- Sound Illustrations of First and Higher Order Ambisonics plus Optimised Decoding Solutions
    - Audible Proofs of Localization Theories
    - All that accompanied by comments, figures, and movies
 

Preamble
Ideal, centred listening position with static sound images
Evaluating sound image robustness with off-centred listening positions - Circular trajectories
(...Probably the more enjoyable demos)

Other (future?) interesting experiments

You can leave me your feedback! (see the bottom of this page)
 

Restrictions of these page
These sound demonstrations deal only with 2D (horizontal) regular loudspeaker configurations! Perhaps I'll extend them to 3D ambisonic rendering in the future!?
This page focusses on perceptual aspects. Almost all comments were presented (in french) in my thesis but I haven't included  some of the basic definitions (eg of the velocity and energy vectors): please refer to usual ambisonic references (eg Gerzon's papers), or to my documents (thesis or powerpoint presentation) for further details and mathematical/physical demonstrations. Moreover, this is not the place for explaining how the decoding is defined: please refer to my documents (thesis or powerpoint presentation) or to future web pages (in english) for that! Nevertheless, you can visualize the different kinds of decoding here.
Finally, explicit biblio references in this page are still rare at present. Please excuse me and refer to my thesis and its bibliography.

General presentation
These sound examples should be considered as a complement to the objective evaluations presented in the 4th chapter of my thesis, or more briefly on the slide entitled "Evaluation et comparison des décodages" in my powerpoint presentation.
All the sound illustrations presented below are processed by a binaural simulation of ambisonic rendering over loudspeakers, the listening position being controlled with regard to the virtual loudspeaker configuration. Thus, these sound files have to be listened over headphones. They are either WAV or MP3 files.

The process (the virtual rendering)
This is a "light version" of the now well known "virtual loudspeaker" principle: one pair of filters (HRTF: Head Related Tranfer Functions) is associated to each virtual loudspeakers as a function of its direction (from the listener viewpoint), thus the signal which should be delivered by the loudspeaker is filtered to yield its contribution to the left and right ear signals (see Figure 0). The filters used are "dry" HRTF, they have been measured by K.Martin & B.Gardner on a KEMAR dummy head and have been made available for years by their authors on the MIT Web Site. Thanks to them!

Figure 0: the virtual loudspeaker method applied to ambisonic rendering.
In practice, some steps of this scheme can be merged to reduce computation cost.

For an experimental purpose, I have also considered off-centre listening positions, which imply directional and amplitude distorsions as well as time delays between the waves virtually impinging the head and coming from the (virtual) loudspeakers. Nevertheless, we ignore the near field effect of the loudspeakers, which mostly affects the (very) low frequency field (by the way, the near field effect of the loudspeaker used for the HRTF measurements has not been compensated).
Here, we consider the situation of a single virtual sound image, ambisonically encoded/decoded, then rendered by the virtual loudspeakers.

Preliminary remarks
Of course, these simulations cannot replace a true listening experience with real loudspeakers in a real room. Many aspects differenciate the virtual listening experience from the real one:
- The HRTF are not measured on your own head and (a priori) don't really yield natural spectral cues for you.
- There's no room effect associated to the virtual loudspeakers.
- You're head is supposed to be fixed, there is no correction of the sound field at your ears when you rotate your head, thus the dynamic localisation mechanisms cannot be satisfied.
Because of these limitations, the virtual listening experience suffers from well known artefacts, such as the lack of externalisation (the sound is heard inside your head), front-back reversals and, finally, localization ambiguities.
That's why we can evaluate the rendering only by a comparison with a reference. This reference is produced as a "direct" binaural simulation of the original virtual sound source: this would be the effect of a single virtual loudspeaker in the expected direction, instead of the effect of the whole set of virtual loudspeakers used for the ambisonic rendering.
 

Here are given auditory illustrations as a complement to the results of the section 4.1 of my thesis (see also the slide entitled "Evaluation et comparaison des décodages" of my powerpoint presentation.)

