#007: Time Alignment in Loudspeakers

August 31, 2017

"Phase aligned". "Phase coherent". "Transient perfect". Maybe you have heard these or other expressions used for describing a loudspeaker. But what does it actually mean? How do they relate to each other? And are they being used correctly in the loudspeaker marketing material?

 

I was very much into loudspeaker crossover filters (or simply "crossovers") about 15 years ago, mainly because I had a really good professor in the course "Filter Theory" (shout-out to Claus Vaarning at University of Southern Denmark, SDU). He made the topic of filters extremely interesting for me. I read a lot of Audio Engineering Society (AES) papers on the topic of loudspeaker crossovers at the time, and I also subsequently wrote one myself [1]. It gave me a lot of insight that has stuck with me ever since. And then recently, I saw a question directed towards Dynaudio regarding sound in cars, and more specifically time-alignment.

Now, I of course do not know exactly what the person is asking about; it is probably about the alignment of the left drivers to the right ones, in order to create the optimal sound stage in the car, but it got me thinking about crossovers, and how their inherent amplitude and phase responses affect such things as delay and time alignment in a normal loudspeaker. This is not necessarily very well understood, even by some loudspeaker engineers, so let's analyse this and try to explain the different terminology used to characterise different loudspeakers.

 

As a starting point we assume that we have a loudspeaker where all acoustic centers are aligned vertically. This will imply that there is a horizontal axis which lines up with the listerner's ear, the so-called on-axis. The listener is assumed to be positioned away from the loudspeaker in order to have the analyses pertain to the far-field response.

The red circles indicate the so-called acoustic centers of the drivers, and although they are static in the drawing, their horizontal position can in fact be frequency-dependent. Here, we will assume that they are not. With this alignment we can discuss such things as lobing (the polar radiation in the vertical plane) and time-alignment.

 

If you go strictly off of the many AES papers on the topic of loudspeaker crossover, there is consistency in at least two terms related to loudspeaker characteristics, and those are 1) constant-voltage and 2) all-pass, to describe the total output of the loudspeaker including all drivers and crossovers [2,3]. A constant-voltage filter simply amplifies the signal without affecting the phase. For simplicity we normalize this amplification to 1, and hence the transfer function of the loudspeaker is written simply as

 

In contrast, an all-pass filter also has a constant amplitude response, but it does have a certain non-zero phase.

 

Either of these two transfer functions will give the same amplitude as the listener's ear, but what is the importance of the phase? Well, let's look a one of the terms used to describe a loudspeaker; "transient perfect". If a loudspeaker is transient perfect, it is able to reproduce any input wave without smearing it in time. So if you send in a electrical square wave to the loudspeaker terminals, you will get a square wave pressure on-axis (in the far-field). This requirement will certainly be met by the constant-voltage transfer function, as the amplitude at all frequencies is the same, and there is no shift in phase. What about the all-pass filter? Well, actually this can also be transient perfect too, if the phase shift is linearly dependent on frequency. This can be examined mathematically via the so-called group delay:

 

If the group delay for an all-pass filter is constant, then it will be able to reproduce any input waveform perfectly; except for the fact that the entire output signal will be delayed in time. This can be a problem if the sound is to be syncronous with a video signal, but otherwise, in my definition, an all-pass filter with linear phase is transient perfect, as is a constant-voltage filter.  Note: As any loudspeaker will have a limited bandwidth, a square wave cannot be exactly reproduced, since the sharp shifts in amplitude require an infinite bandwidth (also look up Gibb's phenomenon for more interesting insight into this), but we will assume that this effect is of lesser importance for the loudspeakers in question than the phase responses.

We will now equate the two terms "transient-perfect" and "time-coherent", whereas the term "time-aligned" could mean both "transient-perfect" or in other cases simply mean that the drivers have been physically aligned, without regards to the crossover.

 

The archetypical constant-voltage crossover is the 1st order filter; a two-way (i.e. two loudspeaker drivers) system with two first order crossover sections. The two crossover sections sum up to unity.

