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Band-pass filter

Starting with the three filter types shown above, which all have real-valued poles and zeros, we now transform them to operate on bands located off the real axis. The low-pass, high-pass, and shelving filters will then become band-pass, stop-band, and peaking filters. First we develop the band-pass filter. Suppose we want a center frequency at $\omega $ radians and a bandwidth of $\beta $. We take the low-pass filter with cutoff frequency $\beta $; its pole is located, for small values of $\beta $, roughly at $p = 1 - \beta$. Now rotate this value by $\omega $ radians in the complex plane, i.e., multiply by the complex number $\cos \omega + i \sin \omega$. The new pole is at:

\begin{displaymath}
{P_1} = (1 - \beta) (\cos \omega + i \sin \omega)
\end{displaymath}

To get a real-valued output, this must be paired with another pole:

\begin{displaymath}
{P_2} = \overline{P_1} = (1 - \beta) (\cos \omega - i \sin \omega)
\end{displaymath}

The resulting pole-zero plot is as shown in Figure 8.15.

Figure 8.15: Two-pole band-pass filter: (a) pole-zero diagram; (b) frequency response.
\begin{figure}\psfig{file=figs/fig08.15.ps}\end{figure}

The peak is approximately (not exactly) at the desired center frequency $\omega $, and the frequency response drops by 3 decibels approximately $\beta $ radians above and below it. It is often desirable to normalize the filter to have a peak gain near unity; this is done by multiplying the input or output by the product of the distances of the two poles to the peak on the circle, or (very approximately):

\begin{displaymath}
\beta * (\beta + 2 \omega)
\end{displaymath}

For some applications it is desirable to add a zero at the points $1$ and $-1$, so that the gain drops to zero at angular frequencies $0$ and $\pi $.


next up previous contents index
Next: Peaking and stop-band filter Up: Designing filters Previous: Shelving filter   Contents   Index
Miller Puckette 2006-09-24