This figure presents a few notations used later:

Are discussed:
Spectral Reconstruction
Low Frequency Lateralisation Effect - Velocity Vector Theory (or Makita's Theory, referred by Gerzon)
Higher Frequency Lateralisation Effect - Energy Vector Theory (introduced by Gerzon)
The Problem of the Spectral Balance - Discussion about the Amplitude/Energy Preservation Criteria
 
 

Spectral Reconstruction
Recall
One major property of ambisonic rendering is the local, centred reconstruction of the acoustic field, provided when a so-called "basic" decoding is applied. The area width of the valid reconstruction is proportional to the wave length, and it increases as higher orders are considered. This is illustrated here: reconstruction of a monochromatic plane wave (reference on the right), order M=1 to 5 from the left to the right, 2D reconstruction at top, 3D reconstruction at bottom.
(To be exact, these represent the Mth order truncatures of the cylindrical(2D)/spherical(3D) harmonic decomposition of the plane wave. For a given order, this is equivalent to an ambisonic rendering using an infinite number of loudspeakers. The extent of the reconstruction area is very similar when using a finite number of loudspeakers).

Figure 1. The red limits correspond to a 20% reconstruction error.

Figure 2.
Left ear energy spectra (color scale in dB) as a function of the azimut and frequency (in octaves: oct-4=1250Hz).
The black line delimits the low frequency domain where a valid reconstruction is achieved (within a 20% error tolerance).

We observe that the reconstruction quality depends on the virtual source angle and also on the loudspeaker angles: it is generally better when the virtual source direction is the same as one loudspeaker.

Auditory experiment
Now, how does it sounds like? Here's a short sound sequence (Tracy Chapman) with a rather rich spectral content. The virtual source position is fixed at -90 degrees (right side). A "basic" decoding is applied to the full bandwidth. On the second line, you can hear the difference (or "reconstruction error") between the reference and the virtual ambisonic rendering.

Remarks and advices
It can be quite difficult to hear noticeable differences between some versions (of the first line) or to determine a "preference" among them, since even the reference example doesn't itself yield a very appreciable "spatial enjoyment" (it's only a monophonic source spatialised without any room effect) and there's no signal degradation (in the usual sense of quantisation noise). Better subjective appreciations should be provided with the a complex sound environment example, and in a "real listening situation" regarding the question of the stability. In the present situation, we must be aware of two objective facts (see Figure 2): (1) the reconstruction quality changes with the virtual source azimuth for a given virtual loudspeaker configuration, and (2) even above the frequency limit (black line), the spectral cues are more or less preserved.
If you listen the "reconstruction error" (second line of the table above), you'll hear a low-cut filter effect with an increasing cut-off frequency as the order increases (though some inversions could at other azimuths, see figure 2).
It's more convenient to listen the different versions one just after the other. As soon as you have downloaded the sound files, you can play them in a "play-list" (e.g. with WinAmp or another player...), for instance.
 

Low Frequency Lateralisation Effect - Velocity Vector Theory (or Makita's Theory as referred by Gerzon)
Recall
We have just seen that thanks to the "basic decoding" and for a centred head, the acoustic field round the head and at the ears conforms to the original/encoded event in a low frequency domain. As is evident, the localisation effect is thus preserved regarding the low frequency ITD (seen as the interaural phase delay ) in a low frequency domain which extends while the system order increases.

Figure 3
Interaural phase delay, also referred as "low frequency ITD" (color scale, in ms) as a function of the azimuth (in degrees) and the frequency (in kHz).
The extent of the low frequency domain where the phase delay is "correct" (regarding the case of a single incident plane wave) is defined by the continue lines (0.02ms error) or the dashed lines (0.05ms error).

But what if another decoding style is chosen? Considering the experience of a plane wave again, the ambisonic rendering keeps this interesting property: the apparent direction and speed of the local propagation remains constant with frequency (this, is because the waves coming from the loudspeakers only differ by an amplitude ratio and arrive synchroneously at the central point). The direction of the synthetic wave front (given by the velocity vector V, defined as the mean of the loudspeaker directions weighted by the associated amplitude: V=SG_iu_i/SG_i) is the same as the original event, and the apparent speed c_V is related to the modulus r_V of the velocity vector V : c_V=c/r_V(see my documents for more details), where c is the "natural" sound speed (about 340 m/s). Since the synthetic wave front is large enough with regard to the head (i.e. in a low frequency domain which extends as the order M increases), the perceived LF ITD (Low Frequency Interaural Time Difference) is given by ITD_synthetic = r_V ITD_natural, by comparison with the case of a natural plane wave experience. For example: if r_V<1, then c_V>c, ie the local synthetic wave "go faster" than a natural plane wave, thus the difference of time arrival between ears (ITD) is shortened. This is verified Figure 4.

