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I’ve been reading lately about analog filters, and in “Electronic Filter Design Handbook”, Williams & Taylor state that the fact that the Bessel filter exhibits no overshoot in its step response, and no ringing, directly relates to the constant group delay in passband.

This statement got me thinking. Obviously, a variation in group delay would cause ringing, since both an impulse and a step contain many frequencies and if each has a different delay, the transient would exhibit ringing.

However having a linear phase does not eliminate overshoots necessarily. For instance, a least squares low pass (linear phase) FIR exhibits overshoot in step response. I presume this is a result of the low pass nature the distorts the sharp edges of the step.

Thus two questions come to mind:

  1. What makes the Bessel filter avoid the ringing and overshoot? I’m quite sure the is damping (in control theory jargon) or Q in filter design jargon. Am I correct?
  2. Since a digital FIR has no feedback, I’m not sure how would Q even be defined. Can we realize an FIR with no overshoot? If so, what is the trade off? Is it only rise time as is the case for all pole filters?
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I think it's intuitive to look at it in the time domain. After all, ringing and over- or undershoot are time-domain phenomena. Of course, damping is an important factor, but looking at the impulse response can give you an immediate idea whether there will be ringing and/or overshoot in the step response.

Note that the step response $s(t)$ is the integral of the impulse response $h(t)$:

$$s(t)=\int_{-\infty}^th(\tau)d\tau\tag{1}$$

So if there are oscillations in the impulse response, there will be ringing in the step response. Of course, low damping (high Q) will result in larger oscillations which are decaying more slowly, hence more ringing.

If there are no oscillations at all, i.e., if the impulse response is non-negative, there will be no ringing. A Gaussian impulse response has this property. Impulse responses of Bessel filters are similar to Gaussians, and they approach a Gaussian with increasing order. Consequently, the step response of Bessel filters exhibits very little ringing.

The flat group delay of Bessel filters is only indirectly related to the relative absence of ringing. What is more important is the relatively soft decay of the magnitude of the frequency response (which in turn is a consequence of the side constraints of a flat group delay). However, we can also have zero ringing with a completely non-linear phase, as long as the impulse response is non-negative. And, vice versa, we can find linear phase filters with a lot of ringing. The ringing is caused by the filter's frequency selectivity, and the sharper the transitions from passband to stopband, the more ringing will occur.

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  • $\begingroup$ Thanks Matt this is great. However, my original interest was since I wondered how can I control the ringing during the design of filters. This is why I looked at frequency domain. Can you comment on filter design techniques which set targets / specs on ringing? $\endgroup$ Commented Sep 1, 2024 at 18:43
  • $\begingroup$ @YairM: If you want to design FIR filters you could use linear or quadratic programming techniques and specify upper and lower bounds on the step response. Such constraints are linear in the filter coefficients. $\endgroup$ Commented Sep 1, 2024 at 19:14
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    $\begingroup$ Thanks Matt. Much appreciated. BTW, I really enjoyed reading your thesis and used your implementation (lslevin) to design LPS with arbitrary phase response. $\endgroup$ Commented Sep 1, 2024 at 20:24

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