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I stumbled apon this old question: Why is a linear phase important?

There the explanation of why linear-phase processing (filtering) is important is very clear. Also the effects on waveforms due to linear and non-linear phase processing are very well illustrated. My question is the following: why then would someone use non-linear phase filters at all? Can someone give examples where one does not care at all about phase? Are there any advantages in using non-linear phase filters and why is not everybody using only linear-phase signal processing?

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  • $\begingroup$ Welcome to SE.SP! Good question! $\endgroup$ – Peter K. Apr 28 at 14:53
  • $\begingroup$ Good discussion guys! $\endgroup$ – Dsp guy sam Apr 28 at 18:58
  • $\begingroup$ Thanks a lot to everyone who answered - now things are much clearer to me. $\endgroup$ – Todor Petkov Apr 30 at 8:17
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Two examples:

Audio: The human ear isn't very sensitive to nonlinear phase (probably because the world filters sound with nonlinear phase filters). It is, however, sensitive to percussive sounds that "pre-ring"*; i.e., that start making sound before the main "bang".

So for high-fi audio, one often wants to use minimum phase filters (which tend to have nonlinear phase response). In particular, the use of IIR filters is common (or at least commonly advocated).

Control Systems: In a control system, absolute phase shift (or delay) rather than nonlinear phase shift is the enemy. FIR filters that have make nice pretty-looking output pulses in response to input pulses make systems that cannot be tuned to the highest possible bandwidth without oscillating -- and systems with pure delay (as you get from a FIR filter) are particularly hard to tune.

So with exceptions (i.e., comb filters can be used to advantage, if you're careful) control engineers want any filtering done with IIR filters, and they tend to want that to be kept to a minimum.

* Or at least there are audio equipment designers that believe this -- I'm parroting things I've read.

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  • $\begingroup$ The only case where I have seen FIRs used in control systems is for repetitive control. $\endgroup$ – Ben Apr 28 at 15:29
  • $\begingroup$ To add to the control aspect, delay introduced by linear phase filters can reduce the phase margin of the closedloop system. So nonlinear phase filters can give better stability margins. $\endgroup$ – fibonatic Apr 28 at 17:57
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    $\begingroup$ @Ben You are referring to something like iterative learning control, where usually only one of the filters is linear/zero phase? But the majority of filters in control usually give better performance when using nonlinear phase filters. $\endgroup$ – fibonatic Apr 28 at 18:04
  • $\begingroup$ @fibonatic researchgate.net/publication/… $\endgroup$ – Ben Apr 28 at 23:10
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enter image description hereHere's a true story that happened in the company I worked before. We designed ultrasound analysis products for non-destructive measurements. One analysis technique is called "time-of-flight diffraction" or TOFD.

In this application, the user must analyze the ringing (or "echos" see the picture attached) to estimate the position of a potential defect in a material. There is usually a high-level of quasi-DC (low-frequency content much lower than the frequencies of interest) that makes it hard to analyze the echos. To remove the "quasi-DC", it is a good idea to use high-pass filters.

However, if you use linear FIR filters, the ringing caused by the filter response will "mix" with the actual ringing you want to measure creating more echos than there actually are. The phenomenon is called FIR pre-echo (at least in some circles)

In this case, using high-pass IIR filters yielded much better results. It removed the quasi-DC without creating any pre-echo, leaving only the real "echos" used for TOFD measurement and the IIR high-pass filters successfully removed the "quasi-DC".

Edit : In Tim Wescott's post above, he calls it pre-ring, same thing as pre-echo.

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Exact linear phase can only be implemented with finite impulse response (FIR) filters. These filters need more computations and memory compared to infinite impulse response (IIR) filters with a comparable magnitude response. Also, if high filter orders are necessary to meet the specifications, the resulting delay of a linear phase FIR filter becomes large (it is approximately equal to half the number taps times the sampling period).

And even FIR filters with non-linear phase are (slightly) more efficient than their linear-phase counterparts because they can have a lower order (fewer coefficients) than linear-phase filters for the same magnitude response.

As for applications, very often you wouldn't care much about phase when filtering (equalizing) audio signals, so this is often done by more efficient IIR filters. There are of course also applications where phase is important, but where a specific non-linear phase is desired. E.g., in phase equalization you want to approximate a specific non-linear phase response for compensating another system's (non-linear) phase, such that the total system (concatenation of some given system and the phase equalizer) has an approximately linear phase.

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If you use minimum phase filter to correct the magnitude of a minimum phase system, you get phase correction «for free».

IIR filters are lower computation and memory cost.

IIR filters lends themselves to parameters and time variant behaviour that suits some applications.

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