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HREF="node130.html">Band-pass and stop-band filters</A>
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HREF="node128.html">Taxonomy of filters</A>
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HREF="node128.html">Taxonomy of filters</A>
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<H2><A NAME="SECTION001211000000000000000">
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Low-pass and high-pass filters</A>
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</H2>
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<P>
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By far the most frequent purpose for using a filter is extracting either
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the low-frequency or the high-frequency portion of an audio signal, attenuating
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the rest. This is accomplished using a
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<A NAME="10062"></A><A NAME="10063"></A><I>low-pass</I> or
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<A NAME="10065"></A><A NAME="10066"></A><I>high-pass</I>
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filter.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.02"></A><A NAME="10070"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.2:</STRONG>
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Terminology for describing the frequency response of low-pass and
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high-pass filters. The horizontal axis is frequency and the vertical
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axis is gain. A low-pass filter is shown; a high-pass filter has the same
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features switched from right to left.</CAPTION>
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<TR><TD><IMG
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WIDTH="491" HEIGHT="318" BORDER="0"
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SRC="img855.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Ideally, a low-pass or high-pass filter would have a frequency response of
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one up to (or down to) a specified cutoff frequency and zero past it; but
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such filters cannot be realized in practice. Instead, we try to find
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realizable approximations to this ideal response. The more design effort
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and computation time we put into it, the closer we can get.
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<P>
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Figure <A HREF="#fig08.02">8.2</A> shows the frequency response of a low-pass
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filter. Frequency is divided into three bands, labeled on the
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horizontal axis. The
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<A NAME="10074"></A><I>passband</I>
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is the region (frequency band) where the filter should pass its input through
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to its output with unit gain.
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For a low-pass filter (as shown), the passband reaches from a frequency of
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zero up to a certain frequency limit. For a high-pass filter, the passband
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would appear on the right-hand side of the graph and would extend from the
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frequency limit up to the highest frequency possible. Any
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realizable filter's passband will be only approximately flat;
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the deviation from flatness is called the
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<A NAME="10076"></A><I>ripple</I>,
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and is often specified by giving the ratio between the highest and lowest gain
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in the passband, expressed in decibels. The ideal low-pass or high-pass filter
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would have a ripple of 0 dB.
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<P>
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The
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<A NAME="10078"></A><I>stopband</I>
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of a low-pass or high-pass filter is the frequency
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band over which the filter is intended not to transmit its input.
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The
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<A NAME="10080"></A><I>stopband attenuation</I>
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is the difference, in decibels, between the lowest gain in the passband
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and the highest gain in the stopband. Ideally this would
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be infinite; the higher the better.
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<P>
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Finally, a realizable filter, whose frequency response is always a
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continuous function of frequency, must have a frequency
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band over which the gain drops from the passband gain to the stopband
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gain; this is called the
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<A NAME="10082"></A><I>transition band</I>.
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The thinner this band can be made, the more nearly ideal the filter.
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