Auditory experiment
The different decoding solutions give various values of r_V , which are constant for any azimuth when the loudspeaker configuration is regular (see Table 3 below). They are all less or equal to unity. r_V<1 means that the LF ITD, thus the lateralisation effect is less than the expected one (see Figure 4): the perceived direction tends to move towards the median plane.
The following table (Table 2) let you hear this effect with a virtual source at -90° (right) in the horizontal plane. The original sound is a low frequency noise (cut-frequency at 780Hz). The predicted localisation effect is given in the next table (Table 3).
Good headphones and a high attention are required to appreciate the lateral changes for high orders...

Theses .wav files are short and have been resampled at 6 kHz. The shortest are only 10 and 29 Ko long. The others (concatenations) are 66 Ko and 104 Ko long.

Comparison with the prediction and interpretation
Here is a table with the corresponding r_V for a regular, 2D (horizontal) loudspeaker layout. The localisation effect is predicted for a given virtual source at -90° (right) in the horizontal plane, and assuming the listener's head is static. q=-90°+arccos(r_V) gives the value of the predicted azimuth, assuming that the source is in the horizontal, front half plane. d=arccos(r_V) gives the angle of the equivalent ambiguity cone (ie the angle between a plausible direction and the interaural axis).

With a rendering over loudspeakers, d can be also interpreted as an elevation angle when the listener rotates his/her head (yaw rotation: the interaural axis remains in the horizontal plane). But in this case, the perceived azimuth is the same as the virtual source. This detection is explained by considering the ITD and its variation together, and by comparison with the perception of a natural, elevated sound source: see this movie (where r_V=cosd), also included in my powerpoint presentation.
When any head rotation is allowed, the listener can detect the direction of the synthetic wave front (thus the direction of the virtual source: azimuth and elevation) completely, but if r_V<1, he/she will never experience such great variations of ITD as he/she would with a natural sound source (ie a "true" plane wave). Therefore, r_V<1 can be interpreted as a lack of precision of the sound localisation.
Having a good lateralisation effect is important not only for the localisation, but also for preserving good spatial impressions (S.I.) such as envelopement, when we consider the rendering of a complex sound field (with lateral reflections and reverberation). In the low frequency domain, S.I. are related to the fluctuations of ITD (phase delay) (Cf Griesinger).

Conclusion
Unlike the "basic" decoding, the "max r_E" and especially "in-phase" decoding are under-optimal for the low frequency rendering and for a centred listening position. Nevertheless, they get closer to the "basic" decoding as the order M increases.
 
 

Higher Frequency Lateralisation Effect - Energy Vector Theory (introduced by Gerzon)

Recall:
The goal of decoding is to provide an accurate localisation effect and to preserve spatial impressions as well as possible. As a sub-goal, it should be able to yield as well as possible the perceptual effect of a plane wave, seen as an elementary acoustic event. We've seen that this can be easily achieved in a low frequency domain.
In the higher frequency domain, where acoustic reconstruction can no longer be achieved at ears, the directional detection of an acoustic event relies on a kind of perceptual artefact, since the reconstructed signals don't correspond to the experience of a true plane wave. Gerzon (see the bibliography in my thesis) introduced a mathematical criterion to characterise the localisation in this domain: the energy vector E (E=SG_i²u_i/SG_i²: the mean of the loudspeaker directions u_i weighted by the associated energy G_i²), which modulus r_E (always<=1) should be maximised for a better localisation accuracy. One can give an intuitive interpretation to this criterion: the closer r_E is to 1, the more the major energy contributions are confined in a small angular sector in the direction u_E of E, and therefore it seems clear that the sound image is more precise and more robust. Though Gerzon had embedded this concept in a strong theoretical context ("General Metatheory of Auditory Localisation", 92nd Conv. AES, 1992), it has remained difficult to explicit the relation between this vector and the perceptive effects. I've tried to give some answers to this question, focussing on lateralisation mechanisms and using the ILD (Interaural Level Difference) and the high frequency ITD (detected as an interaural group delay) for that.

