Can anyone answer my questions about the filters?

[palette]

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.(I've learn a lot about filters from this article: http://www.apogeeweb.net/article/35.html, but I still have some confusion)

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 first 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)

 

[john12ax7]

I think you are missing an important concept,  in that filter design is really a trade off of different parameters.  There is no "best", it depends on the application.

For example,  Butterworth is known as "maximally flat", because the frequency response in the passband is flat.  Whereas Chebychev has better attenuation in the stopband, but this comes at the expense of ripple in the passage. So which is better? It depends.  It's a trade off balancing act,  which is a lot of what engineering is about.

 

[ruffrecords]

The other thing to rememeber is that the ideal is impossible. It will have an infinite delay.

The filters you mention are all 5th order and good enough for most purposes. If you want to make a better version you need to increase the order.

Cheers

Ian

 

[abbey road d enfer]

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.

Not better. All these different formulae are for different alignments, each one obeying a different constraint. B'worth is maximally flat in the bandpass, Chebichev allows steeper roll-off at the detriment of linearity in the bandpass (hence the 0.1dB and 1 dB ripple families), Bessel is for best transient response (minimum overshoot), Cauer is for extra-steep roll-off with a big compromise on transient response and near-cut-off behaviour. There are also "linear-phase" alignments where the phase response is flat within the bandpass (but not outside - this is possible only with digital FIR filters).  All these different alignments are useful; none has supplanted the others.

I believe this should answer both your questions.
Now, your idea of the perfect filter (maximum gain in pass band, a vertical or instant cutoff and 0 in the stop band) is called a  "brickwall" filter. It's a mathematical concept that cannot be achieved in real-life.
As it has been mentioned earlier, this filter would have some characteristics that would make it impossible to use, but also I don't think it would be a significant improvement in any operational performance.