# What is the filter with the less phase shift?

I have to analyze a dynamic signal but there is too much noise so I applied low pass filter but then there is too much phase shift.So what is the most reactive filter I can apply to my signal ?

Best regards

• What do you mean by most reactive filter? The fastest?
– VMMF
Apr 12 '18 at 3:59
• What do you mean by "dynamically", real time? You also tag with "continuous-signals" and "digital-filters". If you want analog, then the Gaussian filter has the smallest group/phase delay, Bessel comes second (but with slightly better attenuation). FIRs are not worth it in this case, even if they are minimum phase (too much hardware vs IIR), and IIR-wise, you can try Thiran (Bessel equivalent), it also has linear phase, if it counts. Otherwise, you can concoct your filter however you want it by manually placing poles/zeroes wherever they suit you best. Can't say more, your question is vague. Apr 12 '18 at 5:20
• By "analyze" do you mean analyzing the signal off-line in software like Octave or Matlab? If it is then you've got numerous options open to you, in particular the function FILTFILT. It is available in Octave and Matlab. Apr 18 '18 at 13:55
• @VMMF exactly, the most reactive to signal changes Apr 18 '18 at 13:57
• Well in that case applying zero-phase filtering will slow you down compared to other IIR filters, but it might be faster that FIR filtering if the order of the FIR filter is high
– VMMF
Apr 18 '18 at 14:06

Technically, the most reactive filter is the all-pass filter with a gain of 1, this filter has no phase shift at all. But it is not a really useful filter.

Here's what you need to take in account :

1 - IIR versur FIR

An IIR filter will have less phase shift than a FIR filter for the same cut-off frequency

2 - Cut-off frequency

For an IIR low-pass filter, a filter with 0.1 Fs cut-off frequency will have a higher phase-shift than a with a 0.2 Fs cut-off frequency. An all-pass filter i.e. a gain of 1 has no phase shift.

3 - IIR filter order

For an IIR, the higher the order, the higher the phase shift.

4 - FIR number of taps

Assuming you a linear-phase low-pass FIR filter, the more taps you have the higher the phase shift. If you want a high cut-off frequency you can get by with less taps. If you want a really low cut-off frequency you will need more taps thus more phase shift.

5 - Minimum phase-filter

There is a class of filter called minimum-phase filter. They are the most reactive (fastest decay) for a given cut-off frequency according to this reference

https://www.dsprelated.com/freebooks/filters/Minimum_Phase_Filters.html

Even though @Ben has already given a wonderful answer I would like to add that there's a technique called zero-phase filtering which allows you to have no phase delay at all. However, as I don't really know what you mean by reactive filter (fast?), I must say that in order to implement this in a causal way you will have to work with a buffer of stored samples over which you apply zero-phase filtering. If your application can tolerate this initial delay then you could perform real time processing (process a buffer while the next is being filled with new samples) and you would get no output delay. See Matlab's filtfilt

If you do offline processing then consider using sgolay filter in matlab or the filtfilt function which is supposed to give zero phase shift. I particularly like sgolayfilt with coefficient 3 and window size 11.

If you want zero phase filtering in real time then predictive filter such as the kalman filter is a good solution. Use a simple velocity filter or second order acceleration filter. If I have measurements of second derivative of the signal I like to use acceleration filter with jerk as the input. This gives me precise measurements of both first and second derivative of the measured signal as well as the signal itself with zero phase shift (or within +-0.2 deg is what I aim for)