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I am reading the Remez algorithm for designing FIR filters. In the first paragraph of section 15.3.1 of Andreas Antoniou's digital signal processing book, I have two questions.

  1. extremals are to be located at the left-hand and right-hand band edges of each band. (why?)
  2. B 1 + B 2 +···+ B J , should be divided intor + 1 − J intervals (Why is it not r+1, but r+1- J?)

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It's because we want our filters to be selective. That means we want the transition band to be as skinny as we can make it. So Passband signals are passed, Stopband signals are stopped, and we have to shrug our shoulders about the signals that are in neither the passband nor stopband.

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So we want $\omega_p$ to be as large as possible and $\omega_s$ to be as small as possible. Then, given a tolerance or limit to our gain error in in the passband, $\delta_p$, we define the passband to be as large as we can make it and satisfy that our gain is within $\pm\delta_p$ from our desired passband gain of $1$. That means when $\omega=\omega_p$ we are at an extremal. It's a minus extremal because the gain is being reduced from the passband gain to the stopband gain.

That means, when you push it to the limit and get the biggest possible passband and the biggest possible stopband (which gets you the smallest possible transition band), it will look like:

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The two bandedges will also be extremals.

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  • $\begingroup$ I can't thank you enough. I am a beginner in signal processing, and your explanation has dispelled many of my learning doubts. $\endgroup$
    – yixin_gui
    Commented Dec 22, 2023 at 5:55

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