Introduction to Ambisonics - Rev. 2015

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1 1

iron@ironbridge-elt.com

Ironbridge Electronics

(Excerpt from the technical papers written during the de

(Excerpt from the technical papers written during the de velopment of Ambisonic Auralizer)velopment of Ambisonic Auralizer)

1.1 1.1 Introduction to Surround and 3D audio techniques

Introduction to Surround and 3D audio techniques

During the development of techniques and technologies in the audio world, engineers have tried During the development of techniques and technologies in the audio world, engineers have tried since the early years of the twentieth century to reproduce recorded sources or live takes in a realistic since the early years of the twentieth century to reproduce recorded sources or live takes in a realistic manner, with the ai

manner, with the aim of giving spaciousness to sound creations.m of giving spaciousness to sound creations.

Research is divided into several branches. Even today most of the sound reproduction systems sold, Research is divided into several branches. Even today most of the sound reproduction systems sold, either consumer or professional, are based on two audio channels. The main reason for that is the high either consumer or professional, are based on two audio channels. The main reason for that is the high cost of amplifiers, signal processors and speakers, which often limited to 2 the number of channels for cost of amplifiers, signal processors and speakers, which often limited to 2 the number of channels for musical productions, television, radio, etc.

musical productions, television, radio, etc.

In cinemas and theaters the widespread use of multi-channel systems started earlier than at home, In cinemas and theaters the widespread use of multi-channel systems started earlier than at home, since it is easier to sell sound systems of medium-to-low quality with relatively low prices for since it is easier to sell sound systems of medium-to-low quality with relatively low prices for home-theatre applications.

theatre applications.

However, only a few commercial post production studios are suitable for multi-channel mixing. However, only a few commercial post production studios are suitable for multi-channel mixing. The vast majority of

The vast majority of post-production control rooms are equipped with 2 (at post-production control rooms are equipped with 2 (at most 3) speakers for stereomost 3) speakers for stereo playback and a subwoofer.

playback and a subwoofer.

Multi-channel audio spread especially over film, theater and video games/virtual reality, whereas Multi-channel audio spread especially over film, theater and video games/virtual reality, whereas the 98% of music

the 98% of music production is still stereophonic.production is still stereophonic.

This is not only due to a problem of costs, but also to the diffusion of the audio formats on which the This is not only due to a problem of costs, but also to the diffusion of the audio formats on which the CD (2-ch, 44100 Hz, 16-bit), and earlier tapes and vinyl, prevailed.

CD (2-ch, 44100 Hz, 16-bit), and earlier tapes and vinyl, prevailed.

Engineers also tried with a very little success to carry the 4-channel information on stereo devices. Engineers also tried with a very little success to carry the 4-channel information on stereo devices. Among these attempts a remarkable solution was the use of subcarriers on vinyl records. Despite the Among these attempts a remarkable solution was the use of subcarriers on vinyl records. Despite the introduction of new formats such as DVD-Audio and Super Audio CD (SACD) or multichannel wave introduction of new formats such as DVD-Audio and Super Audio CD (SACD) or multichannel wave files (Wave Ex), music is still stereo almost in its entirety.

files (Wave Ex), music is still stereo almost in its entirety.

Another issue that discouraged sound engineers from working on multichannel mixes has been the Another issue that discouraged sound engineers from working on multichannel mixes has been the need to keep compatibility of the finalized audio when the number of the channels is scaled down. For need to keep compatibility of the finalized audio when the number of the channels is scaled down. For example, it is a good practice to check how a stereo mix sounds when its channels are summed to example, it is a good practice to check how a stereo mix sounds when its channels are summed to mono. The compatibility with monaural listening is a crucial problem and it should not be mono. The compatibility with monaural listening is a crucial problem and it should not be underestimated. Because of the sum of the left and right channels, you may have phase cancellations underestimated. Because of the sum of the left and right channels, you may have phase cancellations that affect countless hours of

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Introduction to Ambisonics – Francesca Ortolani

Introduction to Ambisonics – Francesca Ortolani – Rev. 2015a– Rev. 2015a

Initially, for example, sound engineers were asked to preserve the sound quality down-mixing from Initially, for example, sound engineers were asked to preserve the sound quality down-mixing from CD (stereo) to TV or radio (in the past these devices were only mono). However, the problem still CD (stereo) to TV or radio (in the past these devices were only mono). However, the problem still exists today in the case of live music in which, because of the size of venues and stadiums, most exists today in the case of live music in which, because of the size of venues and stadiums, most listeners do not benefit from stereophonic

listeners do not benefit from stereophonic listening.listening.

