Earspeakers Calibration and Acoustic Measurements

The next step in my ongoing study of the phantom center and diffuse sound colorations on stereo speakers (see Part I, Part II) will be about the “ideal” conditions for reproducing the phantom center that can be created in a normal, non-anechoic room by means of earspeakers. These listening devices are also known in research circles as “free-field” or “extraaural” headphones. Their main difference from regular headphones is that they create much less occlusion of the ears. This design factor allows them to be used in situations where we need to compare sound from headphones with external sounds directly. With earspeakers this is possible because the listener does not have to take them off in order to be able to hear the external sound practically unaffected by the headphones (although, as we will see, this is not entirely true).

And this property of earspeakers is really unique. Even open-back circumaural headphones of lightweight construction, for example electrostatic headphones, do attenuate high frequencies severely (for example, see the paper “Comparing the effect of different open headphone models on the perception of a real sound source…”) and also affect sound localization of external sources (see the paper “The Influence of Headphones on the Localization of External Loudspeaker Sources”).

I read about a study in which the researchers actually tried to compensate for the headphone-induced attenuation by applying a reverse filter to the speaker signals. I tried that myself with the Sennheiser HD800. However, I was not satisfied with the result for two reasons:

1. Applying significant high-frequency amplification (an inverse filter for the occlusion from the headphones) to the sound from speakers evokes stronger room reflections and the overall result does not sound fully natural.

2. Since having headphones on the head also affects other components of the HRTF such as ILD and ITD, external sounds filtered by open-back headphones still are not perceived the same as without them, even with the spectrum being compensated. In particular, this affects the phantom center because it relies on the symmetry of the stereo pair.

So, having your ears unoccluded while experimenting with the phantom center is actually a good idea. The problem is that all models of acoustic headphones that leave your ears unoccluded do look really strange, if not completely weird (see this open-access paper for photos). There are only a couple of commercial models, namely: AKG K1000 (discontinued), Sony PFR-V1 (discontinued), MySphere 3.2 (still in production; however, is rather expensive). These are very niche products, and the discontinued models are hard to find for a reasonable price.

However, AR/VR researchers do love earspeakers because of their property to allow hearing both natural and synthesized acoustic reality at the same time, creating “augmented reality.” And researchers come up with various designs and ideas for substitutes. One interesting example is a modification of AKG K702 headphones with custom earpads that are cut out in front and in the back (see “DIY Modifications for Acoustically Transparent Headphones”). I don’t have the K702, but I do have the K701 which is really the same model, just without the detachable cable. So along with the K1000 and the PFR-V1 I will try these as well.

I emphasized “acoustic headphones” in the paragraph above because another well-known way of having ears unoccluded is to use bone conducting headphones. I considered them initially, but then rejected them for a number of reasons:

1. I don’t know how to calibrate them properly, as bone conduction works differently from acoustic transmission.

2. They usually use wireless (Bluetooth) connection which adds a lot of latency.

3. According to this research, “BC [bone conducting] earphone can’t provide enough interaural level difference (ILD)” and thus can affect perception of phantom sources, too.

But what about all these wireless earbuds with the “transparency” feature? Sure, they have lots of microphones inside and outside, and a DSP. In theory, they could implement an acoustic pass-through that is indistinguishable from wearing no earbuds at all. However, the question is why anyone would need that (apart from VR researchers). The goal of engineers working on wireless earbuds is to ensure that the user does not get hit by a car while listening to their favorite podcast, and that the user can communicate with a flight attendant while having airplane engine noise in the background—that’s it. For these scenarios, earbuds don’t actually need to replicate the true transfer function of an unoccluded ear. In fact, the DSP may artificially boost up certain frequency bands in order to improve the external voice clarity, which is the opposite of what I need for my research.

We see that achieving high-fidelity acoustic transparency is non-trivial. So the goal of the exploration described here is to measure the key aspects of earspeakers in order to ensure that I’m aware of their shortcomings. Also, since I bought the K1000 and PFR-V1 second-hand, they are a bit old (it’s practically impossible to buy them in new condition because they have been discontinued) and thus may have some aging-related problems. Another task is to check how the “transparentized” K701 compares to real earspeakers.

This is what I measured for all three headphones:

• electrical impedance;
• usable frequency range;
• distortion;
• cross-talk;
• acoustical transparency.

Also, since I tried using miniDSP EARS (or is it HEARS?—that’s what is written on the label that I see on my device) for some of the measurements, there are some notes about this peculiar acoustic tool.

Electrical Impedance

This measurement is needed to ensure that the drivers are matched. For this, I used QuantAsylum QA490 unit. My AKG K701 shows good alignment:

However, in the PFR-V1 due to its age the left coil shows lower impedance than the right one:

Thankfully, the difference in the impedance seems to be constant across the frequency range which means we can compensate for that by increasing voltage (volume level) into the right speaker. I have found that I need to add +2 dB to the right channel.

The AKG K1000 is more interesting because it uses a 4-pin XLR plug. I don’t have an adapter from it to a TRS. I first tried to use QuantAsylum QA460 (a speaker amplifier), however I found that the nominal impedance of the K1000 is 120 Ohm, and that’s too high for the default value of the current sensing resistor in the QA460—0.01 Ohm. QuantAsylum’s Matt recommends that the resistor value should be 100 times less than the load.

I did not want to resolder the resistor just for this measurement, so instead of the QA460 I used Dayton Audio DATSv3. It worked fine and has shown that the K1000 also has an imbalance between its speakers. Unlike the PFR-V1, the imbalance of the K1000 is not uniform and is mostly pronounced at mid to low frequencies:

Sorry, DATS uses a bit less readable palette. So, for the K1000 I ended up adding a low shelf filter at 2.36 kHz, Q 0.8, gain +2 dB, and a peak at 3.3 kHz, Q 5.5, gain +3 dB. All this alignment was checked acoustically on miniDSP EARS and Neumann KU-100.

Note that although I read in many places that AKG K1000 are “hard to drive”, and sometimes people have to connect them to speaker amplifiers, I did not have any issues with driving them. I used two headphone amplifiers: the SMSL SP200 with “high gain” setting driven from the XLR input, and Monoprice Monolith (THX AAA 788) driven from its digital input. Neither of the amplifiers had any problems driving the K1000 to 100 dB SPL and even higher. That’s the level which is enough for my needs since that’s 100 dB near your ears, not at a speaker one meter away.

Usable Frequency Response

Since earspeakers operate in a free-field condition—they do not create a pressure chamber on the listener’s ear—all constraints of loudspeaker drivers apply to them. Due to the small surface area of the driver, they can’t be efficient at low frequencies. Typically, small (3"–4") loudspeaker drivers that must produce bass are designed to have wider excursion so that they can push and pull air more efficiently. However, it’s hard to design a miniature headphone driver this way.

In an attempt to overcome this limitation, Sony PVR-V1 has a “bass duct” which is intended to be placed near the ear canal entrance:

This creates challenges for measurement because artificial pinnae are simplified and are made of silicone thus their retention force is much lighter that of a real ear. So I tried measuring them both on the miniDSP EARS and the Neumann KU-100. The results are actually consistent and show severe bass limitation:

Note that for this measurement I used the miniDSP EARS with “raw” calibration file. I did not intend to produce a measurement comparable with data from other rigs, I just wanted to figure out where they start to roll off low frequencies. The result looks similar to free-field measurements performed long ago by Rin Choi, modulo the peak between 4–5 kHz which comes from EARS. Basically, the PFR-V1 earspeakers are only usable down to 650 Hz (-6 dB from the midrange) as trying to compensate for bass loss via equalization will just drive up distortion in the driver.

