Hand-in-Hand Phase Response and Group Delay
The phase characteristic of a loudspeaker is often neglected. This article will help us learn more about the benefits of understanding phase responses.
A complete frequency response has both level and time information. These are referred to as the magnitude and phase of the frequency response. Together they describe the transfer function of a system component (Figure 1). The phase characteristic of the response is often neglected or ignored, but it can provide a wealth of information about the component. Two loudspeakers can have identical magnitude responses but different phase responses.
Viewing the Phase Response 
Signals are always delayed when they pass through a system component. This delay can be independent of frequency (a pure delay) or it can be frequency-dependent. The latency through a digital processor is an example of a pure delay, as is the time-of-flight from a loudspeaker to a microphone position. There may also be a frequency-dependent delay in excess of the pure delay. Signal delays can be viewed on a plot of amplitude vs. time in the time domain (the impulse response) or phase vs. frequency or group delay vs. frequency in the frequency domain. These two plots display the same information in a different way. In the frequency domain, the group delay vs. frequency plot is preferred when the delay is sufficient to produce severe wrapping of a phase vs. frequency plot.
When analyzing the response of a component, it is useful to remove the pure delay between the stimulus and response, leaving only the excess delay due to filters, etc. The amount of pure delay to remove can be determined from observing the impulse response in the time domain or the group delay response in the frequency domain. Either can be used to get close enough to “fine tune” looking at a phase vs. frequency plot, which is the most resolute way of observing the residual delay. The stimulus can be delayed (usually within the analysis program itself) to yield the least slope in the phase response.
Why We Care
A component with flat magnitude and phase response can preserve the shape of waveforms that pass through it. The square wave or an impulse is often used to evaluate the fidelity of a component since both have broad spectral content. Real-world components have finite bandwidth (a band pass filter) and therefore cannot perfectly reproduce either.
Loudspeakers and other band pass filters exhibit a frequency-dependent delay (or phase shift) due to their high pass and low pass characteristics. This is can be viewed on a plot of group delay vs. frequency, a plot that is useful when used in conjunction with the phase vs. frequency plot. While a fixed delay shows up as a constant slope on a phase vs. frequency plot (linear frequency scale), it shows up as a fixed vertical displacement on a plot of group delay vs. frequency, leaving the relative frequency-dependent arrival times on the plot.
When digitally sampling signals the delay can also be specified in terms of the number of samples. A sample rate of 48kHz has a time-per-sample of 1/48000 or about 20uS.
An Example
To illustrate these concepts I measured a single 3.5 in. transducer with no crossover network. I produced a phase and group delay plot using three different time references relative to the arrival of the HF components of the impulse. The first reference is before the arrival by 40 samples. The second is after the arrival by 40 samples. The third is at the impulse arrival. Note the gross slope changes in the phase vs. frequency plots. Also note that the group delay plots are identical, but are only displaced on the vertical 
(time) axis (the afore mentioned fixed vertical offset produced by a pure delay).
The phase vs. frequency plot displays the phase relationship between two signals that result from it being observed at two different points in a signal chain. These can be the input and output of any component (i.e. mixer, equalizer, amplifier) or the total the phase shift produced by the entire signal chain.
Working Together
The group delay and phase response plots can be used together to characterize the total delay of the signal through a system. The group delay plot can identify the amount of pure delay that must be removed to leave the excess delay, which is usually viewed as a phase response.
Of course the impulse response in the time domain can also be used to characterize the pure delay. This requires an inverse FFT to change domains, where as the group delay plot does not.
Group delay is relative to the delay setting of the analyzer, so negative values (impossible in absolute time) are possible. It can be thought of as a simplified and lower resolution representation of the phase response. In the same way, a positive sloped phase curve implies a time reference that trails the signal. Such a relationship is said to be acausal and can never exist in absolute time.
Optimizing Loudspeakers
Both phase and group delay plots are of particular interest to loudspeaker designers in quest of the best attainable transient (sudden change) response in their loudspeaker designs. They can provide quantitative evaluation of the loudspeaker’s response as determined by component selection and placement. It is expected that better transient response equates with higher fidelity, which is a desirable attribute of a sound reproduction system.
In practice, the group delay response can be used to evaluate signal alignment in a multi-way loudspeaker. The phase response can be used to determine if a peak or dip in the amplitude response can be conjugated with a minimum phase equalizer filter, where the electrical filter has the opposite magnitude and phase response of the aberration. Both are potentially very high resolution views of a system response, and will reveal the fine structure of the response of any system component. Neither response is relevant to the measurement and evaluation of auditoriums, where the less resolute magnitude response (frequency domain) and impulse response (time domain) are preferred.
Minimum Phase
For a given frequency response magnitude, there exists an infinite number of possible phase responses. This is because the magnitude response is “time blind” and yields no information regarding the time/phase relationship of the various parts of the spectrum.
Bandpass filters always produce some frequency dependent phase shift. The minimum phase response is the 
one that exhibits the least possible phase shift given the shape of the magnitude response. The minimum phase response can be determined from the magnitude response. It will have a flat phase response where the magnitude response is flat. The phase angle will be leading for the high pass portion of the spectrum and lagging for the low pass portion. The magnitude response can be Hilbert transformed to reveal the minimum phase response. It is interesting to compare the measured phase response with the response predicted with the Hilbert transform. If they are the same, then the device is said to be minimum phase.
Non-minimum Phase
It is not unusual for multi-way loudspeakers to have a non-minimum phase response as a system. This means that the magnitude response is flat, but the phase response wraps around the display. This is usually caused by the crossover network, which exhibits a frequency-dependent phase shift to the signals passing through it. The significance of this effect has long been debated, but it is generally agreed that if a minimum phase response can be achieved, then it would represent an improvement in the fidelity of the loudspeaker.
Solutions
There exist several ways to make a multi-way loudspeaker have a minimum phase response. The first is to use first order Butterworth filters to form the crossover. These combine to produce a minimum phase response, but the 6dB/octave slope is generally insufficient to protect the drivers from out-ofband energy.
The second solution is to use a digital crossover with FIR filter topology. FIR filters can modify the magnitude response without affecting the phase response. The penalty is increased latency through the network.
A third solution is to use all pass filters to correct the non-minimum phase response. John Murray of Pro Sonic Solutions has provided an excellent tutorial of this technique in an article written for Live Sound International. The advantage of this technique is that it allows a minimum phase response to be achieved with multi-way loudspeakers using analog and IIR filter topologies, at the expense of a small increase in HF latency.
Conclusion
I have included a measurement road map (the classic domain chart) on the following page to describe where these various responses fit into the big picture. pb