If you look closer at the earlier graph showing phase against frequency on a linear scale, you will see it is not quite a straight line. So, if the Butterworth filter has a constant group delay at low frequencies, what is the cause? It is the imperfect phase response as you approach the passband. The reduced amplitude of the higher harmonics will also distort the signal. So, a filter with a cut-off frequency considerably higher than the input square wave will still have an effect on higher harmonics and so distorts the resulting signal, causing the overshoot. So, even by the 99th harmonic you still have sine component which is 1% of the fundamental. The fifth harmonic amplitude is one fifth of the fundamental. The third harmonic component is one-third of the fundamental. So, for a 100Hz square wave the fundamental is a 100Hz sine wave. The amplitude of the harmonics only reduce quite slowly as frequency increases. However, if you perform a Fourier transform on your waveform you will find that the sine harmonics that exist to make up a square wave have no limit. If all frequencies were delayed by the same time (and the amplitude was unchanged) then the square wave wouldn’t have any overshoot. So, why do we get this overshoot and what is the solution? The cause is the delay through the filter and the fact that all frequencies are not delayed by the same time. This overshoot also means the signal takes longer to stabilize so if you were hoping to sample the signal and get an accurate value, you need to wait longer until it has stabilized. For example, a 4 pole 2dB ripple Chebyshev will have 28% overshoot. Higher order filters have higher overshoot and high ripple Chebyshev filters have even higher overshoot. If you look up various filter types in your electronics books then you will find that is exactly what is to be expected from a 4 pole Butterworth filter. This is over 11% overshoot which is quite significant.

If you put a 100Hz square wave through your 1kHz 4-pole Butterworth filter, you will get some overshoot: The effect of these phase shifts can be significant with fast signals. If you are sampling at a high rate then the delay won’t necessarily matter but if you are taking a limited number of samples and expecting to be able to predict when the peak signal will occur then you will need to take the delay into account. You would know when the light source was turned on and off and may have an almost instant optical system response time, but the time you need to sample the returned signal would be subject to your delays. One example might be measuring the response of a system with a modulated light source. If your sample time was based on some signal you generated and you expecting to be sample a signal response then again you need to take into account the delay. However, if you were sampling a signal that was changing and were relying on knowing the time of your sample then you need to know that the signal you are sampling has been delayed, in this case by 416us. A signal well below the cut-off frequency will simply be delayed by 416us but otherwise be fine, as illustrated below for a 500Hz sine wave. So, why should you care about this delay or phase shift? Well, if you are monitoring a constant sine wave then it probably won’t matter much. That is to be expected – if you double the frequency the phase shift doubles. If you double the frequency you would double the phase shift from a fixed time delay. Now you can see that in the area of constant group delay the phase actually changes linearly with frequency as would be expected. Changing the x-axis to a linear scale makes it clearer.

A group delay can be considered as a varying phase shift so plotting that results in:Īs this is on a log-lin scale it is not obvious what is happening. So, even well below the cut-off frequency there is an almost constant group delay of around 416us.

With a 1kHz bandwidth you will see a group delay response below. Take a simple 4 pole Butterworth filter as an example. The time you sample the signal will not necessarily be the time of the actual event before signal processing. One way in which it might affect you is if you are sampling a signal. It may not matter that it is there, but you need to be aware just in case. You may or may not have thought about group delays in your electronics signal processing chain, but it is almost certainly there.

Phase shifter vs simple delay