The driver of the AKG K1000 is much better, and its bass starts to roll off only below 90 Hz. Although Hans Zimmer would not endorse use of these headphones for listening to his music, this range is probably enough for classical music.

This is also obtained on EARS with “raw” calibration. Note that apart from the lack of standardized calibration, both miniDSP EARS and Neumann KU-100 are perfectly adequate for measuring earspeakers because their operation does not depend on replicating the true impedances of a human ear canal.

The AKG K701 with cut out earpads loses its sealing completely and thus starts to roll off the bass early:

For comparison, this is the FR of the left speaker for all three earspeakers:

All in all, I think the K1000 is the most adequate earphone in terms of the usable frequency bandwidth.

SPL Calibration of miniDSP EARS

Yet another myth I had read, apart from the K1000 being hard to drive, is that the miniDSP EARS is impossible to calibrate for loudness. This is how I performed their SPL calibration. I found that the silicone ears can be easily removed, opening access to the microphone capsule:

The diameter of its wrapping grommet matches the size of the Beyerdynamic MM-1 microphone so I used its adapter for coupling the acoustic calibrator:

I found that the capsules are well-matched. However, the sensitivity factor specified in the calibration files is wrong which makes the SPL level reported by REW to be off by about 20 dB. I have switched the EARS to 0 dB gain and edited the calibration file to specify correct sensitivity factor: 2.7 instead of 0.1 dB.

One interesting thing that I did not know before is that on Windows at least, REW monitors the software microphone gain and adjusts the reported SPL number accordingly. This is a good thing because it allows the user to adjust the gain so that the sound captured by the EARS at high SPL does not get clipped in the digital domain.

Distortion

I was mostly interested in whether my earphones have any significant distortion that can affect loudness perception. By “significant” I mean “higher than the room noise level.” Since earphones offer no isolation from the room noise, this criterion works well. I ensured that the room noise was 42 dB or below (C weighted) while I was measuring.

When measuring using the KU-100, the distortion products for all three headphones were below the noise floor in the mid-to-high frequency range. I couldn’t measure bass distortion adequately due to higher level of ambient noise in this region. Since with the open design the drivers have to work hard, I would expect that they distort. However, it is known that our hearing system is much less sensitive to low-frequency distortions, and they also do not play an important role in my experiments, so that’s not a problem.

However, with the miniDSP EARS there were peaks of the 2nd harmonic between 4–5 kHz. It was suspicious to me that this happens the same way for all three headphone models. I also tried placing Genelec 8331A speaker close to EARS and measuring it, and it has shown the same peak. So I think what happens here is that the construction of the silicone pinnae of EARS creates a resonance in that region and that overloads the microphone even when the resulting peak is at modest 100 dB SPL. That means, EARS are not suitable for measuring headphone distortion.

Cross-talk

That’s a practically interesting measurement. Since earphones are “open” designs, there is cross-talk by definition. Will this create a coloration of the phantom center? I measured it on the KU-100 in order to create a realistic head shadowing condition.

What I have found is that the level of cross-talk is actually quite low. If we consider the cross-talk from my speaker setup and hypothetically reduce it by what can usually be achieved with application of Cross-Talk Cancellation (CTC) in a non-anechoic room, the cross-talk from earspeakers is still below that level.

By the way, while exploring this, I have discovered a nice feature of REW that I did not know about for all these years of using it. On the SPL and phase graph it’s possible to select a frequency range by dragging the mouse cursor with Shift key pressed, and this shows min, max and average SPL for that region. So calculating the average cross-talk level consists of two steps:

1. Take the difference in dB (A / B) between the contra-lateral and ipsi-lateral transfer functions (“Trace Arithmetic” in REW). Since there are HRTF-induced notches and peaks all over the frequency range, applying 1/3-octave smoothing makes sense.

2. On the calculated SPL graph, select with Shift the frequency range of interest (I usually use 100–14000 Hz to avoid areas where SNR is lower) and read the average SPL.

If we look this way on the SPL graph of the difference between the ipsi- and contra-lateral sound for my desktop speakers (measured on the KU-100), it shows the effect of the head shadowing (the more shadowing—the better). The average level of shadowing is about 11 dB, and it is slanted towards high frequencies:

Below is a table comparing the average shadowing among earspeakers:

Earspeaker	Avg. shadowing, dB
AKG K1000	19
AKG K701 DIY Transp.	48
Sony PFR-V1	36.5

If we consider the PFR-V1, their shadowing is 25 dB better than on external speakers. Considering that CTC algorithms for loudspeakers can at best remove about 20 dB of cross-talk, Sony already performs better.

The AKG K1000 is the worst performer, probably due to large size of speakers and completely open design. On the graph below we can see that at 3 kHz its shadowing drops and matches the shadowing from speakers:

(The numbers on the legend correspond to the 3 kHz point).

So, probably it is a good idea to apply CTC to the K1000. It appears especially effective because traditional nearfield CTC requires head tracking since the head is constantly moving relative to the speakers. However, since the K1000 always moves together with the head, the CTC filter remains constant.

However, so far I did not succeed with creating a CTC filter for the K1000. I tried using a method based on actual HRTF measurement. That’s because the proximity of the earspeaker to the head makes the transfer function of ipsi-lateral crosstalk highly dependent on the parameters of the head.

Also, unlike the louspeaker situation, the wavefront which reaches the opposite ear is not planar—it is spherical. Because of that, when measuring the actual transfer function at the opposite ear of the KU-100 I can see a lot of group delay deviations, and the resulting impulse response does not even have any distinctive peak—it looks more similar to ripples on a sea surface. Because of that, achieving correct time alignment is insurmountably challenging. I will try harder next time.

Acoustical Transparency

This is an important consideration since I plan to switch between playback in earspeakers and loudspeakers (that’s the whole point of using earspeakers in the first place!). As I mentioned in the beginning, even open-back headphones create significant alterations to the frequency response of external sources, as well as to their ITD and ILD. Earspeakers are a bit more transparent but not entirely.

A paper by C.  Porschmann “How Wearing Headgear Affects Measured Head-Related Transfer Functions” measured how wearing the AKG K1000 affects the HRTF of the KU-100. I also measured my earphones on the KU-100, using Genelec 8331A as an external sound source:

I compared the measurements from the paper with mine (using the SOFA files they have provided) and discovered that although the IRs for bare KU-100 look quite similar to my measurements—modulo the effect of a different distance from the speaker to the head—the measurements with AKG K1000 on the head look significantly different. The peaks and notches simply do not match. I started exploring this discrepancy and realized that the acoustical shadowing that the K1000 creates is highly dependent on the angle at which they are opened.

I think this leads to the important realization that measuring these occlusion transfer functions must always be done for the current setup and can’t be considered as “generic.” I suppose, the external speaker directivity also may heavily influence the result, thus occlusion transfer functions measured with a Genelec speaker will also differ from those measured using an LXmini. Because of that, let’s consider the compensating transfer functions for these three headphones just as a guideline.