Obviously, passing from surround to stereophonic or monophonic systems is even more

Obviously, passing from surround to stereophonic or monophonic systems is even more critical.critical.

Over the past few decades, since the second half of the twentieth century, sound artists tried to give Over the past few decades, since the second half of the twentieth century, sound artists tried to give more and more spatial dimension to sound and their own artistic creations. This has led to the more and more spatial dimension to sound and their own artistic creations. This has led to the development of techniques aimed at rendering 3D sound, which would have been alternative to classic development of techniques aimed at rendering 3D sound, which would have been alternative to classic surround - according to its several standards imposed upon the market by Dolby - and stereo surround - according to its several standards imposed upon the market by Dolby - and stereo techniques.

techniques.

These include binaural audio, Wave

These include binaural audio, Wave Field Synthesis, OPSI and Ambisonics.Field Synthesis, OPSI and Ambisonics. In particular Ambisonics can coexist with stereo and surround

In particular Ambisonics can coexist with stereo and surround sound systems such as sound systems such as 5.1 or 7.1, etc..5.1 or 7.1, etc.. This 3D audio technique was introduced in the 70s by the team led by Gerzon, Fellgett and Barton This 3D audio technique was introduced in the 70s by the team led by Gerzon, Fellgett and Barton supported by the National Research and Development Council and the British Technology Group. It is supported by the National Research and Development Council and the British Technology Group. It is compatible with a wide variety of speaker array configurations (either regular-symmetrical or compatible with a wide variety of speaker array configurations (either regular-symmetrical or irregular-asymmetrical with various shapes). In-depth explanation of the physical principles on which irregular-asymmetrical with various shapes). In-depth explanation of the physical principles on which Ambisonics is founded are

Ambisonics is founded are given later on; given later on; for the moment it is useful for the moment it is useful to know that this technique is ito know that this technique is inn part an extension of the basic principles of Mid-Side miking technique [1] patented by Blumlein. This part an extension of the basic principles of Mid-Side miking technique [1] patented by Blumlein. This technique uses sum and difference signals between a microphone (from the family of cardioids) with technique uses sum and difference signals between a microphone (from the family of cardioids) with

Figure 1.1 In order to

Figure 1.1 In order to have a correct perception of thehave a correct perception of the stereophonic sound, the speaker pair and the listener must stereophonic sound, the speaker pair and the listener must bebe located at the vertices of an

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Ambisonics should not be confused with "traditional" surround. First of all, Ambisonics allows Ambisonics should not be confused with "traditional" surround. First of all, Ambisonics allows including information relative to the height (classic surround techniques are 2D instead). The including information relative to the height (classic surround techniques are 2D instead). The principles of acoustics on which this technique is based will be explained in detail in the next section. principles of acoustics on which this technique is based will be explained in detail in the next section. Furthermore, for example, considering a classic quadraphonic system, while the phase difference Furthermore, for example, considering a classic quadraphonic system, while the phase difference between the signals received at the front speakers is processed quite effectively by the auditory system between the signals received at the front speakers is processed quite effectively by the auditory system (at least for low frequencies), this is not the case for the rear speaker pair, so classic surround systems, (at least for low frequencies), this is not the case for the rear speaker pair, so classic surround systems, quadraphonic or larger systems, do not allow a good source localization.

quadraphonic or larger systems, do not allow a good source localization. This is due to the fact thatThis is due to the fact that sources in classic surround are recorded ac

sources in classic surround are recorded according to cording to "discrete" channels, that is, independent on each"discrete" channels, that is, independent on each other, and the differences in level between channel pairs are used [2], [3].

other, and the differences in level between channel pairs are used [2], [3].

Hence the layout of the loudspeakers relative to the listener becomes crucial: you can experience it Hence the layout of the loudspeakers relative to the listener becomes crucial: you can experience it even in a simple stereo system where, if the listener and the speakers are not perfectly placed at the even in a simple stereo system where, if the listener and the speakers are not perfectly placed at the vertices of an

vertices of an equilateral triangle, the exact equilateral triangle, the exact source localization is source localization is lost. lost. Testing assessed Testing assessed that thethat the quality of the

quality of the ghost imagesghost images11 between speaker pairs is poor if these are spaced by an angle greater thanbetween speaker pairs is poor if these are spaced by an angle greater than 60 degrees (i.e. the equilateral triangle mentioned above).

