Iowa Hills Software Digital and Analog Filters
What is Group Delay Home
This page explains group delay, and its effects, without any math.
The basic idea of group delay is reasonably simple; the negative derivative of phase with respect to frequency. But for most people, this doesn't mean much, which is unfortunate, because group delay is an important way to describe a filter's pass band characteristics. Our purpose here is to show why it is important.
Why the Name "Group Delay"
We all understand ordinary time delay (phase shift), but what is group delay. Consider a simple example of a square wave, which as you know, is composed of a large group of frequency components. A square wave is square only because its frequency components are in proper phase alignment with one another. If we pass a square wave through a device and expect it to remain square, then we need to ensure that the device doesn't misalign these frequency components. A Group Delay measurement shows us how much a device causes these frequency components to become misaligned.
This may sound like a trivial matter, but as we show below, misaligning the phase between a group of frequency components can destroy a signal. In short, we want all frequency compontents to experience the same amount of delay (in seconds, not phase angle) as they pass through a device.
Why Does a Filter Generate a Phase Shift
Let's start by answering this frequently asked question. To start with, a better question would be, is it possible to construct a device that doesn't generate a phase shift, and the answer is no.
Lets consider the simplest device possible, a piece of wire of length L where the signal can travel at the speed of light. Clearly, it takes L/c seconds for the signal to get through this device. The simple fact that time is required for a signal to pass from a device's input to its output means that it generates a phase shift. It's just that simple.
As compared to other electrical devices however, filters have significant amounts of phase shift, and the longer the filter (i.e. the more poles it has, or the more taps it has) the more delay it has. In the case of digital filters, this delay is usually discussed in terms of samples, not seconds.
Group Delay Describes Phase Shift
Let's start by making it clear that group delay is a measurement of time, so let's compare group delay to ordinary time delay (phase shift).
Here we compare the time delay to the group delay of a low pass filter. It turns out that the time delay and the group delay of a filter take on similar values in the pass band where the filter's phase response is linear. For an FIR filter with constant group delay, the time delay and the group delay are equal at all frequencies. For a filter with non-linear phase however, such as the Butterworth filter shown here, the delay is close to the group delay in the pass band.
This plot makes it obvious that we cannot in general equate time delay and group delay, but this comparison helps to simplify the topic a bit. i.e. Group Delay is a measurement of time, and is similar in value to ordinary time delay in the filter's pass band where the phase is most linear. Here is another example using the more phase linear Gauss filter.
In loose terms, Group Delay is the amount of time required for a signal to propagate through a device.
More importantly however, the group delay curve indicates how much a device will distort a signal. A device with constant group delay (linear phase) equates to minimal distortion, but as we will show with examples below, non linear phase is a contributor to distortion, not the sole reason for it.
Group Delay vs. Signal Propagation Time
We said that in loose terms, the group delay is a measurement of the time needed for a signal to propagate through a device. To demonstrate this, we send a square pulse through a low pass filter. This particular filter has a group delay of 40, and as can be seen, the mid point of the pulse's leading edge occurs at the output at t=40. It's just that simple.
Note: For those not familiar with digital filters, t=40 means 40 sample times. If the filter's clock is running at 1 MHz, t=40 means 40µs. This signal propagation behavior is the same for analog filters.
We used a low pass filter and a square pulse in this example because the effect of group delay is so readily seen, but keep in mind that a sine wave would have the same 40 sample delay. Since this filter has the same group delay at all frequencies, a sine wave at any frequency experiences the same delay time.
Why is the Step Response Important?
Please remember that all signals passing through a device will experience the same distortion, but we use a step response to measure distortion for two reasons. First, a step input contains all frequencies, so using it is a bit like doing a frequency sweep, but in the time domain. Second, since you know the exact shape of the input signal, any variations to that shape make the distortion readily apparent. Distortion would be much harder to detect if we used an audio signal, for example.
Why is Group Delay Important?
In most cases, we are not particularly concerned with a filter's delay (i.e. the amount of time it takes for a signal to propagate through the filter). We are concerned however that each of the signal's frequency components experience the same delay so that their phase relative to one another is maintained. Or said a bit differently, we want our signals to maintain their shape as they pass through a device.
In principle, the filter shown above has perfect group delay, but the pulse was still distorted by the filter. The filter slowed the pulse's rise time and gave it overshoot. So there is more to be concerned with than just group delay. As you know, the square wave entering the filter had an infinite number of frequency components, and the filter did its job by attenuating the higher frequencies. So, in this respect, there is no way for a low pass filter to not affect a square wave, so we must expect some distortion.
This plot compares the distortion caused by two filters. Both of these FIR filters have the same ideal group delay, but they attenuate the high frequencies differently. This difference has a clear effect on the distortion of the pulse.
Here we show the converse of the example above where two filters have the same magnitude response, but have different group delay responses. The effect of nonlinear phase is clear. The step response for the filter with non linear phase has significantly more overshoot and ringing than the filter with constant group delay. Also note the time delay differences, which of course, coincides with the different group delay values.
An Extreme Example of Group Delay Distortion
You are probably familiar with the term "all pass filter", and you have probably wondered why a device that passes all frequencies is called a filter.
Let us explain that the pass band of an all pass filter is defined by its group delay response, not it's magnitude response. The key to using an all pass filter is to make sure that the frequency content of the signal is within the filter's pass band. In this example, the square wave is virtually destroyed by the filter because it has significant frequency content well beyond the filter's pass band (where it group delay is flat).
BTW: All pass filters are used to delay a signal in order to align it with another signal. This IIR example would delay the signal by 10 samples. If we had used an FIR all pass, which would have constant group delay at all frequencies, there would be no distortion.
Group Delay as a Measure of Phase
Now let's show group delay as it is defined, the negative derivative of the phase.
It should be clear that as the tap count is increased for an FIR filter, the group delay increases as well, simply because the filter is getting longer. Similarly, as the pole count is increased for IIR and analog filters, the group delay increases, again, because the filter is getting longer.
Or said a bit differently, as the filter gets longer, the slope of the phase curve gets steeper. Here we compare the slope of the phase for two FIR filters with different group delays. The first has 10 taps and the other 14 taps. The 14 tap filter is longer so its phase has more negative slope (more group delay) .
Negative Group Delay
We want to make it clear that, in general, the group delay for a physical device (as opposed to a pure mathematical construct) cannot be negative. If it were, the signal would appear at the device's output before it entered the device.
It is possible however to construct an active circuit (not a passive circuit) with negative group delay. One simply needs to a place a pole in the right hand plane, which of course makes the circuit unstable. The circuit will want to oscillate at the pole frequency, which is fine if you are designing an oscillator, but not very useful otherwise.
While not negative, the ideal Hilbert Transform has no group delay, because its phase is a constant ± 90 degrees (making the derivative zero), but it cannot be implemented in hardware. Only in mathematics can we construct a filter with constant phase. The phase of a filter implemented in hardware must vary with frequency. Therefore, if we want to implement a Hilbert Transform in hardware, we must give it some delay.
Our purpose here was to try to give some meaning to the term Group Delay. The more signal processing experience you get, the more attention you will pay to group delay curves. It isn't a complicated subject, but it does take a bit of experience to understand its significance.
To understand this topic better, try downloading our free FIR Filter Design software and experiment with low pass filter design. In particular, use the minimum group delay switch to toggle between linear and non-linear group delay and watch the effect on the step response.