 

Since the order is only one, it is difficult to realize since the driver's inherent response must be included in the total response, and protection of the drivers become an issue. However, several companies have touted the benefits of 1st order crossovers, for example Dynaudio, Green Mountain Audio, and Thiel Audio. The former and the latter companies have in later years to a large degree abandoned this philosophy, especially with so many active loudspeakers coming out, and perhaps driver protection was an issue.

Green Mountain Audio discusses such terms as "phase-coherent" and "time-coherent", and from my interpretation of their text "time-coherent" means specifically "contant-voltage", as the phase shift is desired to be exactly zero in their analysis. They also state that another name for "time-coherent" is "minimum-phase", which for an all-pass amplitude response is only true in the special case of zero phase; in general an all-pass filter cannot be minimum-phase (since all of its phase is excess phase), but an all-pass filter with linear phase can however be "time-coherent"/"transient perfect", i.e. capable of reproducing any input signal shape (with a time delay). The term "phase-coherent" on the other hand seems to simply mean that the speaker drivers (including crossover responses) are all in (the same) phase (possible only at the crossover frequency) whatever that phase may be, i.e. linear or non-linear. An example of such a phase-coherent filter would then be the Linkwitz-Riley filter, which we will discuss later. In other on-line litereature "phase-coherent" has other meanings, so be careful if you see these terms, and try to analyse the actual mathematics presented yourself.

Regarding all-pass filters with linear phase, these can only be realized digitally, so therefore it makes good sense to equate "transient perfect" with "constant-voltage" only, when discussing passive loudspeakers. Later in this post we will however have a brief look at some loudspeakers with DSP in-built.

Other configurations of the constant-voltage network has also been explored in the literature, see e.g. the Kido-Yamanaka multiway filter, the multiway filler-driver by Bang & Olufsen [4], and the two-way filters of Small [2].

 

For a very concise take on transient responses, I recommend this post (the web page is dated, but the content holds up), which has definitions that should be consistent with mine.

 

With digital active loudspeakers becoming more and more popular, new methods for obtaining good compromises between frequency response, transient response, and radiation pattern are availble. This is illustrated below for the new KEF LS50 Wireless based on the LS50, which was already a very attentively designed model.

http://uk.kef.com/ls50-wireless

 

The figure shows how a square-wave input signal will result in a pressure which is distorted in time using a standard loudspeaker, whereas for the LS50 Wireless a much better approximation of a square-way is obtained at the listener's ear. Now, with a co-axial driver the acoustic centers are already better aligned than in a standard loudspeaker, so the starting point is good. There is no detailed explanation of DSP time correction, but the temporal response looks very clean. It would be interesting to see how well the standard analog LS50 performs compared to the active version when it comes to temporal response, to see how much of the effect comes from DSP alone.

 

Another interesting loudspeaker was recently reviewed in Stereophile; the Kii Audio Three. Via DSP it is possible to change its time delay characteristics as illustrated below:

https://www.stereophile.com/content/kii-audio-three-loudspeaker

 

The first plot shows the step response for a certain setting where time coherence is prioritized. The plot looks very nice as it reproduces the step closely, but of course it dies out eventually, since a loudspeaker cannot uphold a DC pressure. Note that the output is fairly delayed overall, about 92 ms. In another setting this delay is much reduced, at the expense of a time smeared step response, as shown in the other plot. This particular loudspeaker actually has more tricks up its sleeve, but you can read more about those in the above link.

 

For both the KEF LS50 Wireless adn the  Kii Audio Three it would be interesting to see how e.g. the polar pattern is affected by the time alignment.

 

Leaving the "time-coherent"/"transient-perfect" characteristics, we next turn to the "phase-coherent" term mentioned earlier. We first define the term as a system where all driver outputs are in phase, i.e. they all have the same phase (as mentioned, the term may mean something different across the literature). The most known and used example of this is the so-called Linkwitz-Riley filter [3]. Linkwitz realized that there are certain problems with the off-axis characteristics of constant-voltage filters, namely that near the crossover frequency, there is a lobing issue as shown below:

From [3].

