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Say I want to design a low pass filter with stop and pass band ripple constrained to $\epsilon$ and transition region $\delta$, what is the smallest FIR Filter length that suffices? Can you please provide a derivation?

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    $\begingroup$ Is this a homework question? Have you tried looking for a solution? $\endgroup$ – ThP Dec 19 '14 at 9:38
  • $\begingroup$ I'm not sure why people have voted to close this question as being unclear. I think it's very clear and also very practical. There's even a Matlab function (firpmord.m) that addresses this question by computing an estimate of the required filter order given the filter specification. $\endgroup$ – Matt L. Dec 20 '14 at 17:59
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Many people have asked themselves this question. Unfortunately, there is no exact solution, just approximations. One relevant paper is Practical Design Rules for Optimum Finite Impulse Response Low-Pass Digital Filters by Herrmann, Rabiner and Chan, where they basically fit a formula to their data obtained from lots of designs. A simplified formula is [1]

$$M=\frac{-10\log_{10}(\delta_1\delta_2)-13}{2.324\;\Delta\omega}$$

where $M$ is the filter order, and $\Delta\omega=\omega_s-\omega_p$ is the width of the transition band.

Also the Matlab function firpmord.m uses some formula to estimate the necessary filter order, but they also say that the required order is often underestimated.

[1] Discrete-Time Signal Processing, Oppenheim, Schafer, Buck (2nd ed.), p. 502

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Try the FDAtool in MATLAB to experiment with various possible solutions.

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