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:
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?
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.
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?
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.
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:
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.
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.