60 degrees (i.e. the equilateral triangle mentioned above).

In quadraphonic sound, for example, speakers are spaced by 90 degrees causing a feel of "hole in In quadraphonic sound, for example, speakers are spaced by 90 degrees causing a feel of "hole in the middle". A

the middle". A homogeneoushomogeneous sound reproduction system is defined as a system in which no direction sound reproduction system is defined as a system in which no direction is treated with any particular preference. Typical cinema surround systems are

is treated with any particular preference. Typical cinema surround systems are not homogeneousnot homogeneous, in, in fact the sound coming from the front stage (screen) is usually controlled more accurately than the rear fact the sound coming from the front stage (screen) is usually controlled more accurately than the rear channels, since a solid match between sound and image is searched with the objective not to distract channels, since a solid match between sound and image is searched with the objective not to distract the audience. We can say, however, that surround systems are

the audience. We can say, however, that surround systems are coherent coherent , within certain limits, in the, within certain limits, in the sense that the sound image remains stable, that is, not subject to significant discontinuities, if the sense that the sound image remains stable, that is, not subject to significant discontinuities, if the listener changes position [4]. The consistency of the front image is guaranteed by the presence of listener changes position [4]. The consistency of the front image is guaranteed by the presence of sounds uncorrelated from the rest of the system. This can be achieved, for example, by delaying and sounds uncorrelated from the rest of the system. This can be achieved, for example, by delaying and spreading the signals sent to the surround system.

spreading the signals sent to the surround system. What we ideally want is that the reproduced soundWhat we ideally want is that the reproduced sound

Figure 1.3 Example

Figure 1.3 Example of Mid-Side configuratof Mid-Side configuration ion - polar- polar diagram (MID: cardioid microphone, SIDE: figure-of- 8 diagram (MID: cardioid microphone, SIDE: figure-of- 8 microphone).

microphone). LEFT = (MID+SILEFT = (MID+SIDE)/2, RIGHDE)/2, RIGHT = (MID-T = (MID-SIDE)/2 [14]

SIDE)/2 [14]

Figure 1.3 Quadraphonic

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In Ambisonics, on the other hand, the signals sent to the speakers contain information from each In Ambisonics, on the other hand, the signals sent to the speakers contain information from each microphone capsule used in the recording with different relations resulting from a decoding matrix. microphone capsule used in the recording with different relations resulting from a decoding matrix. The effect of spatialization here is much more robust than in traditional surround techniques, in the The effect of spatialization here is much more robust than in traditional surround techniques, in the way that the sweet spot, i.e. the op

way that the sweet spot, i.e. the optimum listening position, is wider. timum listening position, is wider. Ambisonics is not Ambisonics is not limited to alimited to a precise number of speakers: the higher the number the better the directional resolution you can get. precise number of speakers: the higher the number the better the directional resolution you can get. The reason for that will be explained next, by introducing the concept of order in Ambisonics.

The reason for that will be explained next, by introducing the concept of order in Ambisonics.

1.2 1.2 The Physics in Ambisonics

The Physics in Ambisonics

A comparison with other techniques

In sound field description, source characterization is one of the most important jobs of In sound field description, source characterization is one of the most important jobs of Auralization

Auralization. Auralization involves. Auralization involves creating audio files from simulated, measured or synthesizedcreating audio files from simulated, measured or synthesized numerical data

numerical data [5]. [5].

For example, it is possible to represent multipole or extended sources by summing a certain number For example, it is possible to represent multipole or extended sources by summing a certain number of monopoles

of monopoles22,, i.e. point sources whose dimensions are much smaller than the wavelength of thei.e. point sources whose dimensions are much smaller than the wavelength of the incident sound wave, or integrating over a distribution of monopoles or infinitesimally small surface incident sound wave, or integrating over a distribution of monopoles or infinitesimally small surface elements. Each source contributes in

elements. Each source contributes in terms of sound pressure to the terms of sound pressure to the acoustic field.acoustic field.