Characterising the lateralisation effect through the energy vector E and its modulus rE
Like the velocity vector, the energy vector concept have first of all to be used when the contributing waves differ only in amplitude and arrive synchroneously at the listening point (as in Figures 5 and 6) (though it can be still pertinent in extended conditions, see later), considering the reproduction of an elementary acoustic event (seen as a plane wave). Hence, this concerns ambisonics as well as two- or multi-channel DI-sterophony.
The demonstrations I propose next are based on: (1) the detection of the interaural group delay (HF ITD), which is supposed to apply with transients or modulated signals; (2) the consideration of ILD, which becomes significant above 1.5 or 2 kHz.

Figure 5.
Static virtual source (in the energy vector direction). Combination of the contributions at ears (S_R & S_L).
ITD prediction and estimation (from envelope detection) as functions of the head orientation regarding the energy vector direction,
and compared with the ideal case of single plane wave.

Figure 6.
Same legend as Figure 5, excepted that:
the head diffraction is no longer ignored,
and the pseudo-angle g (of the equivalent ambiguity cone) is plotted instead of the ITD.

The auditory experience
The spatialised sound is a noise burst (low frequency cut at 1.2kHz) with a 10ms bell shaped amplitude envelope.
Caution: it's an especially hard listening test! (And it is not a very pleasant sound!) A very (very) high attention is required to hear differences most of time. Close your eyes if this helps you.
Training sound file
I suggest you to experience this situation, so that you get used to perceive the lateral displacement of the noise burst: the burst is repeated 19 times and moves from right (-90°) to front (0°) in steps of 5° (direct binaural simulation of the single sound source). Listen to it (262Ko, fs=32kHz, duration about 2s). If you can (using eg CoolEdit), train yourself on the first part of the sound file (where the directional discrimination is less evident).

The virtual source is located at -90° (right hand). Like the simulations of Tables 1 and 2, the virtual loudspeaker configurations are those represented Figure 2. For an efficient comparison, you can play the "concatenated" versions directly (last column and line).

You see that basic decoding and in-phase decoding give the same values of r_E. But maybe you have found that the in-phase decoding gives a better lateralisation effect. This is clearly indicated by quantitative evaluations of ITD and ILD. Although r_E indicates a same "energy concentration rate" for both decodings, the energy associated to the loudspeakers is more continuously distributed towards the virtual source direction with the in-phase decoding.

The interpretation of r_E as a function of the listening constraints (static, yaw rotation or any rotation allowed) is very similar to what what has been described with r_V in the low frequency domain (see above). Nevertheless, special comments should be added. Unlike what happens in the LF domain, the ear signals are "smeared", compared to what would occur with a single plane wave experience, and the estimated cues (ITD and ILD) are much less regular/continuous with regard to the azimuth.
As a consequence:
(1) The lateralisation quality is generally less good than the predicted values (given in the table).
(2) Although there is a global improvement of lateralisation while increasing order and choosing the supposed optimal decoding, it is not necessarily perceived in the same way considering a fixed azimuth.
(3) Besides ITD and ILD, there's another important aspect that can change the perceived localization quality: the spectral coloration/balance (see also a bit later). This can make the appreciation of the lateralisation itself difficult.

Conclusion
The "max r_E" decoding is optimal in the HF domain for a centred head, whereas the "basic" and the "in-phase" ones are under-optimal. In the present test conditions, this is more perceptible with 1st order systems, less so with higher order systems. The advantage of "max r_E" decoding (and also the "in-phase" decoding) over the basic one appears more evident in terms of stability/robustness in off-centre listening conditions (also in a relatively low frequency domain, in these conditions), as you will experience it in this part.
Considering the rendering of a complex sound field, the "max r_E" decoding will help the lateral separation, thus, a better interaural decorrelation in the relatively high frequency domain, and a better preservation of the spatial impressions.

