Understanding Relationships: Bringing Clarity To Phase, Frequency And Time
The path to making optimal choices with sound reinforcement systems...

May 12, 2014, by Ken DeLoria

One of the most difficult aspects to comprehend in the field of sound is the relationship of phase response, frequency response, and how each relates to time. Understanding these relationships can help a lot in making optimal choices when you’re deploying and utilizing sound systems.

What the deuce is phase? (Sorry, I was channeling my inner Stewie Griffin.) The term phase response is actually short for “phase versus frequency,” in the way that we typically use the term in our industry. Likewise, when we speak of frequency response, we usually mean “amplitude versus frequency.”

In both cases, whenever the measured data is graphed, the “X” axis of the response curve – which represents frequency – will run horizontally across the graph.

The vertical “Y” axis represents intensity in an amplitude-versus-frequency response measurement (usually scaled in dB), and it likewise represents phase in a phase-versus-frequency response measurement (typically scaled in degrees).

Do time and phase representations differ? It’s a question I get asked quite a lot. The answer seems a bit elusive, but makes perfect sense once we dissect the characteristics. A phase “lead” or phase “lag” is a way of expressing a time differential in relation to a reference point – which usually is an ideal flat-line phase response.

For example, an equalizer that’s set to “flat” will present a flat phase and frequency response curve. But when a boost or cut is introduced, the frequency response will reflect the newly introduced response curve, as will the phase response (Figure 1).

Figure 1: A representation of a nearly perfect loudspeaker. (Phase is offset from 0 degrees for clarity.)

The Ø (zero) crossing point in which the phase response is absolutely flat will always be at the center of the peak or dip of the graphic or parametric EQ (PEQ) filter. But before and after the center frequency, there will be a phase lead (or lag) depending on whether the filter has been set to boost or cut.

Note that this assumes the use of traditional IIR (Infinite Impulse Response) filters father than newer FIR (Finite Impulse Response) filter types. (See sidebar, IIR vs. FIR.)

Signal – Not Time – Delay
Unlike a digital delay that delays all frequencies in the audible spectrum equally, and is rightly referred to as a signal delay device, a phase lead or lag is a more subtle concept. This can be difficult to grasp, which is why we’re going to explain it carefully.

However, there is no such thing as “time delay.” Only a signal can be delayed in relationship to the universal clock that’s always ticking, mind you.

But before we move forward, ponder this: When we speak of phase, as it relates to the usage of pro audio systems, we’re almost always interested in phase “offset.” We don’t really care about the phase lag from a loudspeaker to a listener (correctly referred to as propagation delay), but we intently care about any offset than might occur when one sound source is referenced to one, or more, other sources.

If multiple sources are not precisely in phase with each other, some of the sonic energy will cancel…and some will add…and this occurs purely as a function of the wavelengths of any given frequency in respect to the physical offset of the radiating devices.

The resultant effect of loudspeakers that are offset from one another is the often mentioned “comb filter” response pattern.

The term comes from the hundreds of additions and subtractions that corrupt the frequency and phase response of the system, which end up resembling the teeth of a comb.

Here’s an example of how phase-offset manifests in the real world: a 600 Hz wavelength is 1.5 feet long (18 inches), whereas a 6 kHz wavelength is just 0.15 feet in length (1.8 inches).

Therefore, if one loudspeaker is offset in relation to another loudspeaker by 0.15 feet, a signal at 6 kHz will cause the two sources to totally cancel each other - at least in theory.

In reality, however, total cancellation is actually unlikely, due to the imprecision of most loudspeakers. Nonetheless, a large percentage of cancellation – possibly as much as 90 percent – will inevitably occur.

But move upwards, let’s say to 6.5 kHz, or downward to 5.8 kHz, and the whole picture changes. That’s because acoustical addition and acoustical cancellation will always be a function of the wavelength of the source material in relation to the physical and/or the electrical offset that’s affecting the relevant signals.

Conversely, the same offset of 0.15 inches represents only 1/10th (0.1) of a phase differential of the 600 Hz wavelength. By no means is this desirable, but it’s not going to cause much more than about a 10 percent cancellation of forward radiated energy. (It will also change the polar response of the system, but that’s another topic for another time.)

This is precisely why we speak of – and why we measure – the phase relationship of multiple sound sources, instead of thinking only about the pure time differential. A phase-versus-frequency response measurement will characterize the arrival time of a sonic wavefront at a given point in space, in relation to the wavelength of each relevant frequency.

That may sound difficult so here’s a simplified analogy:  A firing squad of five shoot at the same target at exactly the same time with exactly the same rifles, but each one is standing a few feet behind the other. Therefore, the projectiles do not reach the target at precisely the same time.

While it might not mean much, insofar as the ultimate intent of the firing squad, it means absolutely everything when a “sound squad” wants all of the sonic energy from each loudspeaker in the sound system to arrive at the listener’s position in perfect phase with all the other loudspeakers.

Unlike rifles, if the sonic energy doesn’t arrive at exactly the same time from all sources, cancellations and additions will occur, causing an imperfect and comprised frequency and phase response.

Phase & Polarity
We often hear phase spoken about in a very basic manner, such as “one loudspeaker is either ‘in-phase’ or ‘out-of-phase’ with another.” This is more correctly referred to as the polarity relationship of the loudspeakers. When polarity is reversed, then all of the energy emanating from a loudspeaker – without respect to frequency – is also reversed.

