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The process of designing filters includes too many parameters like bandwidth, transition, degree of attenuation in side-bands, phase delay, causality, the magnitude of ripples, and so on. First of all, we want to approach the ideal filter structure so we can design many different filters to obtain the desired ideal filter.
How can we compare all of these filters in terms of quality? Is there any way to measure the performance of filters? I am looking for a mathematical solution to this problem.
The answer can be given without being at all specific to filters. In fact, one has to answer that in terms of engineering in general:
Whenever you have alternative solutions, the only measure of quality that really matters, is fulfillment of the requirements.
So, you'll have to know what you'll use that filter for, and evaluate how well the alternatives perform with respect to that specific application. Nothing else helps you. There's arbitrarily many properties that a filter can have, and for most of them, it's not clear whether a smaller or bigger value is better, and they all are intrinsically linked to other parameters. There's no single "best" solution, all depends on the application.
So, define your application. Define how that sees performance. For example, if your FIR is used in a wireless communications system, then the FIR that makes the system have the least bit error rate might be the "highest quality" one for this one system under this specific metric that you care about. For a different system, a different FIR will be the best one.
to obtain the desired ideal filter.
That's like asking: how do design the ideal car? There is no such a thing: The reason why there are so many different car models, is because people have very different needs and requirements. Each car model represents a different trade off between size, cost, efficiency, loadability, towing ability, aesthetics, performance, etc.
The art of filter design is NOT to design the actual filter (which is rather trivial), but to first understand and formulate the specific requirements of your application and specifically the trade offs and then tweak the design parameters until you have the best possible set of trade-offs.