Besides the data from the paper on headgear, another paper which had introduced the idea of the “transparent” K701 (“DIY Modifications…”) also has measurements of them and the K1000 on the KU-100. However, these measurements are from certain directions: front, back, and top only. Note that these two papers calculate the transparency in an opposite way. The headgear paper divides “reference” (bare head) HRTF by the HRTF of the head with the gear on, while the K701 paper carries the division the other way around. In my view, since we need to compensate for the occlusion by applying a reverse filter to the external speaker, we need to follow the headgear paper approach.

Below are occlusion graphs for each earspeaker, for frontal (0°), side (42°), and rear (138°) directions. For side directions, this is for the ipsilateral ear only. Also, for compatibility with the graphs from papers, my graphs are time windowed for 3 ms and 1/3 octave smoothing applied.

We can see that Sony PFR-V1 is the most unoccluding earspeaker, with AKG K1000 coming next, and the “DIY Transparent” K701 modification is actually not so transparent for rear sources.

Conclusions

Unfortunately, none of the earspeakers is an ideal one. Here is their comparison on the acoustic parameters:

Earspeaker	LF cutoff, Hz	Shadowing, dB	Transparency
AKG K1000	~60	19	Fair
AKG K701 DIY Tr.	~300	48	Poor
Sony PFR-V1	~700	36.5	Good

Thus, for simulating anechoic listening on speakers, it makes sense to use the Sony PFR-V1 as much as possible, by limiting the sound sample choice according to the headphone bandwidth. One caveat here is that by cutting out this frequency range we sufficiently limit the bandwidth below 1.5 kHz where the auditory system uses ITD for sound source localization.

The AKG K1000 can be used for wider selection of samples and does not have any issues providing ITD cues; however, we need to take care about reducing its cross-talk, and compensating for partial loss of transparency when comparing its output with external loudspeakers.

Spectral Correction of Phantom Audio Sources, Part II

Stereo Setup with LXdesktop

Now let’s experiment with real speakers in a real room. For this experiment, I re-arranged my LXdesktop setup into a more conventional placement and orientation. Because my desktop setup sits close to the walls, I positioned the speakers to minimize strong reflections in the first 15 ms. The best configuration delayed the first strong reflection to 12 m, resulting in a 42° speaker angle and 74 cm listening distance. Below are ETC graphs for the left and the right speaker:

Next, I loaded the original LXmini filters into a convolution plugin hosted by Reaper. I used a physical center speaker—a Genelec 8331A—to find the right room curve, first equalizing it for a flat response. Then, starting from the “Harman target curve” that I’ve got from ASR I slightly adjusted the high frequencies roll-off in order to compensate for the near field speaker location. The resulting “nearfield Harman” target curve looks like this:

I then used this curve to equalize both LXdesktop speakers using simple IIR filters generated by REW. I used microphone oriented towards the ceiling in order to avoid coincidental high frequencies bump for the center speaker:

(This photo is from early experiments, the LXdesktop speakers are in their original “wide” setup). After experimenting with the Genelec, I realized that it sounds a bit different from LXdesktop likely due to difference in the radiation pattern. So I built a third LXdesktop speaker just to avoid this difference. The resulting test setup is below:

This is how all three speakers are tuned with respect to the target curve (with 15 cycles of Frequency-Domain Window (FDW) applied):

As you can see they are set up to measure as close as possible to each other.

Notes on Loudness Equalization Approaches

A phantom center and a discrete center equalized to the same linear frequency response do not sound the same. We are interested in matching their perceived sounding, not the frequency response of their linear model as measured by an omnidirectional microphone. If I try to equalize the centers based on the conventional frequency response measurement using anthropometric KU-100 or even binaural microphones, the perceived sound will still not match exactly. After all, non-linear effects, the radiation pattern, and the resulting room reflections of the sound from the speakers play very important role in the final perception of the sound image, and they are not the same between a physical center produced by a single speaker and the phantom center produced by a pair of speakers. Even more distinctive would be the sound for the diffuse field scenario.

We can try to approach the equalization in a controlled way from subjective perspective. The method proposed by D. Griesinger for headphone equalization and implemented by his Sonic Focus app can be employed for matching the perceived sound of speakers. It’s actually even more straightforward for them. Griesinger’s idea is that a human can listen to short 1/3 octave filtered bursts of pink noise and compare their loudness. For the purpose of headphones tuning, the human compares loudness between the current band and the reference band (500 Hz) which results in building an equal loudness contour—a subjective EQ. Others and I have tried this approach (see this MSc thesis by T. Kinnunen). It requires significant effort and yields inconsistent results.

With speakers, however, we don't need to build a loudness contour. We can simply switch rapidly between the reference and target speakers to compare the loudness of each 1/3-octave band directly—I will refer to this later as “modified Griesinger's method.” The results of this tuning are consistent, however it still requires time and effort to go over every 1/3 octave band and meticulously tune the equalizer for it. In addition, one needs to go back and forth multiple times because the bands of human hearing do not necessarily match this fixed 1/3 octave quantization and tuning one band may affect the perception of loudness for adjacent bands. However, I like the overall idea with using perceived loudness as the metric because it takes into account many factors such as non-linearity of speakers and the effects of the room and of the comb filtering.

Can we measure loudness objectively, similar to frequency response? Of course, we can to some extent because loudness is a very important metric for hearing safety, and acousticians do work hard on making the measurement process as automatic and repeatable as possible. One of the methods which is an international standard is the Moore-Glasberg model (ISO 532-2) which can be applied to tones, broadband noises, and complex sounds with emphasized spectral components. The method simulates the highly non-linear, level-dependent compression of the cochlea. There is an implementation of this method within MATLAB’s Acoustic Toolbox—the acousticLoudness function. As you can see from examples, for ISO 532-2 it is possible to obtain a nice graph of loudness per frequency, which looks similar to the frequency response, but in fact is a psychoacoustic measurement. The paper by V. Gunnarsson which I referred to in the first part of this post also employs ISO 532-2 loudness model for assessing the discrepancy between phantom and physical sound sources.

There is a catch: loudness is measured in sones, which can not be directly converted into the decibel values needed for equalization. There are only “rules of thumb,” for example “a 10 dB increase in physical energy roughly doubles the perceived loudness.” But that depends on the frequency range that we are working with and possibly the room conditions. In addition, as I mentioned, the loudness model is non-linear by itself (in order to simulate human hearing correctly). Thus, the character of the sound largely affects the resulting loudness perception.

Gunnarsson sidesteps the issue of translating sones into decibels for equalization by not actually using the loudness model to calculate the correction filters. Instead, he calculates the filters using the composite summed power spectra of both ear signals. Please refer to his paper for more details on this approach.

I decided that instead of using a proxy measurement for loudness, we can automate loudness matching. In some sense, this is similar to performing manual loudness matching by ear as in the modified Griesinger’s method. But instead of doing this manually, we solve a minimization problem for the difference between two loudness curves by applying equalization and adjusting the gain of its bands until we reach the desired tolerance. Basically this is still “matching by ear” but performed by a computer.