Criterion for defining the optimal transition frequency above which the "max r_E" decoding should be applied: defining when this decoding yields globally better localization cues than the basic one! This seems more pertinent that just applying the basic decoding up to a frequency corresponding to an arbitrary error tolerance for the acoustic reconstruction of ear signals.
Anyway, the transition from one decoding to the other should not be abrupt and is rather smooth in practice (with usual shelf-filters).
With slightly extended area: the frequency limit decreases as the area width increases.
 

The Problem of the Spectral Balance - Discussion about the Amplitude/Energy Preservation Criteria
The problem:
In the low frequency domain, an amplitude preservation criterion applies since the acoustic reconstruction is OK for the listener. In the higher frequency domain, the wave summation at ears is no longer fully controlled, and it is usual to consider that the contributions are "statistically added in energy". Hence, an energy preservation criterion applies in this bandwidth, which implies a level compensation in this bandwidth.
The problem is that the energy summation is not actual for each frequency: one observes a kind of comb filter effect instead (at least in the current, very strict conditions: no diffuse field and fixed, centred head). This comb filter effect becomes more perceptible and critical as the level compensation is increased, which occurs when the number of loudspeakers increases (compared with the "minimal" recommended number N=2M+2). The level compensation is also a function of the kind of decoding: for a given order M and a given number N of loudspeakers, max r_E decoding needs more compensation than the basic one, and less than the in-phase one. (For more quantitative results, see the table 3.10, chapter 3, of my thesis).
In critical situations, one feels, besides the comb filter effect, an unpleasant spectral unbalance.
Sound demonstration:
I'll try to put some sound examples on this subject later...
Conclusion:
For individual, centred listening, it's better to use the minimal, recommended number of loudspeakers N=2M+2.
In the context of a virtual rendering (over headphones), one could imagine an optimised process for the spectral balance, based on a global equalisation.

Recall:
Most of time with ambisonics, all loudspeakers are fed even for the rendering of one single sound image. With regular loudspeaker configurations, the directional distribution of the signals (ie the equivalent pan-pot law) is given in a synthetic way by a single directivity pattern (a combination of spherical harmonics up to order M) which is characteristic of the decoding style (basic, max r_E or in-phase, or, indeed, any other) and the system order M (Cf Figure 3.14, sec3.3, chap3 of my thesis) (click on the present figure to see a moving source, and concentrate either on one lobe or on one loudspeaker axis).
With basic decoding (the figure shows 2nd order, basic decoding), and a little less with max r_E decoding, the loudspeakers in the opposite direction to the virtual source have a non-negligible participation (secondary lobes of the directivity pattern). You may find it curious, but this is the optimal feeding for a centred listening position considering the constraints of a limited ambisonic order and a limited number of loudspeakers! One feel that such a feeding distribution will not provide a very robust sound imaging if the listener is placed at a distance from the centre, and especially if he/she is close to some loudspeakers. D.Malham had proposed a modified decoding (called "in-phase" decoding) for 1st order systems, which minimises the participation of loudspeakers as they're at a greater distance from the virtual source. I extended this decoding solution to any higher order (see my thesis; the 2nd order "in-phase" decoding is also presented under the name "controlled-opposite" by R.Furse on his pages): their equivalent directivity patterns are all characterised by the lack of secondary lobes (see this figure).

What happens off-centre:
Regarding the sound imaging at off-centre listening points, two aspects have to be discussed: (1) the directional and energetic distorsion of the perceived contributions at this new listening point; (2) the loss of temporal synchronism of the contributing waves at this point.
(1) - This first aspect depends only on the relative distances between each loudspeaker and the listener (the distorsions remain identical if we change the configuration radius R and the distance of the listener from the centre in the same ratio): the "perceived" amplitude (convergent unfilled arrows) associated to each loudspeaker changes in inverse ratio to its distance from the listener (in the present figure, the new listening point is on the left at 75% of the radius R). One can compute a local energy vector (black arrow in the figure, very small on the off-centred point) which will partially predict the localization effect at this point: apparent direction u_E and "quality" r_E(the black circle has a unitary radius, compared with this modulus r_E). Moreover, the perceived global energy is also affected (represented by the radius of the magenta circle) compared with the one perceived at centre (unit circle).

