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In this paper on page 4 (or in slide page, 23), it describes the equation for calculating the minimum required order for elliptic filter for design specification. However, it says it is only the estimate, and I wonder how the equation is actually derived. Can I trust the equation in all applications, or should I give some doubts before using it?

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The estimate is just that: an estimate. If the estimate yields $N=2.4$ then you're pretty safe with a $3^{\rm rd}$ order filter. If the estimate yields $N=2.95$, then a 3rd order might work, but there is a high likelihood that a $4^{\rm th}$ order is required. In cases where the estimate is just under a whole number, the way to be safe is to either increase the estimated order by 1, or to design both a $N$ and $N+1$ order filter to see if both meet your requirements.

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