As I understand it, the ideal frequency response for a filter is that we want maximum gain in pass band, a vertical or instant cutoff and 0 in the stop band, however this is ideal and can never be achieved in real life.
Along came Butterworth and found a relationship / formula for choosing the right combination of inductors and capacitors to be able to create filters that matched a certain spec and resembled the frequency response that we want – but not ideal.
With the exact same filter structure along came Chebyshev and he worked out an even better formula to get an even tighter / more tuned relationship to give us a filter that was closer to idea than Butterworths relationship, for example a much more steeper transition band than the same design using Butterworths.
Then again, along came Cauer with his own method of calculating the relationships and values needed to get an even better and closer approximation to an idea filter. With the steepest transition yet.
With all these, they haven't discovered a new "structure" or found that by making a different connection you get a better filter, but instead each one devised a better and more accurate formula for choosing the values of your components to get your filter close and closer to ideal.
So my question is this.
Lets say we are dealing with a 5 order low pass filter. (The type doesn't really matter). With Butterworth we have a good filter, then still using a 5th order and same structure Chebyshev was able to "tune" it even more to create and even better response, and then the same with Cauer.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
And then when someone is able to devise or work out a relationship linking all these values together and with the spec thats required it would then be a classified its own filter type such as Butterworth etc?
My second question is then,
Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
Similar to saying, if NP = P was proven it would mean we know for sure that there are solution to the problems but we haven't worked out yet how to get there / havnt worked out an algorithm for it. (Sorry if thats a bad example or its wrong but its the best analogy I can think of for now)
Best Answer
The difference between the filters you name is not that each new one invented made a closer approximation to the ideal filter, but that each one optimizes the filter for a different characteristic. Because there's a trade-off between different characteristics, each one chooses a different way to make this trade-off.
Like Andy said, the Butterworth filter has maximal flatness in the passband. And the Chebychev filter has the fastest roll-off between the passband and stop-band, at the cost of ripple in the passband.
The Elliptic filter (Cauer filter) parameterizes the balance between pass-band and stop-band ripple, with the fastest possible roll-off given the chosen ripple characteristics.
It depends what you mean by "best possible" or "closest model". If you mean the one with the flattest response in the pass-band, you'd end up with the Butterworth filter. If you mean the best possible roll-off given a fixed ripple in the pass-band, you'd end up with the Chebychev design, etc.
If you chose some other criterion to optimize (like mean-square error between the filter characteristic and the boxcar ideal, for example), you could end up with a different design.
The filters you named (Butterworth, Chebychev, Cauer) are the best, for the different definitions of "best" that define those filters.
If you had some other definition of "best" in mind, you could certainly design a filter to optimize that, with existing technology. Andy's answer names a couple of other criteria and the filters that optimize them, for example.
Let me add one other question you might ask as a follow up,
Why don't we in practice design filters to optimize the mean-square error between the filter characteristic and the boxcar ideal?
Probably because the mean-square error doesn't capture well the design-impact of "errors" in the pass-band and stop-band response. Because the ideal response has 0 magnitude in the stop-band it's hard to define a "relative response" measurement that has equal weight in both regions.
For example, in some designs an error of -40 dB (.01 V/V) relative to the ideal 0 V/V response in the stop-band would be much worse than an error of 0.01 V/V in the passband.