Lowpass Filter Design by Pole Placement

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+