Which IIR filters approximate a Gaussian filter? So it recently dawned on me that Bessel filters, despite being listed along with the other common types, are really an oddball that belongs in a different "class", and I'm trying to learn more about it.


  The rectangular magnitude response represents the ideal frequency domain response, for the transition band is zero and the stopband has infinite attenuation. The Gaussian magnitude response, on the other hand, represents the ideal time-domain response, in that no overshoots occur in the impulse response and the step response. Many of the responses attained in practice are approximations to these ideal ones  source


So a brickwall filter is convolution with a sinc function, and has these frequency domain properties:


Flat passband
Zero stopband
Infinite roll-off rate/no transition band


It's non-causal and unrealizable because of the infinite tails in both directions.  It is approximated by these IIR filters, with the approximation improving as order increases:


Butterworth (maximally flat passband)
Chebyshev (maximum roll-off rate with stopband or passband ripple)
Elliptic (maximum roll-off rate with stopband and passband ripple)
Legendre  (maximum roll-off rate with monotonic passband)






The Gaussian filter is convolution with a Gaussian function, and has these time domain properties:


Zero overshoot
Minimal rise and fall time
Minimal group delay


It's unrealizable for the same reasons as the sinc function, and can be approximated by these IIR filters, more closely as order increases:


Bessel (maximally flat group delay)  according to 1 and 2


Here's Bessel filters of increasing order along with a Gaussian dashed line I picked merely because it seemed to fit the trend ($e^{-{1 \over 2}(\pi \omega)^2}$):



So my questions are:

Is everything right so far?  If so, are there other IIR filters that approximate the Gaussian?  What are they optimized for?  Maybe one that minimizes overshoot?

If you search for "IIR Gaussian" you can find a few things (Deriche? van Vliet?), but I don't know if they're really the same as a Bessel or if they optimize for some other property, etc.
The Deriche and van Vliet filters are heuristics.  In both cases they choose the locations of poles and zeros to minimize either the RMS difference or the maximum difference of the filter's impulse response from a Gaussian.

Both filters are causal-anti-causal pairs.  So I think they have no phase error or group delay, but you need to be able to run them backwards on the data as well as forwards.  This makes them popular in image processing, but perhaps restricts their applicability elsewhere.

That they are heuristics is evidenced by the wealth of papers that tweak them.  For example a google search (while I was looking for the link to the Deriche paper) turned up this one that tries to solve the problem that the Deriche derivative-of-gaussian filter doesn't have an exactly 0 DC response.  There are also some interesting issues about initializing the boundary conditions correctly.

I have found the following overview a good resource:
Dave Hale, Recursive Gaussian ﬁlters, Colorado School of Mines Center for Wave Phenomena report CWP-546.