There is another hurdle. The ISO 532-2 model expects a measurement microphone with a flat transfer function and internally applies its own pinna gain. But we are using a KU-100 dummy head to correctly capture stereo cross-talk, meaning our measurement is already pre-emphasized by the KU-100's physical pinnae. Since the loudness model is non-linear and mimics the effects of masking and cochlear compression, the presence of double pinna filtering will affect the result in a non-obvious way. Because the loudness model is non-linear, this double pinna filtering corrupts the result. Unlike a purely Linear Time-Invariant (LTI) system, we cannot assume the pinna filter will cancel itself out when we subtract the two curves.

That means, before feeding any readings from KU-100 into MATLAB’s acousticLoudness, first we need to remove the pinna filtering. Thankfully, there is an easy approach to this called free-field equalization (FF-EQ). We basically measure the transfer function of KU-100 for the frontal sound source, smoothen it, and use it as a calibration function. That means, the output from a physical center will be seen a flat line—well, not entirely flat because we do not intend to remove the frontal notch fully, as this will inevitably introduce coloration and skew the results from the loudness model. After applying FF-EQ, the outputs from all other directions around KU-100 will be relative to the frontal source.

I used Claude Code to help me to write the automated loudness matching MATLAB script. Claude wrote a function loudnessmatch which takes as parameters:

• the SPL level of the “reference” FF-EQd binaural recording (because perceived loudness depends on it, recall equal-loudness contours);
• the reference recording itself (for example, the recording from a physical center speaker);
• the recording of the source system that we need to equalize to the reference.

And then the function uses a 29 band 1/3 octave IIR equalizer (similar to what Griesinger’s DGSonicFocus app employs), and carefully minimizes the difference in calculated loudness between the reference and EQ’d source recordings. The function outputs both the values of the equalizer bands and the impulse response of the filter.

Anechoic Simulation of Loudness Matching

Before jumping forward to measure and adjust loudness between physical speakers I decided to run a quick test on the same Ambisonics simulation of KU-100. It was interesting to compare the difference between impulse responses (linear model) with the equalization needed to match loudness (non-linear human hearing model).

This is what we have for the physical vs. phantom center:

And this is for the “diffuse field” (actually, two rear speakers):

I used 80 dB SPL as the source loudness. In theory, when playing audio at a sufficiently different level (e.g., 65 dB SPL or 95 dB SPL), the target loudness matching curve would change (again, see the equal-loudness contours). Thus, the loudness matching approach is sort of an optimization compared to purely linear matching.

As we can see, for the physical vs. phantom center loudness matching has the midrange peak attenuated by about 2 dB and also represents a more regularized curve. For the “diffuse field,” loudness matching also smoothens the dip and makes it slightly less pronounced. Also, the region between 6–10 kHz is amplified instead of being attenuated as frequency response measurement suggests. I guess, this has something to do with equal loudness contours. Anyway, it’s interesting to figure out how does its prediction compare with what I can hear.

Physical Center vs. Phantom Center on Speakers

Matching the frequency response of stereo speakers to a center speaker does not make the phantom center sound like a physical center. To confirm this, I matched their SPL levels by using a measurement microphone while playing correlated signal from the stereo pair.

Then I started playing various mono recordings of acoustic instruments (from Alan Parsons’ “Sound Check” CD) and pink noise, and noticed obvious differences in the sounding of the physical and the phantom center:

• the phantom center sounds wider;

• it gets automatically “anchored” to the center speaker even when in fact the center is muted (I had to check myself a couple of times at first—I was sure that I forgot to mute the center);

• if I close my eyes, the phantom center might be perceived as elevated, depending on the sound being reproduced;

• the tonality is indeed different, which is especially obvious for the pink noise signal. The interesting thing is that the phantom center sounds “softer” and more pleasant to my ears. Maybe that’s why a lot of music producers still prefer it to the physical center. Going a bit ahead, this is caused by differences in IACC, I will explore that in subsequent posts.

So, yes, the “phantom image problem” is real and even with speakers that have narrow, directed radiation pattern the sound from the opposite speaker creates comb filtering. However, even the single, physical center is not immune to the comb filtering problem because the sound from it reflects from nearby surfaces and from the listener’s torso! This can be easily confirmed by stepping towards and away from the speaker. One may argue that this comb filtering is “natural”, however it is not helping me to “unhear” it!

Linear Model

First, let’s now compare the result from the anechoic simulation of KU-100 from the first part of the post with a measurement on a KU-100 in my room:

Below is the comparison between them, the KU-100 room measurement is averaged for the left and the right ear, and I have applied Psychoacoustic smoothing in REW to the result. I also included the Linkwitz EQ curve:

We can see in a room the segment of 300–3000 Hz matches the Linkwitz EQ curve closer than of the anechoic measurement using an ideal point source. Obviously, the room has its influence on the bass region making sources at different locations to produce different sound pressure, so that region is uneven. The high frequency bump is also observed in the room, and the Linkwitz EQ is missing it.

I also measured the same setup using in-ear microphones, and derived the compensation the same way. The result is close to the KU-100 measurement despite the fact that KU-100 is just a head without torso:

What about stereo phantom source to physical center compensation in a real room? For a reference, I used graphs from the paper by B. Shirley et al. “The Effect of Stereo Crosstalk on Intelligibility…” which Toole cites in his book (and also re-uses their graph for illustrating the “phantom image problem” in a non-anechoic room). The graph below compares the following curves: compensation needed for KU-100 and for me in a real room, and Shirley’s result (recall that graphs in my posts are all “EQ graphs,” thus where the original transfer function has a dip, on my graph you see the inversion of it—a hump):

We can see that in my room the location of the cross-talk interference dip is at a different frequency range, only partially overlapping with the Shirley’s result, and it is also wider.

Loudness Model (Non-Linear)

Now a more interesting and practical comparison in terms of perceived loudness. I used an excerpt of 5 seconds of pink noise recorded from each of the sources using KU-100 with SPL of 80 dB in front of the head. The noise was then “free field equalized” (FF-EQ) using the compensation curve obtained during the anechoic experiment. After my loudness matching script has produced the compensation EQ, I loaded it into the Reaper and repeated the recording and measurement in order to ensure that the EQ indeed compensates modelled loudness. It all worked out quite smooth! Below is the graph comparing loudness alignment needed for the phantom center vs. physical center of the anechoic simulation, real KU-100 in my room, and also the linear measurement of KU-100 in my room:

We can see that EQ by loudness is more gentle. It’s interesting to note that loudness matching for the anechoic simulation looks more similar to simple linear matching from room measurements of KU-100.

And then, since my head is a bit different from KU-100, I used DGSonicFocus to check the loudness of each band between the phantom and physical center as perceived by my ears. I have noted some slight differences and applied some compensation:

The differences are quite minor, so I ust conclude that the ISO 532-2 model works really well in this case.

Listening Impressions

Of course, after this meticulous matching of the phantom center it’s interesting to compare how does it sound. Since we only considered the perceived loudness aspect and attempted to fix the spectral coloration, there are still other factors that can contribute to perceived sonic differences.

Comparing full stereo recordings is pointless here; a single physical center cannot convey the intended width and depth of the mix. So I only checked mono dry recordings of vocals and instruments from that Alan Parsons’ test CD.