According to source distribution, a specific spatial radiation pattern is created depending on the According to source distribution, a specific spatial radiation pattern is created depending on the position and the distance of the sources. In other words, this is expressed by

position and the distance of the sources. In other words, this is expressed by Huygens’ principle Huygens’ principle saying saying a wavefront can be considered as a secondary distribution of sources. For example, the 3D audio a wavefront can be considered as a secondary distribution of sources. For example, the 3D audio technique

technique Wave Field SynthesisWave Field Synthesis is based on this and works as the acoustic equivalent of holography. is based on this and works as the acoustic equivalent of holography. In practice, the sound field can be considered emitted by the original source or by a secondary In practice, the sound field can be considered emitted by the original source or by a secondary source belonging to the wavefront. In mathematical terms, this is equivalent to saying that we can source belonging to the wavefront. In mathematical terms, this is equivalent to saying that we can obtain the sound pressure on the area

obtain the sound pressure on the area A A, knowing the sound pressure, knowing the sound pressure

p

₀₀ and its gradient and its gradient ∇∇rr

p

₀₀_{on the}_{on the} boundary of

boundary of A A, by calculating the Kirchhoff-Helmholtz integral:, by calculating the Kirchhoff-Helmholtz integral:

( ( ))

( (

))

00 0 0

1

00

,,

4

jkR jkR A A

p

R

ee

p

p rr

p

p n

n

jkR

dS

rr

A

R

π

R

− − ∂ ∂





= = ∇

_

∇ ⋅ ⋅ −− ⋅⋅ ++

_

∀∀ ∈∈





∫∫

r r r r rr uurr rr _r_r (1.1) (1.1) where

where k k is the wave-number and is the wave-number and R R

r r

is the vector connecting the source with the listening point [6]. is the vector connecting the source with the listening point [6]. A detailed analysis of integral (1.1) shows how each secondary source is composed of a monopole A detailed analysis of integral (1.1) shows how each secondary source is composed of a monopole (relative to the pressure gradient signal) and a dipole (relative to the pressure signal). However, there (relative to the pressure gradient signal) and a dipole (relative to the pressure signal). However, there are slight conceptual differences in the formulations by Kirchhoff-Helmholtz and Huygens. The are slight conceptual differences in the formulations by Kirchhoff-Helmholtz and Huygens. The former is more general. The shape of the boundary does not depend on the wavefront, in addition the former is more general. The shape of the boundary does not depend on the wavefront, in addition the Kirchhoff-Helmholtz integral itself carries information relating to both amplitude and phase of the Kirchhoff-Helmholtz integral itself carries information relating to both amplitude and phase of the acoustic signal, whereas in Huygens’ principle it is assumed that the secondary sources are located on acoustic signal, whereas in Huygens’ principle it is assumed that the secondary sources are located on equiphase surfaces. In practice, we can conclude that the Kirchhoff-Helmholtz integral generalizes equiphase surfaces. In practice, we can conclude that the Kirchhoff-Helmholtz integral generalizes

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In practice, Kirchhoff-Helmholtz integral is used as represented by

In practice, Kirchhoff-Helmholtz integral is used as represented by Figure 1.4Figure 1.4 (Wave Field Synthesis): (Wave Field Synthesis):

The listening area is surrounded by pairs of transducers composed of a pressure microphone and a The listening area is surrounded by pairs of transducers composed of a pressure microphone and a velocity/pressure gradient microphone. In section 1.8 some basics on these types of microphones are velocity/pressure gradient microphone. In section 1.8 some basics on these types of microphones are given. So, the recorded field is due to the sources external to the microphone array.

given. So, the recorded field is due to the sources external to the microphone array.

Then, the Kirchhoff-Helmholtz integral can be interpreted considering that each secondary source can Then, the Kirchhoff-Helmholtz integral can be interpreted considering that each secondary source can be split into two elemental sources:

be split into two elemental sources: DIPOLE

DIPOLE SOURCE: SOURCE: fed by fed by a a pressure pressure signalsignal

p

₀₀ MONOPOLE

MONOPOLE SOURCE: SOURCE: fed fed by by a a pressure pressure gradient gradient signalsignal ∇∇rr

p

₀₀

During playback a specular action is operated: arrange speakers having physical characteristics as During playback a specular action is operated: arrange speakers having physical characteristics as shown in Figure 1.4 in the place of the microphones, that is, replace the pressure mics with acoustic shown in Figure 1.4 in the place of the microphones, that is, replace the pressure mics with acoustic dipole speakers (these speakers radiate both forwards and backwards) and replace the pressure dipole speakers (these speakers radiate both forwards and backwards) and replace the pressure gradient mics with monopole speakers (closed speakers radiating only forwards thus having a gradient mics with monopole speakers (closed speakers radiating only forwards thus having a directional characteristic). The geometrical layout of the microphone array and the speaker array has directional characteristic). The geometrical layout of the microphone array and the speaker array has to be the same.