 

It is seen that radiation pattern is "drooping" down towards the floor for two-way constant-voltage crossover, meaning that if the listener moves his/hers head up and down, the frequency response is affected near the crossover frequency. Also, even with your head on-axis, the power response and reflection characteristics of the loudspeaker could be adversely affected; a fact that is of course never mentioned in the marketing material. Linkwitz and Riley realized that by having all (noncoincident) drivers be in-phase, the radiation pattern could be mirror symmetric on the on-axis, as illustrated below:

From http://www.rane.com/pdf/ranenotes/Linkwitz_Riley_Crossovers_Primer.pdf.

 

The Linkwitz-Riley filter is characterized an all-pass filter with non-linear phase, and so is not transient-perfect for any of its possible (equal) orders. However, Linkwitz himself has conducted several experiments that show that this is not audible for typical music signals. Another crossover that falls into the same category, i.e. all-pass with non-linear phase and symmetric radiation pattern was investigated by the late Steen Duelund, see also [1]. In [1] I also show the output from such filters for a square input:

From [1].

 

Admittedly, it does not look nice... The perfect square input becomes temporally distorted for the two-way Linkwitz-Riley as well as for the three-way Duelund output. However, you listen with your ears (+brain), not your eyes. And the subjective experience for music signals is not a distorted, unrecognizable mess; instead there seems to good agreement that the Linkwitz-Riley filter is one of the best comprimises available, at least for analog designs.

 

A more lax definition of "transient perfect" can also be found in the literature, e.g. in this primer from RANE. Here, a crossover is considered perfect if the transient response of the low-frequency driver with its respective crossover section does not overshoot the input signal. In that case, the Linkwitz-Riley second order cross-over can be considered transient-perfect, eventhough it is an all-pass filter, whereas the LR4 and higher cannot be considered transient-perfect. I do not particularly like this definition, as it does not consider the entire system, and also "perfect" should in my book mean perfect, not "somewhat perfect". On the other hand, his definition and motivation are clear enough, just be careful when comparing definitions across literature.

 

Where does the terminology confusion come from? I think it mainly comes from (falsely) assuming that 1) a phase shift is the same as a time shift and defining time alignment from that , with an inherent assumption that only the crossover frequency is important and 2) that lobing errors come only from the physical offset between drivers, without considering the crossover+drivers. This seems to be case for the Wikipedia description of time alignment in loudspeakers; it could be that moving one driver relative to the other(s) is advantageous, but it requires a thorough analysis to figure this out. As already mentioned, phase and time are linked via the group delay, and any definitions should respect this fact.

 

So is it actually worth chasing this illusive time-alignment/time-coherence/transient-perfect characteristics? Well, as shown it can certainly come at a cost when it comes to protecting the drivers, lobing errors, and in general it can be difficult to achieve such a characteristics when the drivers are included [5]. There are those who swear by it, but the jury is still out when it comes to the audibility of this transient perfection. I have no strong opinion about it, as I am more interested in spreading knowledge about the mathematics and physics involved, but I would claim that it falls under "micromanagement", whereas other characteristics such as frequency and power response have been demonstrated to be much more important. On the other hand, if you can get it "for free" as an option in a DSP-controlled loudspeaker, then why not play around with it.

 

I hope that I have cleared up some confusion about time alignment in loudspeakers, otherwise feel free to comment or write me. Oh, and by the way, there is also a term called "absolute phase" which has been the reason for many discussions and disagrements, but I will not open that can of worms, at least in this post.

 

[1]: "Active All-Pass Crossover Networks with Equal Resistors and Equal Capacitors", René Christensen, Journal of the Audio Engineering Society, 2006, 54(1/2)

[2]: "Constant-Voltage Crossover Network Designs", Richard H. Small, Journal of the Audio Engineering Society, 1971, 19(1)

[3]: "Active Crossover Networks for Noncoincident Drivers", Siegfried H. Linkwitz, Journal of the Audio Engineering Society, 1961, 24(1)

[4]: Engineering Brief, Eric Baekgaard, “A Novel Approach to Linear Phase Loudspeakers,” May 1977 issue of the AES Journal

[5]: "Loudspeaker-Crossover Systems: An Optimal Crossover Choice", Robert M. Bullock, III, Journal of the Audio Engineering Society, 1982, 30(7/8)

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