The most striking difference is of course in the perceived width. A phantom center sound source is always perceived as wider than a physical speaker. For some instruments, especially playing in the lower frequency range this creates much more pleasant feeling of envelopment by its sound. However, for physically smaller instruments like flute or tambourine this widening sounds artificial and make the sound image more blurred. A mono recording of a snare drum completely lost its dry character when played over stereo speakers. I suppose, such sharp transients interact more actively with the room than sounds of more stationary character.

Another difference that is easy to spot is the image stability. In my near field setup moving the head laterally completely changes the location of a phantom center source, while the physical center unsurprisingly remains stable.

A more subtle difference is that dry instruments played via physical center speaker sound a bit “punchier”, with more “in your face” sound. I guess, whether it’s good or not depends on whether one wants “realism” vs. “relaxation.”

As a side note, I think, these differences make it clear why music producers (as opposed to movie dialogue and FX producers) all express greatly varying opinions on the use of the physical center in modern multichannel and object-based recordings. On one hand, it improves the clarity and the “presence” of the rendered source, but on the other hand it could be too much “real” to the point of being disturbing and unpleasant. That’s why it becomes an art in itself how to support the discrete source with phantom source, or how to play with the “sound width” parameter in object-based representation in order to achieve the desired character of the reproduced source.

Finally, the loudness-based equalization almost completely fixed the problem of phantom center elevation. Even with eyes closed—to avoid visual anchoring of the sound to the center speaker—both physical and the phantom center sources are perceived at the same height. With the only exception of the pink noise signal! This is a true acid test for source similarity. Not only the tonality difference due to comb filtering could still be heard, but the perceived height was also not matching, with phantom center sounding elevated compared to physical center. We probably have reached the limit of what can be achieved with a phantom source.

Diffuse Field Simulation on Stereo Speakers

The reference source was the same pair of stereo speakers but facing the back of KU-100 (I have turned it around). The test signal was decorrelated pink noise, at the same output level. Matching loudness in this case required building a more divergent EQ curve. We could see that from the anechoic simulation example, and in the actual room the result is almost the same. Let’s compare the EQ curve obtained in the room with the anechoic curve and the diffuse field compensation curve proposed by Gunnarsson:

We can see that the curve from Gunnarsson’s paper is indeed diffuse field because it lacks the head interference dip.

Making any direct comparisons between the “real” source and its rendering via stereo speakers for this case is quite hard in my setup. However, I relied on Griesinger’s original method of creating an equal loudness curve. I created one for physical behind the head location of speakers (staying with my back turned to them) playing uncorrelated (between left and right) noise bands, as a “reference,” and then turned facing them, and corrected the algorithmically produced curve to yield a similar loudness contour. Here is my final curve:

I suppose, diffuse field (in my case, it’s not even a proper diffuse field since it’s not all around me) is perceived a bit differently from point sources, and because of that the loudness model does not exactly match my personal perception.

Then I tried several tracks that feature off-stage, behind the back, and very diffuse sources:

• left/right imaging test (Track 10 from “Chesky Records Jazz Sampler & Audiophile Test Compact Disc, Vol. 1”);
• tom-tom drum naturally panned around (Track 28 “Natural stereo imaging” from “Chesky Records Jazz Sampler & Audiophile Test Compact Disc, Vol. 3”);
• music instruments performed all around the microphone (Track 47 “Generic Image and Resolution Test” from “Chesky Records Jazz Sampler & Audiophile Test Compact Disc, Vol. 2”);
• F16 and Tornado jets flying by overhead (Track 88 from the “Sound Check” CD by Alan Parsons and Stephen Court);
• Track 1 from “Ambient 1: Music For Airports” by Brian Eno;
• the recording of rain featured at the very end of the movie “Memoria” (2021).

The employed diffuse field compensation noticeably improves presentation of “off-stage” sources and makes overall sound more enveloping. I have combined it with the phantom source EQ by putting the diffuse sound compensation into the side channel of the Mid-Side equalizer. It might sound a bit unusual because the Side channel by definition emphasizes anti-correlated rather than uncorrelated signals. However, commercial music producers typically avoid having strongly anti-correlated signals because they: a) sound really weird, and b) disappear completely when the stereo track gets downmixed into mono. Thus, typically the side channel contains just hard panned left and right channel sources (with the right channel being inverted) and also weakly anti-correlated sources, that is—the ambience.

The thought about hard panned sources had brought me to one realization which required a bit more experimentation.

The Effect of Mid-Side EQ on Hard-Panned Sources, and Up/Downmix Alternative

In commercial stereo recordings it is not unusual to encounter “engineered” (as opposed to “live recorded”) music where the producer panned some instrument to one channel exclusively (also known as “hard panning”). These kinds of mixes were used actively in the early days of stereo, for example on such tracks as:

• “Anna (Go to Him)” by Beatles from one of their first albums (1963);
• “The Time Has Come By” by The Chambers Brothers (1967);
• “Space Oddity” by David Bowie (1969).

But even stereo albums from more mature recording era could use hard-panned instruments, at least in some elements, for example in Madonna’s “American Life” song (same titled album from 2003) we can hear a synth and a guitar hard-panned to left and right channels, respectively, during some moments of the song.

What happens to such hard-panned sources when we employ our Mid-Side EQ for phantom center and diffuse sound compensation? If we just used to compensation for the phantom center, the answer would be trivial—hard-panned source will get reduced amount of the phantom center compensation because it’s applied to the half of the signal—the part from the Mid component, while the Side component is left untouched. But if we also put our diffuse field compensation EQ into the Side channel, then the result becomes less predictable. Linear-phase EQ only affects signal amplitude, but applying different EQs to two instances of the signal and then summing them back introduces severe spatial bleeding and frequency-dependent panning shifts. For instance, this is what happens to hard-panned signals with our spectral correcting EQ (the effect is symmetric for left and right channels):

We can see that application of this EQ introduces “bleeding” and coloration. The situation around the region between 4–5 kHz—this is where the side channel EQ has a deep wide notch—is especially dire. Not good! Can this be fixed? One approach is to abandon diffuse field compensation and use phantom center compensation exclusively, and also reduce the swings of EQ as much as possible balancing phantom center “correctness” vs. the effect on hard-panned sources.

Another approach is to use more sophisticated tool for stereo signal decomposition which actually understands the correlation between signals and perform what is called “Primary-Ambient Extraction” (PAE). In other words, use a modern multichannel upmixer (as opposed to older matrix-based upmixers). PAE algorithms typically utilize time-frequency masking (analyzing the Short-Time Fourier Transform). This allows the algorithm to dynamically route correlated energy to the center and uncorrelated energy to the sides on a per-frequency-bin basis. However, as we saw from my experiments on using upmixers to improve headpone rendering, these algorithms can add artefacts for complex dynamic signals (especially noise-like).

In order to check the differences between the Mid-Side and this approach, I fired up HALO Upmixer and configured it for 5.0 upmixing with hard-panned center extraction and producing an “exact” upmix which can be then precisely downmixed back into stereo with a ITU-R BS.775-compliant downmixer, for example, Nugen’s own HALO Downmixer.