to be the same. Each speaker is fed with the signal picked up Each speaker is fed with the signal picked up by the respective microphone.by the respective microphone. Similarly we can surround the source instead of the listener with the microphone array [6]. Similarly we can surround the source instead of the listener with the microphone array [6].

Such a system guarantees (ideally) the exact reproduction of the field within the listening area and, Such a system guarantees (ideally) the exact reproduction of the field within the listening area and,

Figure 1.4 Application of the Kirchhoff-Helmholtz integral in holophony/WFS Figure 1.4 Application of the Kirchhoff-Helmholtz integral in holophony/WFS

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Possibly, in that case, we encounter

Possibly, in that case, we encounter spatialspatial aliasingaliasing. As in the time domain, spatial aliasing occurs. As in the time domain, spatial aliasing occurs when the signal is sampled in space taking an insufficient amount of points. Aliasing is revealed by the when the signal is sampled in space taking an insufficient amount of points. Aliasing is revealed by the appearance of fictitious sources.

appearance of fictitious sources.

The maximum frequency above which spatial aliasing occurs is calculated as (Nyquist Theorem): The maximum frequency above which spatial aliasing occurs is calculated as (Nyquist Theorem):

max max

2

_trans_trans

cc

f

d

= = _(1.2)_(1.2) where

where

d

_trans_trans is the distance between two transducers and is the distance between two transducers and cc is the speed of sound.is the speed of sound. A signal of frequency above

A signal of frequency above

f

_max_max produces a time difference of arrival at two transducers greater thanproduces a time difference of arrival at two transducers greater than the signal period, while a signal of frequency below

the signal period, while a signal of frequency below

f

_max_max is such that, being the time difference in theis such that, being the time difference in the range of the signal period, the phase difference at the transducers allows an unambiguous time range of the signal period, the phase difference at the transducers allows an unambiguous time difference evaluation.

difference evaluation.

Basically, some simplifications on the Kirchhoff-Helmholtz integral and its use are operated. We try to Basically, some simplifications on the Kirchhoff-Helmholtz integral and its use are operated. We try to minimize the number of transducers to represent the most important secondary sources and we try not minimize the number of transducers to represent the most important secondary sources and we try not to use both the monopole and dipole transducers: what we normally do is to send signals recorded with to use both the monopole and dipole transducers: what we normally do is to send signals recorded with cardioid or figure-of-8 microphones to monopole speakers. Note that the superposition of a monopole cardioid or figure-of-8 microphones to monopole speakers. Note that the superposition of a monopole and a dipole gives as a result a cardioid polar characteristic. Finally, with the aim of limiting the and a dipole gives as a result a cardioid polar characteristic. Finally, with the aim of limiting the number of spaced microphones used, it is preferred to build "virtual microphones" through the number of spaced microphones used, it is preferred to build "virtual microphones" through the processing of the recorded signals by weighting the amplitudes and delays in the time of arrival processing of the recorded signals by weighting the amplitudes and delays in the time of arrival appropriately, in order to improve the resolution of the system.

appropriately, in order to improve the resolution of the system.

