EQ, Phase & Time

Equalizers (EQs) are in electronic terms filters, and I will be using the terms interchangeably in this document.

Most EQs and filters used in audio are what is known as "minimum phase". That is to say they have a phase response that is completely determined by their amplitude (often called frequency) response. If you know the amplitude response of any minimum phase device then the phase response can be calculated from the amplitude response.

Conversely, if you know the phase response of any minimum phase device then you can calculate the amplitude response. In a way, the phase and amplitude responses can almost be thought of as two different ways of looking at the same thing, as long as the device or system in question is minimum phase.

Minimum phase amplitude (top) and phase (bottom) responses.

Phase is related to time, but a pure time delay does not involve any phase shift. A pure time delay or "group delay" is constant with frequency. Phase shift varies with frequency and can "advance" or "retard" as the frequency changes. If you see a display where phase appears to be constantly advancing or retarding then almost certainly the group delay has not been correctly subtracted, and the apparent constant change of phase in a single direction is really due to a display error that did not properly account for the group delay.

The vast majority of analog filters and EQs are minimum phase, with the only notable exception being "all-pass" filters.

All Pass filter response. Note the amplitude response is flat but the phase response is not. This is NOT a minimum phase device.

Most digital filters and EQs used for audio are of what is known as "Infinite Impulse Response" or IIR type design. As the name implies, if an impulse is applied to the input of such a filter, then at least in theory, at the output the results of that impulse could stretch out to infinite time. IIR filters are almost always designed to emulate common analog filter circuits and as a result are also minimum phase.

Far less common in audio are digital filters of "Finite Impulse Response" or FIR types of designs. As the name implies, such filters when a impulse is sent to their input will produce a resulting output that only lasts for a known and finite amount of time. FIR type filters can be either minimum phase or non-minimum phase, and in fact it is possible to get any amplitude response with any phase response. It is possible to make a "linear phase" FIR filter with a flat phase response but with any amplitude response desired. Linear phase FIR filters are rarely used in live audio (although you will find them in some other applications) because they add a delay to the audio passing through them. The lower in frequency such a filter is designed to act, the longer the required delay. One company made a linear phase EQ with almost 1/2 second of delay through it which made it unusable for live sound applications.

Better loudspeakers and microphones will tend to have a mix of minimum phase and non-minimum phase characteristics. In general the better sounding the loudspeaker or microphone the closer to completely minimum phase they are. Even cheap loudspeakers or microphones will often be mostly minimum phase.

When it comes to acoustic effects, these are much less likely to be significantly minimum phase since they often are due to the combination of multiple signals with delays due to the time it takes for sound to travel through air.

Another thing to know is that if two or more minimum phase devices are cascaded one after the other, then the total system will also be minimum phase.

Minimum phase amplitude and phase response that is opposite to that of the first response above.

Both of the above responses cascaded.

Where this starts to get interesting is when an EQ is placed in front of a loudspeaker. As long as the loudspeaker is at least mainly minimum phase, then as we smooth out and flatten the system amplitude response with the equalizer, then we are also smoothing out the phase response of the system and converting any time anomalies into closer to a pure group delay.

Many folk get concerned with the "phase errors" or "phase distortion" or "time anomalies" or "time incoherence" created by filters and EQs without realizing that such effects are actually a good thing since they will cancel out the corresponding phase response of the loudspeaker and as the EQ is adjusted so that a smooth and relatively flat amplitude response is obtained, the EQ is also providing the same benefit to the phase response of the system.

Now this is not to say that if you use an EQ to smooth out the response you measure with an RTA out in the room this will result in a smooth phase response as well. There are some things that just can't be fixed with an EQ. In particular, acoustic cancellations of portions of the amplitude response can't be equalized because the response is due to two or more signals arriving with different acoustic delays.

Part of the problem is the time blind nature of a RTA. If you were to use a test instrument such as a TEF which allows you to just look at the direct sound from the loudspeaker and ignore the reflections of that sound coming back from the room, then as you adjusted the EQ to get a smooth and relatively flat amplitude response from your loudspeaker, then you would at the same time be fixing the phase response of the system.

Another advantage to using a test instrument that allows you to isolate the direct sound from the loudspeaker when adjusting the system is that to a large degree that is how the ear hears. Particularly at higher frequencies, the ear tends to ignore the reflections back from the room when determining the tonal balance of the system. At low frequencies the ear considers a wider range of time arrivals when deciding the tonal balance. This difference is the idea behind the Smaart analysis system which has a much wider time window at low frequencies than at high frequencies.

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Last edited on 12/31/2003
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