In such a case, two theoretically perfect loudspeakers would perfectly cancel each other’s entire output across the full audible spectrum. However, this won’t actually occur due to the physical imperfection of virtually all known loudspeakers.

What does in fact occur is that the low frequencies will almost totally cancel each other because the physical imperfections of the loudspeakers are relatively minor in relation to the long (and therefore forgiving) LF wavelengths, while the shorter wavelengths of the high frequencies will partially cancel and partially combine, causing havoc with the MF and HF frequency and phase response.

It’s worth noting that a phase offset of multiple sound sources is akin to how a flanger works. The effect of “flanging” was originally created by slowing down a reel of tape on a tape recorder that started out in sync with another tape recorder playing an identical track.

Flanging was accomplished by applying your finger on the edge of the reel to alter the speed of the tape machine (hence the term). The small offset in time created the well-known sonic effect that’s now easily replicated by the use of modern analog and digital electronics.

But it’s also important to note that a flanger is an intended sonic effect.

Conversely, loudspeaker offset – which also creates a form of flanging (albeit not wandering up and down like the tape recorder variety), is rarely intended to be of positive value.

Let’s look again at one of the numerous definitions of phase. Phase-versus-frequency is the relationship of an alternating signal (all sonic energy is comprised of alternating signals) at a given frequency, in relation to the time that it takes the sonic energy to propagate to a given point in space.

When all frequencies arrive at the same time at the same point in space, then the phase response is said to be linear or flat. If some segment of the frequency spectrum arrives earlier or later than another, than the phase response is not flat and phase offset has occurred.

If we were to move one of a pair of matched loudspeakers a few inches rearward while maintaining a fixed listening position, the effect on the phase response of the energy arriving at the listener’s position would be much greater in the higher frequencies than in the lower frequencies, simply because the wavelengths are much shorter in the high-frequency range than in the low-frequency range.

Thus, phase can be said to be wavelength versus time. Once this concept becomes clear, it also becomes easier to understand what’s happening with your sound system on a practical level.

Known & Stable Source
Whenever a signal, musical or otherwise, occurs in time and space, it inherently possesses a measureable frequency response and phase response. A sine wave at a fixed frequency exhibits only a simplex frequency response – that of a single frequency – along with possibly some distortion-related overtones. Therefore, the phase response of a single sine wave is generally considered to be a simplex matter.

But speech and music are different. They inherently comprise a complex series of waveforms which make the picture a lot more complicated. This is why measurement equipment is so valuable, because a good measurement system provides a known and stable source, that when acquired by the measurement engine, is capable of accurately characterizing a sound system in a short time span.

Every time an IIR is introduced into a signal path (PEQ, shelving, HP, LP, crossover, etc.), a corresponding phase lead and lag is introduced as well. Phase and frequency are two sides of the same coin. One cannot exist without the other. (Again, refer to the sidebar.)

Figure 2: An example of a simple PEQ boost.

The phase lead, or lag, begins at the lower skirt of the filter and ends at the upper skirt. At the maximum peak or trough of the filter, the phase response is always exactly at zero (Figure 2). Intuitive? Hardly.

But if you spend some time measuring the frequency and phase response of your favorite equalizer as you adjust the settings (highly recommended), you will eventually be able to decipher the response of the acoustic signature of your sound system in a typical venue environment. It takes practice to understand what you’re seeing, but it’s time well spent.

The bottom line is that learning to use measurement equipment, particularly affordable systems like the industry-leading (Rational Acoustics) Smaart package, are essential in developing a solid understanding of what’s really taking place in regard to the system you’re trying to optimize.

While many might argue that “what you hear is more important than what you measure,” this position is usually based on the use of imperfect measuring apparatus—or uninformed interpretation of the measured data.

Peaks & Dips
Offering an alternative perspective, one would need to listen to chromatic scales across the entire audible frequency spectrum, for hours on end, to identify the subtle (and not so subtle) peaks and dips that are always present in the frequency domain.

I sometimes hear about engineers who EQ the system to the tonality of the drums or to the band’s sound check. I then ask what happens if the band decides to play in a different key? Or if the gig is a festival and some acts are playing in, let’s say E Major, while others play in Bb, Ab, or Eb minor – what might have been missed? It’s not far-fetched to imagine that holes or peaks in the spectral response won’t be identified until the next act takes the stage.

But even more so, trying to convert what you hear from the system into making improvements in the time/phase domain is next to impossible without the use of accurate instrumentation.

Ken DeLoria is senior technical editor for Live Sound International and ProSoundWeb, and has had a diverse career in pro audio over more than 30 years, including being the founder and owner of Apogee Sound.

Sidebar: IIR vs. FIR

As in most aspects of life, there are exceptions to everything. Unlike IIR filters, the relatively new breed of filters known as FIR can alter the frequency response of a system without altering the phase response.

However, there is a price to pay. Since phase and time are impossible to separate, the price you pay is that any change in the frequency domain will result in a corresponding effect in the time domain, i.e., additional signal delay.

For some applications like cinema or AV track playback, this might be perfectly acceptable. For other applications, such as stage monitors or front-fill loudspeakers in small theatres, even a small degree of signal delay may not be appropriate.

Each situation must be carefully considered and addressed in regard to the intended end-result. And that is exactly the approach that helps us all further our craft.