A constraint to the techniques based on the theory of the Kirchhoff-Helmholtz integral (with any A constraint to the techniques based on the theory of the Kirchhoff-Helmholtz integral (with any simplistic changes) is the fact that within the listening area, surrounded by the speaker array, primary simplistic changes) is the fact that within the listening area, surrounded by the speaker array, primary sources should be absent, i.e. the array is able to reproduce only the external sources. This is a false sources should be absent, i.e. the array is able to reproduce only the external sources. This is a false problem as it is possible, however, to reverse the phase of the signals feeding the array relatively to the problem as it is possible, however, to reverse the phase of the signals feeding the array relatively to the secondary sources and reproduce, in this way, the internal sources, too. Therefore, we can create a secondary sources and reproduce, in this way, the internal sources, too. Therefore, we can create a concave wave front instead of a convex one [7].

concave wave front instead of a convex one [7]. Another way to describe a sound fiel

Another way to describe a sound field, especially in the case of sources witd, especially in the case of sources with spherical symmetry, ish spherical symmetry, is based on the decomposition of the sound field into

based on the decomposition of the sound field into spherical harmonics spherical harmonics.. Ambisonics is founded onAmbisonics is founded on this second descriptive approach. Spherical harmonics are also used in issues concerning quantum this second descriptive approach. Spherical harmonics are also used in issues concerning quantum mechanics, gravitational fields and can be found in 3D graphics applications and lighting engineering. mechanics, gravitational fields and can be found in 3D graphics applications and lighting engineering.

The starting point is to express the acoustic wave equation in spherical coordinates

( (

r

,, ,,

θ

θ ϕ

ϕ

))

,, where

where rr is the radius,is the radius,

θ

is the azimuth andis the azimuth and

ϕ

ϕ is the elevation.

is the elevation. The acoustic wave equation in the time domain is:

The acoustic wave equation in the time domain is:

( (

))

( (

))

2 2 2 2 2 2 22

,, ,, ,,

1

1 ,, ,, ,,

p

p rr

t

0

0 p

p rr

t

cc

t

θ

θ ϕ

ϕ

θ

θ ϕ

ϕ

∂∂ ∇ ∇ − − == ∂ ∂ (1.3)(1.3)

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The acoustic pressure field, due to external sources, can be developed into Fourier-Bessel series, The acoustic pressure field, due to external sources, can be developed into Fourier-Bessel series, whose terms are weighted products of the directional functions

whose terms are weighted products of the directional functions

Y

_mn_mnσ σ

( (

θ

θ ϕ

,,

ϕ

))

-- spherical harmonicsspherical harmonics –– with the radial functions

with the radial functions

j

_m_m

( ( ))

kr

- - sphericalspherical Bessel’s functions of the first kind Bessel’s functions of the first kind ::

(

)

) (

(

)

) (

( )

)

(

( ))

0 0 00 ,, 11

2

1

mm _m_m _m_mn_{n m}_mn_n

,,

m m nn mm

p

p rr

m

jj jj

kr

B Y

B

σ σ

Y

σ σ σ σ

θ

θ ϕ

ϕ

∞ ∞ = = ≤≤ ≤≤ =± =± = =

∑

++

∑

r r (1.4) (1.4) with

with m = degreem = degree and and n = ordern = order (the meaning of(the meaning of

σ

is the is the spinspin and it will be obvious looking at the and it will be obvious looking at the pictures further on),

pictures further on), k k is the is the wavenumberwavenumber

k

2 2 f

f

cc

π





= =









. Equation (1.4) . Equation (1.4) represents the solurepresents the solution of thetion of the

Wave Equation in the special case of plane wave. Wave Equation in the special case of plane wave.

As shown later, the ambisonic signals in the transform domain [7] are represented by the coefficients As shown later, the ambisonic signals in the transform domain [7] are represented by the coefficients

mn mn

B

σ σ and behave like Fourier coefficients in a Fourier series. and behave like Fourier coefficients in a Fourier series.

Note that, unlike WFS or Holophony, the sampling and the reconstruction of the sound field in Note that, unlike WFS or Holophony, the sampling and the reconstruction of the sound field in

Figure 1.5 Bessel spherical functions of the first kind. Figure 1.5 Bessel spherical functions of the first kind.

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Let's see the spherical harmonic functions in detail, analysing how ambisonic signals are obtained Let's see the spherical harmonic functions in detail, analysing how ambisonic signals are obtained from these functions. Spherical harmonics are

from these functions. Spherical harmonics are defined as:defined as:

( (

))

( (

))

(

))

( (

))

0, 0, (ignore (ignore if if 0)0)

cco

os

s

iif

f

1

1 !!

