it is possible to give input that will cause an IIR filter to go into self-sustaining oscillations.
How could a stable IIR filter go into oscillation? And if it's unstable, I'd think most inputs would drive it into oscillation (but I haven't done much with IIR filters myself).
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That's why FIR filters are much more common in control systems. Unconditional stability is a big plus. FIR filters also have linear phase response. However, the transport delays introduced by an FIR filter can be problematic, too.
FIR filters can also have non-linear phase response. A minimum-phase FIR filter has most of the energy at the beginning, just like an IIR filter. I'm assuming this is what you mean by "transport delay".
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Why use an IIR filter? An IIR filter with a given number of taps can be made to have much steeper roll-off than an FIR filter with the same number of taps.
This is about the only reason, efficiency. Otherwise, a (much higher order) FIR can virtually match response, since one can always take the impulse response of the IIR, window it, and make an FIR out of that.
