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Prewarping IIR Filters

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  • 24-01-2004 7:55pm
    #1
    DMT
    Closed Accounts Posts: 1,669 ✭✭✭


    I'm trying to prewarp the critical frequencies for an IIR filter but I found two conflicting formulae for prewarping. They are:

    ωd = 2*tan^1(ωa*T/2)

    and

    ωd = (2/T)*tan(ωa*T/2)

    Which one is correct?


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    #2
    Capt'n Midnight
    Moderators, Recreation & Hobbies Moderators, Science, Health & Environment Moderators, Technology & Internet Moderators Posts: 91,754 Mod ✭✭✭✭Capt'n Midnight


    Sounds like this should be on the Trek forum..

    Are you sure about the ^ exponential - is there a chance the tow formulae are different versions of the same thing ?


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    #3
    Sev
    Registered Users Posts: 1,328 ✭✭✭Sev


    I've no idea an IIR filter is, or what wd or wa are.. but just looking at the equations. I get the impression d and a are somehow interchangeable. If you rearrange the first equation to get wa on its own it kinda looks like your second equation, without the other T. Is that a coincidence? Perhaps as captain midnight says... maybe forms of the same thing?


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    #4
    cerebus
    Registered Users Posts: 447 ✭✭cerebus


    My signal processing theory is a bit rusty, but I'll take a shot at this.

    I believe the two formulas you provide describe the relationship between analog and digital frequency when using the bilinear transformation between the s-plane and the z-plane. It looks like there might be a couple of typos in your second equation (it should be the inverse of the first)

    W = 2/T . tan(w/2)

    and

    w = 2. tan^-1((W.T)/2)

    where W = analog frequency and w = digital frequency. (You used wa and wd instead of this notation I think?)

    What this means is that if you want an IIR filter with a critical digital frequency of wi you need to design an analog filter with a critical analog frequency

    Wi = 2/T . tan(wi/2)

    That's the prewarping operation.

    I guess you can prewarp in either direction, i.e. if you want to build a digital filter with a defined set of analog critical frequencies, you use the reverse operation to work out the digital critical frequencies you need to design to. I don't know if this gets done very often though... I've usually seen prewarping applied in the design of analog filters to satisfy digital specs.

    I hope this helps - and I could well have mucked this up, so if anyone spots a mistake please feel free to correct it.


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