In this processing chain we apply the phantom center compensation to the extracted center channel, and the diffuse field compensation to the side left and side right channels, and then we do a downmix back into stereo:

I’ll call this Up-Downmix EQ, or “U/D EQ” for short. It works better in terms of reducing coloration of hard-panned sources. For comparison:

We can see that the amount of coloration to hard-panned sources and their “bleeding” has been reduced significantly, however it was not eliminated entirely—why? Here is why:

• The PAE-based upmixer does not just route source left channel into the left channel of 5.0 upmix (if we think about it, even a matrix based upmixer would put the source L channel both into the L and C channels because it would make C as L+R). Instead, a PAE upmixer spreads it among all three frontal channels (L, C, and R) with phase relationships that cause them to cancel completely in the right channel of the downmix. This is with the assumption that the intermediate 5.0 representation was not tampered with.

• However, since our purpose is actually to apply EQ to some of those 5.0 channels, we disrupt this balance, and as a result, in the downmix the components from L, R, and C do not cancel themselves perfectly in the right channel of the downmix.

Note that HALO Upmixer’s behavior with hard-panned sources depends dramatically on whether the opposite channel contains true digital silence (literal zeroes) or acoustic silence (e.g., a -120 dB copy of the primary channel). In the digital silence case, HALO Upmixer puts the signal (say, from the left channel) into left front, center, and left rear channels, effectively creating the same problem for hard-panned sources as Mid-Side EQ. However, if the opposite channel actually has a copy of the primary channel, although reduced as much that it will never be heard via an electro-acoustic system, the algorithm detects correlation, and spreads the signal in the frontal pane, as I had described in previous paragraphs. We can see this difference by looking at HALO’s own visualization of the acoustic field:

As for the “bleed” that the equalization of the intermediate 5.0 representation creates—it’s really negligible. In order to confirm that, I measured using KU-100 a “clean” hard-panned channel and compared it with the result of Up/Downmixing processing. The results are very close:

This is likely the best we can do to leave hard-panned sources untouched while providing spectral compensation for both the phantom center and diffuse field. I have compared whether this upmix/downmix equalization approach with my initial Mid-Side EQ approach, and I think it sounds a bit “fuller” considering sides on tracks with lots of ambience, for example, the rain recording mentioned earlier. My hypothesis here is that since side sources are less colored by the frontal source signature, they are indeed perceived as coming from sides, while with M/S EQ they are more colored with the diffuse curve and their location becomes more ambiguous.

Another benefit of the upmix/downmix approach is that it allows more flexibility for controlling the resulting sound field as HALO Upmixer in particular allows changing the angles of rear speakers—this effectively balances the energy between virtual front and back, and also includes psychoacoustic shelving filter for the rear channels which helps to adjust conveniently the perceived height of off-stage sources. For sure, to some extent we can emulate that in the Mid/Side EQ by adjusting gain and shelving of the side component, but this can bring in its own issues.

For now, as I see it, we have reached the limit of how phantom center reproduction can be compensated purely by spectral correction without eliminating the physical sources of the divergence: the comb filtering and the fact that the speakers are located on sides of the listener instead of in front. It would be interesting to check what happens if we try to fix the root of the problem by employing true binaural rendering.

Spectral Correction of Phantom Audio Sources, Part I

I understand that the title of this series of posts sounds maybe too scientific, so first I would like to clarify what I’m talking about. Let’s state one of the most prominent challenges of audio reproduction. Complex sound scenes (for example, music performances by groups of people) always contain multiple sound sources. The number of individual sound sources in an orchestra is vastly bigger than the number of loudspeakers that anyone’s audio system has. Even if we consider a small band and a surround sound system with multiple speakers, the problem still exists because the performers are not necessarily arranged at the same locations as the loudspeakers. In addition, in movies sound sources often change their location dynamically. Because of that, when we play a recording on an audio system, it has to create phantom audio sources originating from locations between speakers. The simplest speaker setup which allows creating phantom sources is the good old stereo, so that’s what we consider here. In the context of stereo playback, we have two problematic aspects of stereophonic playback: the phantom center and the reproduction of diffuse sound fields.

Phantom Center

The phantom center phenomenon is very “unnatural” in the sense that it very rarely can be achieved in nature since it requires two almost identical synchronized (in-phase) sources located symmetrically in front of the listener. Yet, for some reason our brain seems to harness it just fine. This is likely because, from a physics standpoint, a physical center produces sound pressure level changes at both ears, which the brain successfully integrates into a single auditory image. This is why the ‘synchronized’ (in-phase) property of the sources is so important. If the signals are out of phase, the solid center image collapses into a spatially ambiguous, unlocalizable haze; and if they arrive at sufficiently different times, binaural fusion breaks completely, causing our brain to perceive them as two separate auditory events.

It’s interesting that since the phantom center has been used for such a long time in stereo recordings, there are still ongoing debates and discussions between music production professionals on the use of physical center channel vs phantom center, even with modern object-based formats such as Dolby Atmos. The main argument for use of the phantom center by audio producers is that the image of the audio source created by it is perceived as being “fuzzier” and “warmer,” whereas a physical center is more “point-like” and can have a “sharper” character.

However, more technically-oriented audio professionals never get tired of pointing out one of the most widely recognized problems associated with the use of phantom images—the comb filtering effect. Since a phantom sound source is created by combining acoustic waves from two or more neighboring speakers, when the wave from each speaker arrives to the listener’s ear at slightly different times, their sum can produce both constructive and destructive interference. This means, some frequencies will be boosted and some attenuated. This, in turn, means the timbre of the phantom image will differ from that of the physical source we are trying to imitate.

The change of the timbre may also affect the perceived location of the source, for example, it can appear to be elevated, especially in the absence of visual anchors—this is a psychoacoustical problem, a consequence of how the auditory system works. As it had been demonstrated by J. Blauert, narrow-band signals are perceived at specific elevations depending heavily on their center frequency, regardless of the actual sound source location.

Besides being perceived as “warmer” and “fuzzier,” another notable difference in the perception of phantom vs. physical sources is better stability of the latter when the listener is moving in the acoustic space, or when the listener simply turns or tilts their head. A phantom source experiences more dramatic change in its tonality because the comb filtering pattern changes immediately, and this affects the resulting tonal balance.

The well known solutions for the comb filtering problem are:

• For stereo setups, one approach to eliminating the coloration due to cross-talk is to attempt to cancel the latter. Cross-talk cancellation can be abbreviated both as ‘CTC’ and ‘XTC,’ I will use the former acronym. There are different implementations of this approach, such as BACCH (see the description in the “Immersive Sound” book) and RACE.

• For multichannel and Ambisonics setups the preferred approach is slight decorrelation of the physical components of a phantom source emitted by each speaker participating in its creation. That’s because there are many speakers around the listener, and tuning each pair of them for CTC becomes impractical.

Besides the comb filtering, there is another interesting problem which affects the phantom center severely. Since the human HRTF differs for the frontal and lateral directions, the phantom center created by lateral speakers may have different tonality from a frontal physical center just because there is a location mismatch: the brain thinks that the sound is arriving from the front of the listener, so it applies reverse frontal HRTF, but it is a wrong filter because the acoustic waves actually arrive from sides. S. Linkwitz thought about this problem and proposed to use a shelving filter based on the spherical head model. Conversely, D. Griesinger argues that “the frequency response is nearly constant as a sound source moves from zero to ±30 degrees in the horizontal plane.” With all respect to him, I disagree with this statement—as we will see, frontal and side HRTFs are significantly different. There is actually also a more recent, very detailed research by V. Gunnarson (paper “Spectral Correction of Audio Objects in Stereophonic Rendering” from 2024) which clearly shows that the phantom center is affected by the differences in HRTF, and this can be corrected using equalization.