(2) - The second aspect introduces the problem of the precedence effect (also known as "Haas effect" with speech signals, and also referred as the "law of the first wave front") (Cf eg Blauert: "Spatial Hearing"). Very basically, it says that the listener tends to localise the sound source in the direction of the earliest wave front. But things are actually not so simple. The localisation effect depends on several parameters: their relative energetic weight (discussed above), the sharpness of the original signal, and the time scale of the wave front succession (thus, unlike the energetic/directional distorsion previously discussed, this effect depends also on the absolute distance). The effects of these last two parameters are illustrated with these sound examples (to be listened over headphones!). This is the case of a 1st order, basic rendering, with a virtual source placed at -90° (right hand), and a listening position at 10% of the loudspeaker radius R on the left half axis (see the figure in the margin). In terms of energy vector distorsion, the "quality" r_E =0.65 is only a bit less than at centre (where r_E =0.67). We move from a little radius (1m) to a huge one (12m).

Because we're unable to integrate these latter aspects in a synthetic prediction tool, we will use the only energy vector for the visualization of the supposed localisation effect. And that's not a bad thing, since we can compare a local (off-centre) energy vector with the reference one (at centre): off-centre, the earliest wave fronts are also those whose amplitude have "increased" (for the listener), thus the possible precedence effect acts on the localisation in the same direction as the energy vector is distorded towards.

Now, let's have a little more fun with the next sound demos!...

The scenario (for the listening test):
I have chosen to present the case of a moving single source, drawing a regular, circular trajectory around the listener. There are two reasons for this choice. First, I was unable to create a complex, realistic ambisonic sound field at any order (it would require using an efficient/realistic room effect processor with a true ambisonic encoding). Second, this is more convenient for appreciating the directional distorsion of the sound image, and its regularity or its fluctuations as a function of the originally expected azimuth.
Here, the sound file is a 10.6 s extract of a Chopin Walz played on a piano (Walz in A flat major, as played by Jean-Marc Luisada). You could imagine you're walzing in a ball-room, turning and turning, but the fact is that you're staying static... Let's consider instead a less usual but more convenient image: you're in the middle of a "merry-go-round" ("un manège" in french) and the sound is coming from a moving wood horse (played rather by a loudspeaker than by a piano!) ...
    In this situation, you're hearing something like that (please configure your player to play in loop). The source draws exactly two circles around the listener, starting on the front and beginning to move left (anti-clock wise). Click here to vizualize the trajectory (click on the appearing image if needed) (the green o represents the source position and the arrow placed at the centre gives the perceived direction). It is possible that you fail to feel the sound image in front of you when it is supposed to be there. But try to imagine a circular trajectory around you (in the horizontal plane), and train yourself to feel that: it will be useful for the next listening tests.

This is the situation that our ambisonic systems will try to reproduce: a virtual ambisonic microphone placed at the centre encodes the sound source (the incident wave is supposed to be plane), and various decoding are applied. 1st, 2nd and 3rd order systems are illustrated here. The loudspeaker configuration is the same for any rendering: a regular octohedral configuration (horizontal rendering).
The rendering is simulated for 4 listening positions distributed on the left half axis, the distance from the centre being 0% (centred pos., shown with 50% and 75% pos.), 25%, 50% and 75% of the loudspeaker configuration radius.

Now you'll confront these movies with the corresponding listening experiences (listen over headphones!)... For these sound simulations, the configuration radius is fixed to R=2 meters.

About the MP3 encoding/decoding:
How to play them? Players and encoders can be found as freeware or shareware from this site.
The signal degradation. This files have been encoded at 128 kbits/s. There are many other kinds of sound (than piano) which are much more critical for this encoding, so the degradation could be much more annoying than there. Nevertheless, if you play (always over headphones!) the files loud enough, you will hear a little babbling in addition to the piano sound... Try to imagine it's happening outside and forget it: concentrate only on the piano! Anyway, I've verified that the MP3 encoding doesn't really affect the directional effect, compared to the original (uncompressed) wav versions.
The sampling frequency is 32 kHz. Each file is about 10,6 s and weights 167 Ko (versus 1331 Ko for the uncompressed wav file).

