||
|
|
Section <A HREF="node78.html#sect5.waveshaping">5.3</A> almost always contain a constant component.
|
||
|
|
This is inaudible, but, since it specifies electrical power that is sent
|
||
|
|
to your speakers, its presence reduces the level of loudness you can
|
||
|
|
reach without distortion. Another name for a constant signal component is
|
||
|
|
<A NAME="10330"></A>``DC", meaning ``direct current".
|
||
|
|
|
||
|
|
<P>
|
||
|
|
An easy and practical way to remove the zero-frequency component from an audio
|
||
|
|
signal is to use a one-pole low-pass filter to extract it, and then subtract the
|
||
|
|
result from the signal. The resulting transfer function is one minus the
|
||
|
|
transfer function of the low-pass filter:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
H(Z) = 1 - {{{1-p} \over {1 - p{Z^{-1}}}}}
|
||
|
|
= p{{{1-{Z^{-1}}} \over {1 - p{Z^{-1}}}}}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="252" HEIGHT="44" BORDER="0"
|
||
|
|
SRC="img912.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
H(Z) = 1 - {{{1-p} \over {1 - p{Z^{-1}}}}}
|
||
|
|
= p{{{1-{Z^{-1}}} \over {1 - p{Z^{-1}}}}}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
The factor of <IMG
|
||
|
|
WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img910.png"
|
||
|
|
ALT="$1-p$"> in the numerator of the low-pass transfer function is the
|
||
|
|
normalization factor needed so that the gain is one at zero frequency.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
By examining the right-hand side of the equation (comparing it to the general
|
||
|
|
formula for compound filters), we
|
||
|
|
see that there is still a pole at the real number <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$">, and there is now also
|
||
|
|
a zero at the point <IMG
|
||
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img262.png"
|
||
|
|
ALT="$1$">. The pole-zero plot is shown in Figure
|
||
|
|
<A HREF="#fig08.13">8.13</A>
|
||
|
|
(part a), and the frequency response in part (b). (Henceforth, we
|
||
|
|
will only plot frequency responses to the Nyquist frequency <IMG
|
||
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img41.png"
|
||
|
|
ALT="$\pi $">;
|
||
|
|
in the previous example we plotted it all the way up to the sample rate,
|
||
|
|
<IMG
|
||
|
|
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img16.png"
|
||
|
|
ALT="$2\pi $">.)
|
||
|
|
|
||
|
|
<P>
|
||
|
|
|
||
|
|
<DIV ALIGN="CENTER"><A NAME="fig08.13"></A><A NAME="10338"></A>
|
||
|
|
<TABLE>
|
||
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.13:</STRONG>
|
||
|
|
One-pole, one-zero high-pass filter: (a) pole-zero diagram; (b)
|
||
|
|
frequency response (from zero to Nyquist frequency).</CAPTION>
|
||
|
|
<TR><TD><IMG
|
||
|
|
WIDTH="542" HEIGHT="245" BORDER="0"
|
||
|
|
SRC="img913.png"
|
||
|
|
ALT="\begin{figure}\psfig{file=figs/fig08.13.ps}\end{figure}"></TD></TR>
|
||
|
|
</TABLE>
|
||
|
|
</DIV>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR>
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2688"
|
||
|
|
HREF="node142.html">
|
||
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||
|
|
<A NAME="tex2html2682"
|
||
|
|
HREF="node139.html">
|
||
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||
|
|
<A NAME="tex2html2676"
|
||
|
|
HREF="node140.html">
|
||
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||
|
|
<A NAME="tex2html2684"
|
||
|
|
HREF="node4.html">
|
||
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||
|
|
<A NAME="tex2html2686"
|
||
|
|
HREF="node201.html">
|
||
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||
|
|
<BR>
|
||
|
|
<B> Next:</B> <A NAME="tex2html2689"
|
||
|
|
HREF="node142.html">Shelving filter</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2683"
|
||
|
|
HREF="node139.html">Designing filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2677"
|
||
|
|
HREF="node140.html">One-pole low-pass filter</A>
|
||
|
|
<B> <A NAME="tex2html2685"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2687"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
<ADDRESS>
|
||
|
|
Miller Puckette
|
||
|
|
2006-12-30
|
||
|
|
</ADDRESS>
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|