On these pages you will find notes on filters, accurate coefficients for specific filters, and software for solving filter problems. See Design Notes on the Circuit Analysis page for additional information.

If you're a little hazy
about how the pole plot shown in the above left figure leads to the response
curve shown on the above right, click on the *Refresh* or *Reload*
button on your browser's toolbar. The animation below zooms in on the filter
response surface in the left half of the complex plane to show the magnitude
response along the frequency (imaginary, *j*w) axis.
The 'zoom in' simply consists of scaling the vertical dimension (magnitude), so the complex
plane scales (s,
*j*w) remain fixed.

**Response of 5th Order Butterworth Filter over the Left-Half
Complex Plane.**

Note the open area under the lower right of the figure which represents the filter response and which illustrates the conventional frequency attenuation curve as the animation progresses.

**Magnitude in Left-Half Complex Plane for
Butterworth Filters of First 6 Orders**

This paper contains transfer functions, equations for amplitude and phase response, equations for group delay for all Butterworth filters up to order 5. Extended tables of Butterworth polynomials, poles, and circuit elements are provided in a separate paper, for filter orders up to 10.

Here are the transfer functions, equations for
amplitude and phase response, equations for group delay, and normalization
constants for all Bessel filters up to order 6.
Extended tables of Bessel polynomials, poles, and circuit elements are provided
in a separate paper for filter orders up to 10. Parameters
are given for *delay normalized* and *frequency normalized* filters.

So called "Optimum Filters", presented by Papoulis in 1958, exhibit maximum rolloff for a monotonic filter of given order. His original paper treated filters of odd order only, and it wasn't until his follow-up paper in 1959 that he completed the presentation for even orders. Tables of Optimum polynomials, poles and circuit elements are provided in this paper for filter orders up to 10. To my knowledge this information has not been published elsewhere. A separate paper describing the algorithms used to compute the filter element values and an example can be found here.

This is a screen shot of a DOS executable which displays a complete summary of the time and frequency domain behavior of any filter, given its representation as a ratio of polynomials in the complex variable 's'. Shown above are the main properties of interest for a 3rd order Bessel filter. The program generates files for each displayed response function, so the actual values used for each of the plots is available after a run. The program can be downloaded here.