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I have read that for FIR filters, the duration of the transient response is equal to the order of the filter. But, for linear phase FIR filters constructed with the windowing method, the transient is N/2. I know that this has to do with the filter impulse response being symmetric with respect to N/2 (or (N-1)/ 2 if N is even), but I don't fully understand why. Could someone provide an explanation to this?

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  • $\begingroup$ Isn't the transient response duration of an FIR, in units of the sample period, simply the number of FIR taps minus one? If an FIR has only one tap, it's just a gain with zero transient response duration. $\endgroup$ Dec 8, 2021 at 21:17
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    $\begingroup$ How can the transient response be anything but the total length of the filter, assuming that it is not zero-padded? Are you sure that you're not looking at non-causal responses, and neglecting that if you model a FIR filter as symmetric and non-causal then there'll be an N/2-point "pre-transient" before the signal does something, then another N/2-point "post-transient" afterward? $\endgroup$
    – TimWescott
    Dec 8, 2021 at 21:40
  • $\begingroup$ No the filter response is causal and has symmetry with respect to N/2. It's a low pass filter designed by windowing, but this applies to every filter. As Richard Lyons pointed out, I may be having trouble understanding group delay: doing the convolution graphically I do not understand why the response of the filter is 0 until N/2 - 1. $\endgroup$
    – Tony2015
    Dec 9, 2021 at 13:53
  • $\begingroup$ I think it would be helpful if you added a plot of a case that you don't understand. I must admit that I'm not sure what you're asking. $\endgroup$
    – Matt L.
    Dec 9, 2021 at 15:21
  • $\begingroup$ "Causal and symmetric with respect to N/2" means it has N non-zero taps. How can the transient not be N points long, then? $\endgroup$
    – TimWescott
    Dec 9, 2021 at 15:52

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You may be confusing "transient response" with "group delay".

I'm not sure what is your definition of transient response but it's important to know that for a linear-phase, N-tap, tapped-delay line (nonrecursive) FIR filter the first valid output sample is the Nth output sample. The delay line has to completely "fill up" with N signal samples before the output samples are valid.

The group delay of such a filter is N/2-1.

My above two paragraphs do NOT apply to recursive FIR digital filters.

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  • $\begingroup$ Yes I may have a confusion with that, but that was precisely my problem, because looking at the convolution between the impulse response and a sinusoid, it's clear that the transient response goes from 0 to M-1. But in the simulations of linear phase FIR filters, the response is zero from 0 to N/2 - 1, and that's the thing that I'm not understanding. Why does this happen? $\endgroup$
    – Tony2015
    Dec 9, 2021 at 13:43
  • $\begingroup$ @Tony2015 I'm suspicious your simulation code may not be correct. If your simulation code is written in MATLAB and you send the code to me (at e-mail [email protected]) I am willing to have a look at it. $\endgroup$ Dec 9, 2021 at 15:25

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