Diffuse Sources

For sure, the phantom center which represents the leading performer is very important in stereo sound reproduction. However, there is also less noticeable but equally important component of the audio scene: the diffuse component which represents “the feeling of the space,” felt mostly unconsciously. However, in live performance recordings this component may jump to the listener’s attention when they are hearing applause. The applause originates from a widely spread source, and reinforced by the hall acoustics, creating a huge diffuse source with an enveloping feeling.

A stereo system trying to reproduce this diffuse source inevitably struggles. The listener’s room may help if it has enough diffusing surfaces, and the speakers are located far enough from the listener, but it’s not always the case. Multichannel system, by design, are much better at reproducing diffuse sources. However, as V. Gunnarson’s paper demonstrates, even multichannel sources win from some room-tailored correction for diffuse sound, and for a stereo system it’s really essential.

One example of such correction is the well known “BBC dip”. This is the speaker equalization which technically was intended to smoothen the transition between the woofer and tweeter which can cause a “power hump” in the upper-midrange making the speaker sounding overly aggressive or “bright” in a reflective environment. As judged by ear, this EQ was known to improve “spaciousness” and “depth” of orchestral recordings. Both of these feelings are communicated to the listener via the diffuse sound field.

The Goals of My Exploration

In my hobbyist research I decided to explore the following questions:

1. In the context of a stereo speaker setup, how is the difference between a physical and phantom center perceived? And what are the major contributing factors to this difference?

2. How should an “ideal” phantom center sound? The ideal phantom center is achieved by making sure that the sound waves arriving from a pair of speakers to the listener’s ears are the same as from a real, physical center. This is hard to achieve in a domestic room due to high level of reverberant, reflected sound. However, we can use earspeakers—a weird kind of headphones that do not block or even cover the ears (because that creates its own problems), but rather are suspended very close to the listener’s ears—in order to simulate speaker playback under anechoic conditions.

3. What can be done in order to make the phantom center produced by stereo speakers sound similar to a physical center, or the ideal phantom center? Are the techniques of stereo speaker sound correction such as CTC and decorrelation actually effective for my speaker setup?

4. Similar questions about the diffuse field reproduction. If I don’t use purposefully built diffusers in my room, how can the reproduced diffuse field be corrected in order to be perceived as more enveloping? Unlike the phantom center situation where the reference can be provided easily, creating a reference diffuse field in a domestic room is challenging.

5. If we consider the often used psychoacoustic metric of Inter-Aural Correlation Coefficient IACC, how does it change between physical and phantom centers? The “Early IACC” (0–80 ms) is associated with “Apparent Source Width” (ASW), while the “Late IACC” (>80 ms) correlates heavily with “Listener Envelopment” (LEV). How is the IACC metric affected by phantom center correction? Also, can we improve the “feeling of space”?

Simulating Phantom Center via Ambisonics Binaural

In order to avoid complications from the issues with room acoustics, let’s first evaluate the ideal anechoic case. In the past, researchers had to simulate physics of the interactions of acoustical waves with a spherical head model, but these days we can perform a more realistic simulation using a binaural renderer. My preferred approach is to encode an acoustic scene containing left, right, and center speaker using Ambisonics and then render it via KU-100 HRTFs. For this purpose, I use IEM Ambisonic plugins: MultiEncoder and BinauralDecoder configured for the 6th order Ambisonics and connected as follows:

3 Channel Source --> MultiEncoder ----> BinauralDecoder --> 2 Channels
                      -42° azimuth (L)                        L/R Ear
                       42° azimuth (R)                        Signals
                        0° azimuth (C)

This setup simulates an anechoic chamber with the KU-100 being in the center of a circle with 3.25 meter radius (this was the distance used by B. Bernschütz when capturing KU-100 HRTFs), with speakers placed at 0° and ±42° in the horizontal plane. I’ve chosen 42° number not because it’s “the answer to everything” but rather because it’s the same angle that I have in my desktop setup.

Side note: although Ambisonics is prone to “spatial aliasing” and in theory requires very high orders for reproducing correct magnitude and phase at high frequencies, the MagLS method used by BinauralDecoder allows to produce correct magnitude only even at high frequencies with relatively low Ambisonics orders.

Our goal here is to check out two things:

1. What is the transfer function (EQ) for compensating the HRTF of a source at 42° on the side of the head to sound like a source in the front of the head (0°). This is to check the EQ curve suggested by S. Linkwitz.

2. What is the EQ for compensating the stereo phantom center to sound like a real, physical center. This way we will double-check the existence of the “phantom image problem”—as Toole calls it, see section 4.3.2 in the 4th edition of the “Sound Reproduction” book, in particular Figure 4.4(d) which demonstrates the phantom center impairment due to stereo cross-talk in the anechoic case.

There are two things about these measurements and derivations that we need to keep in mind:

• The KU-100 HRTF captured by B. Bernschütz and used by BinauralDecoder are symmetric. This is not the case for any real KU-100 since its pinnae although being closely matched are still not absolutely identical, and this affects slightly the measurements at high frequencies. But this symmetry actually simplifies our task since we only need to consider speaker on one side.

• When comparing physical and phantom center, their levels must be aligned. If we just align the levels of speakers, the phantom center will have bass twice as loud as the physical center, because of summing. Unlike the sound waves at midrange frequencies, the bass audio waves are largely unaffected by the presence of a head or even a full human torso, and they just combine mostly in phase which gives them a considerable boost. I suppose, Toole and his colleagues took this fact into account as their transfer function looks flat in the bass region.

So, here is the answer to the first question about the difference between a side source and the frontal source, and I’ve overlaid it with the EQ curve suggested by Linkwitz:

As we see, in general his curve complies with the physical measurement. I would not expect these curves to match completely because Linkwitz was tuning his curve in a real room. However, I must note that his curve is missing an important energy bump after 11 kHz that I can actually hear when comparing phantom vs. physical center by ear—more on that later.

This is the EQ graph demonstrating the impairment of the phantom center compared to the physical center. I have overlaid it with the transfer function graph from Toole, but I had inverted his graph because he is showing how the phantom center is impaired, while I’m showing how the sound of the physical center could be equalized in an anechoic chamber. Note that I totally understand that minimum phase compensation, like traditional EQ, can’t overcome destructive wave interference, however, I’m using EQ curves instead of transfer function curves everywhere for consistency.

Note that Toole’s data is only up to 5 kHz. The exact location of the EQ “hump” for correcting the dip does not match, probably due to differences in the geometry of the KEMAR and KU-100, but the effect is very similar.

When I checked the correction curve for the phantom center in the anechoic case (D/R = ∞dB) by Gunnarson, it shows a decline by -2 dB in the bass region, and the text confirms that this compensates for the summing of bass from the frontal pair. That means, we can’t compare these curves directly with our curves from the last picture.

As you can see, different methods for measuring or calculating compensation curves for the phantom center yield different results. There is even more disagreement about the diffuse field equalization.

Simulating Diffuse Field via Ambisonics Binaural

Simulating enveloping diffuse field using two speakers is definitely more challenging than simulating a discrete center. It is not even entirely clear what should be our “reference sound source.” We can imagine an ideal isotropic diffuse field which envelops the listener from all directions, but this would be impossible to reproduce using a pair of speakers placed in front of the listener, even with acoustic help from a good listening room.

