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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 radians and a bandwidth of . We take the low-pass filter with cutoff frequency ; its pole is located, for small values of , roughly at . Now rotate this value by radians in the complex plane, i.e., multiply by the complex number . The new pole is at:

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

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.

The peak is approximately (not exactly) at the desired center frequency , and the frequency response drops by 3 decibels approximately 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):

For some applications it is desirable to add a zero at the points and , so that the gain drops to zero at angular frequencies and .

Next: Peaking and stop-band filter Up: Designing filters Previous: Shelving filter Contents Index