,,

2

1

2

2 ssiin

n

ssiin

n

iif

f

1

1 !!

m mnn nn mmnn n n

n

m

m n

n

Y

m

P

n

m

m n

n

σ σ

_θ

_ϕ

_δ

_ϕ

θ

σ

θ

σ

== = = ++ − −



= = ++ −− ××

_

= = −− + +

_

(1.5) (1.5) where

where

P

_mn_mn

( ( ))

ξ

are associated Legendre functions of degree mare associated Legendre functions of degree m and order and order nn,,

δ

_pq_pq represents Kronecker represents Kronecker delta and

delta and is equal is equal to 1 to 1 ifif p p == qq_{, else it’s equal to 0.}_{, else it’s equal to 0. The associated Legendre function is}_{The associated Legendre function is defined as:}_{defined as:}

( ( ))

( (

22

))

₂₂

( ( ))

1

1 (

( )

22

)

₂₂

( ))

(

22

1

2

2 !!

m m n n nn nn mm nn m m mn mn nn mm mm mm nn

d

P

d

m

d

ξ

+ + + + − − = = −− == −− −− _(1.6)_(1.6)

where

ξ

==

cos

ϕ

_.

In Ambisonics some kind of normalization

In Ambisonics some kind of normalization of Legendre functions often takes place of Legendre functions often takes place [8].[8]. Schmidt Semi-Normalization

Schmidt Semi-Normalization is defined by: is defined by:

( (

))

(

))

( (

))

( (

))

( (

))

0, 0, 0 0

!!

2

1

2

2 !!

!!

1 1 if

if

0

2 2 if

if

1

m mnn nn nn n n

m

m n

n

m

m n

n

N

m

ee

m

m n

n

m

m n

n

ee

n

ee

n

δ

− − −− = = ++ −− == + + ++ = = == = = ≥≥ (1.7) (1.7)

The harmonic functions can be rewritten in

The harmonic functions can be rewritten in Schmidt Schmidt semi-normalized semi-normalized form (SN3D) by substituting form (SN3D) by substituting (1.7) into (1.5): (1.7) into (1.5):

( (

))

(ignore (ignore if if 0)0)

cco

os

s

iif

f

1

1 ,,

ssiin

n

ssiin

n

iif

f

1

m mnn mmnn n n

n

Y

P

n

σ σ

θ

σ

θ

θ ϕ

ϕ

θ

σ

== = = ++



= = ××

_

= = −−



%% _(1.8)_(1.8)

The set of spherical harmonics forms an orthonormal basis in the sense of the spherical scalar product, The set of spherical harmonics forms an orthonormal basis in the sense of the spherical scalar product, that is:

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So, they can be linearly combined in order to define functions on the surface of a sphere. So, they can be linearly combined in order to define functions on the surface of a sphere.

In such a position, Equation (1.4) has to be arrested to a certain order

In such a position, Equation (1.4) has to be arrested to a certain order M M (because of manageability), (because of manageability), also known as

also known as order of Ambisonics.order of Ambisonics. Writing again for convenience:Writing again for convenience:

(

)

) (

(

)

) (

( )

)

(

( ))

0 0 00 ,, 11

2

1

mm _m_m _m_mn_{n m}_mn_n

,,

m m nn M M

p

p rr

m

jj jj

kr

B

B Y

σ σ

Y

σ σ σ σ

θ

θ ϕ

ϕ

∞ ∞ = = ≤≤ ≤≤ ==±± = =

∑

++

∑

r r (1.10) (1.10)

We have seen that components

B

_mn_mnσ σ are tied tightly to the acoustic pressure field and its higher-order are tied tightly to the acoustic pressure field and its higher-order derivatives about the origin.

derivatives about the origin. In a vector form we have: In a vector form we have:

Figure 1.6 High Order Ambisonics (up to 3rd

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The following example reveals how ambisonic signals can be achieved from

coefficientscoefficients

B

_mn_mnσ σ ..

For the time being,

we stop the Fourier-Bessel series at orderwe stop the Fourier-Bessel series at order M M ==11_{obtaining the signals called W,}_{obtaining the signals called W,}

X, Y, Z , which we will define better in the next sections: X, Y, Z , which we will define better in the next sections:

( ( ))

[ [

]]

1 1 33 T T M M DD

B

== ==

WXYZ

1 1 00 00

B

−− ==

W

_relative_{relative to}_{to PRESSURE}_{PRESSURE signal}_signal

1 1 11 11

B

++ ==

X

1 1 11 11

B

−− ==

Y

_{relative to PRESSURE GRADIENT signals (or to acoustic velocity)}_{relative to PRESSURE GRADIENT signals (or to acoustic velocity)}

1 1 10 10

B

++ ==

Z

As can be immediately seen from the 3D illustration of the spherical harmonics (

As can be immediately seen from the 3D illustration of the spherical harmonics ( Figure 1.6Figure 1.6), in), in order to achieve a higher directional resolution, the order of Ambisonics must increase.

order to achieve a higher directional resolution, the order of Ambisonics must increase. Attention!