Train yourself with the reference file first (single source r=0%) if you have not already done so, in order to really imagine that the source is drawing a circular (anti-clock wise), horizontal trajectory around you.

How to synchronize sounds and movies:
The simplest, but less satisfying method is: while playing your sound file (table 8) in a loop, open the associated movie (table 7) and start it when you hear that the sound file rewinds. I've adjusted the frame rate correctly (2 movie loops = sound file duration).
An other and more elegant method: use the great and easy shareware called "VideoMach" (http://www.gromada.com) or an equivalent software; open the movie twice and the associated sound file once; File|Define Output, choose "Video and Audio (same file)" and choose a filename.avi; in the same box: Video|Format Options, choose preferably MicrosoftVideo1; close the boxes (OK); then File|Start Processing. This produces a rather big file (because the sound is then uncompressed, that's why I can't put this on this page) but with sound and images automatically synchronized.

Now compare one sound simulation with the associated movie (synchronize them if possible): concentrate either on the directional effect (energy vector, black arrow), or on the energy fluctuation (magenta circle). Perhaps you will agree with my own feeling: I find that the listening experience corroborates very well the graphical visualization (and vice-versa)!
At a critical listening position, the sound image evolution is always smoother with the in-phase decoding than with others (especially basic), but higher order max r_E rendering is quite robust too, and offers a more precise imaging.

Conclusions
    Basic decoding is (very) under-optimal for (very) off-centred listening positions. In-phase decoding is the more robust decoding (smooth and continuous evolution of sound imaging) when the listening area extends to the loudspeaker periphery. The sound imaging becomes globally more precise and robust as the system order M increases.
    For a given order M, there's a limit distance from the centre below which the max r_E decoding is more suitable than the in-phase one, and above which it's the contrary. This distance probably increases (for a given radius R) as the order M increases.
    For a given order M and a given limit distance, an optimal decoding could be an interpolation between max r_E and in-phase decoding.
    Further listening tests should be carried through for more quantitative results and conclusions.
    The present sound examples have featured decoding solutions applied on the full frequency band. But the optimal decoder may use several decoding solutions distributed over the frequency range (with a "shelf-filtering"). I've proposed (Figure 3.4, chap.3 of my thesis, see also the powerpoint presentation) that in some cases, the main three kinds of decoding could be involved at the same time: the "basic" one in a (very) low frequency domain, the "max r_E" one in the intermediate (low/middle/high? frequency) band and the "in-phase" one in the middle/high frequency band.

    Lastly, we can expect that another problem will appear with complex sound fields: because of the level unbalance occuring off-centre, it's likely that the sound masking (of one side of the sound panorama by the other one) will lead to a loss of spatial informations and impressions (including envelopment).
 

Off-Centre, Low Frequency Localization - Prediction by the Energy Vector
I have shown (see my PhD thesis) that the energy vector should be a low-frequency localisation predictor at an off-centred listening position, the low frequency domain being such that the detection is based on interaural phase delay and that the phase coherency of the loudspeaker contributions is no longer observed at the listening point (so, above a frequency in inverse ratio to the distance from the centre).
Maybe I'll build some experiments to let you hear that?...

If my head were acoustically transparent ... -
Centred Head, HF Localization Only Based on ITD Detection - Prediction by the Energy Vector
I have given a partial proof (see my PhD thesis, or above) that the energy vector can predict the (HF) localization effect, assuming the head is centred, and considering only the HF ITD detection (interaural group delay). Mathematically, this proof works the best with the hypothesis of an acoustically transparent head. In the real life, one must take the head diffraction effect into account, of course. This implies an ILD (Interaural Level Difference), which is an other important cue for the detection of lateral events in the HF domain. It is clear that the closer the energy vector modulus is to 1 (ie the more the energy contributions are concentrated in the same direction), the more significant ILD is for lateral virtual sources, but there's no simple prediction law.
Anyway, it's difficult to verify perceptually how the group delay detection itself helps for the localization. I'm curious to know how I could detect directional events if my head were acoustically transparent... there wouldn't be any diffraction, thus neither ILD.