# Lowpass Filter Design by Pole Placement

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An ideal lowpass filter, drawn in blue, passes all frequencies less than and rejects all frequencies greater than . Such a filter cannot be implemented physically because it is non-causal. However, we can design causal filters, drawn in orange, that approximate the ideal lowpass filter. This Demonstration lets you design such a filter by locating poles (of the transfer function), drawn with red s, in different arrangements.

Contributed by: Aaron T. Becker (December 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Four options are given for filter design by pole placement. All filters are scaled so that the maximum magnitude is about 1.

The first, "vertical line", places poles equally spaced on the imaginary axis between and , with real component . The associated magnitude spectrum favors frequencies in the to range, but the pattern has undesirable resonances at frequencies corresponding to the pole locations along the complex axis.

The second, "triangle", places poles equally spaced along the imaginary axis between and , with real component , where is the imaginary component of the pole. The associated magnitude spectrum again favors frequencies in the to range, but the pattern has undesirable resonances at and .

The "Butterworth" filter, introduced in 1930 by British engineer and physicist Stephen Butterworth, places the poles along the circumference of a circle of radius in the left half-plane. The resulting filter is causal, BIBO stable, and flat for low frequencies. The roll-off rate is .

Switching to placing poles along an "elliptical"* *circumference enables faster roll-off, but introduces ripple in the low-frequency range.

Finally, in "manual" mode, you can drag, add, or delete poles and see the resulting effects in the frequency domain.

Reference

[1] F. Ulaby and A. Yagle, *Engineering Signals and Systems*, Allendale: NTS Press, 2012.

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