Another issue is with the diffuse field transfer function—it’s usually too smooth because it’s an average over all the sources on the sphere. Whereas, the frontal transfer function always has a deep notch somewhere between 8–10 kHz due to destructive wave interference. As I noted above, equalization can’t compensate it, especially under ideal anechoic conditions. So it’s unlikely that it’s even possible to fully converge these transfer functions.

If we consider the original goal of the diffuse field spectral correction, starting from the “BBC dip,” we can see that its original purpose was to compensate for the difference between the acoustic space of a listening room and a concert hall (many thanks to late Linkwitz for the scan). Gunnarson in his work proposes to use the diffuse field compensation as a way to align the sound of speakers with “a reference ideal diffuse sound field,” (as I mentioned, this is impossible for a stereo setup) however he also mentions that the BBC dip was intended for the same purpose.

So in my Ambisonics simulation I tried a couple of things. First I tried creating a lot of sources behind the listener, spread across the entire rear hemisphere, each playing its own random pink noise. When listened in headphones via BinauralRenderer it sounded quite enveloping. However, trying to equalize the frontal sources to have the same spectral profile yielded unsatisfactory results.

Then I restricted the simulated diffuse field to two uncorrelated rear sources, placed in symmetry with the front sources, that means at ±138°. This configuration looked similar to the classic Quadraphonic sound rectangle. I recalled that the main acoustic flaw of quadraphonic was inability to create stable side sources, and also the “hole in the middle” due to wider angle of the front speakers, but reproduction of surround diffuse images was just fine. Interestingly, when the front sources were equalized to have the same spectral profile as these rear sources, they started sounding much more like the “full rear hemisphere” setup that I have tried initially.

For comparison, here are my compensation curve, the BBC dip, and the diffuse field compensation curve for stereo speakers (D/R = 0dB) which Gunnarson sees as the “more detailed correction” than the former:

Indeed, we can see that Gunnarson’s curve (green) has the same dip as the BBC EQ curve (yellow), and in general it follows the trend of the KU-100 simulation curve (blue), albeit it is much smoother.

Notes on Equalization Approaches

In previous sections we have touched the question of equalization of the center image. Let’s consider how it can be achieved in practice. If we just apply our hypothetical “phantom center EQ” to the entire stereo signal, that will inevitably affect all directions, not just the phantom center. Ideally, we need to apply our EQ to the phantom center only. For that, the recording ideally needs to be object-based, that is composed of individual audio tracks with attached coordinates that are used by a renderer which is aware of the actual speaker configuration that is being used. However, as I checked on the MPEG-H authoring plugin (version 4.0.0 from 2020) they actually do not apply any spectral compensation to objects at the frontal location when rendering into stereo speakers.

As for binaural rendering, the situation is different. Since a binaural renderer employs some kind of HRTF, the results will be similar to our anechoic simulation in the previous section. However, use of non-matching HRTF with headphones lacking individual calibration can easily produce tonal colorations on its own. Because of that, some binaural renderers use more conservative curves. For example, the paper “A Practical Approach to the Use of Center Channel in Immersive Music Production” by K. Richard et al. compares Dolby Atmos binaural renderer vs. “true” binaural rendering using KU-100 HRTFs. From the illustrations we can see that Dolby binaural renderer uses much smoother curves that can be considered more like “head-related equalization” rather than actual HRTFs.

For non-object-based recordings (that is, the majority of commercial recordings), the only way to faithfully extract center objects is to perform some neural network-based (or “AI”, to say it more fashionably) process of stems extraction, and effectively re-synthesize the acoustic scene. But this process is too complicated for me to try, and likely has its own caveats. A more realistic approach is to apply some kind of upmixing into multichannel, at least into LCR and use the resulting center channel as the approximation for center objects, process this center with “phantom center EQ” and downmix back into stereo. Or start with a multichannel mix in the first place.

The interesting thing is that even multichannel and object-based mixes can have “phantom center” sources. As the K. Richard’s paper states, “phantom center images are often preferred over a discrete center, because of the added spaciousness, envelopment, etc.” For example, if we look at Pink Floyd’s “Dark Side of the Moon” 5.1 mix from 2003, and analyze the correlation between left and right channels on the section of the “Time” track with leading vocals (“Taking away the moments that make up the dull day”), we can see that all three channels: Left, Center, and Right are mutually correlated, and the levels of Left and Right are actually higher than of the Center:

That means, the producer intentionally wanted to achieve that classic feeling of the phantom center vocal, however it has reinforced it a bit with the physical center in order to avoid leaving a hole in the middle if the listener has a wide home theater-like setup. The spectrums of the left and right channels are not corrected for HRTF and are practically identical to the center:

I would hypothesize that since the level of the center channel here is much lower than of the left and right combined, it gets psychoacoustically integrated with the phantom center, and the resulting spectral discrepancy is left unnoticed. Similarly to humans, automatic stereo to surround upmixers also rarely pull all correlated components into the center channel (they can do that, but the user has to enforce this setting), spreading them instead across the front channels.

So, even use of a multichannel source (be it the actual multichannel mix, or an upmix of a stereo source) still requires some work to find the correlated components that form the phantom center acoustic image. But as I noted in the post on LCR upmixing, extracting three channels from two is an ill-posed problem. While modern upmixers are excellent, they rely on active steering and decorrelation which inevitably alters the phase relationships of the original stereo mix, often introducing artifacts on complex or uncorrelated signals.

Paradoxically, the cheapest and most reliable tool—mid/side processing—can provide better fidelity by avoiding introducing phase artefacts because it does not create any new channels. By simply summing the signals from the left and the right channels of a stereo recording we get a 6 dB boost for strongly correlated components. Note that it does not completely isolate the center, thus our equalization will affect side-panned sources as well, just to a lesser degree.

Many equalizers can work in the “M/S mode”—they transform left/right stereo into mid-side, apply the EQ to these signals, and then transform them back into stereo. However, if they use minimum-phase EQ filters (IIR filters being a typical example), the change in the phase that these filters inevitably have on the M/S components creates leakage between channels during the reverse transformation into stereo, as I have illustrated previously. Thus, a much cleaner approach is to use a linear phase M/S equalizer which only affects the magnitude of the signals. Note that it’s not without drawbacks, too—linear phase filtering can add substantial latency and also may add pre-ringing artifacts.

But linear filtering is what I use in practice, anyway. If the intended equalization is relatively simple (like the Linkwitz EQ or the BBC dip), then a plugin like ToneControl by Goodhertz can suffice. However, for a more “surgical” kind of EQ, I use LP10 by DDMF. Of course, there is always an option to use generic convolution plugins with a custom-made linear phase FIR filter, that’s in case the 10 bands provided by LP10 are not enough, or when we need to optimize the latency.

Note that Linkwitz did not mention that he used anything like M/S EQ for his proposed filter. I suppose, he was applying it to the whole stereo signal? Same thing for the “BBC dip” which is also considered as the “speaker EQ.” This makes these approaches more like tweaks on the room/speaker target curve, rather than actual phantom source correction.

I think, this part was long enough, so I will stop here. In the next part of the post, we will explore how real stereo speakers behave in a real room.