Attention! So as to avoid confusion, it should be noted that order So as to avoid confusion, it should be noted that order M M of Ambisonics is different of Ambisonics is different from order

from order nn defined in Legendre functions. We can rather say that it refers to the ambisonic order indefined in Legendre functions. We can rather say that it refers to the ambisonic order in terms of degree

terms of degree mm in Legendre functions. in Legendre functions.

Ambisonics is not only a 3D audio technique. Sound field representation can be specialized in 2D Ambisonics is not only a 3D audio technique. Sound field representation can be specialized in 2D environments. For this purpose, the sound field should be decomposed according to a system of environments. For this purpose, the sound field should be decomposed according to a system of cylindrical coordinates for a horizontal-only reproduction system.

cylindrical coordinates for a horizontal-only reproduction system. One has: One has:

(

( )

)

(

( ))

( (

))

( ( ))

(

)

( (

(

)

(

))

( ( ))

1 1 11 11 00 00 00 1 1 1 1 11 11((22 )) 11 11((22 )) 00 00 00 1 1

,,

2 2 cco

oss

2 2 ssiin

n

,, 0

0 ,,0

0

m mnn mmnn mm m m D D DD m mnn mmnn mmnn mmnn mm m m

p

p rr

B

B JJ

kr

B

m

B

m

m JJ

kr

B

B JJ

kr

B

B Y

Y

B Y

B

Y

JJ

kr

θ

∞ ∞ − − = = ∞ ∞ − − −− = = = = ++ ++ == = = + + ++

∑

r r (1.12) (1.12)

Also in this case we get an orthonormal basis as seen in the set of 3D equations. The functions denoted Also in this case we get an orthonormal basis as seen in the set of 3D equations. The functions denoted

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The sound field represented by Equation (1.10) and

The sound field represented by Equation (1.10) and unpacked unpacked in 2D form in Equation (1.12) can be re- in 2D form in Equation (1.12) can be re-written in the explicit form for a plane

written in the explicit form for a plane wave as (referring to Figure 1.7 and [9])wave as (referring to Figure 1.7 and [9])

( ( ))

(

)

(

)

(

))

(

( )

)

(

( )

) (

( )

) (

( )

)

(

( )

) (

( )

) (

( ))

cos cos 0 0 1 1 0 0 1 1 11

2

2 cco

oss

2

2 cco

oss

cco

oss

2

2 ssiin

n

ssiin

n

jkr jkr mm m m m m m m mm m m mm m m mm

p

p rr

P

P ee

P

P JJ

kr

P

jj JJ

kr

m

P

JJ

kkrr

jj JJ

kkrr

m

jj JJ

kkrr

m

ϕ ϕ θ θ θ θ θ θ θ θ θ θ

ϕ

ϕ θ

θ

ϕ

θ

ϕ

θ

∞ ∞ − − = = ∞ ∞ ∞∞ = = == = = == ++



_

−−



_

==





= =

_

+ + ++

_





∑

r r (1.14) (1.14)

Equation (1.14) can be transformed into

Equation (1.14) can be transformed into matrix form:matrix form:

( ( ))

T T

p

p rr

rr ==

P

_θ_θ

_B

_{B h}

_h

_(1.15)_(1.15) where where 1 1 1 1 22 22 ii 22 22 ii T T B B BBσ σ



θ









 





B B

Figure 1.7 Sound wave impinging on listening position [9] Figure 1.7 Sound wave impinging on listening position [9]

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On the other hand, the information about sound field variations relatively to the listening point is On the other hand, the information about sound field variations relatively to the listening point is included in vector

included in vector

h

. Source directivity changes with . Source directivity changes with the emitted frequency (directivity increases withthe emitted frequency (directivity increases with increasing frequency).

increasing frequency).

Furthermore, the use of microphones of the family of cardioids simplifies expressions (1.10) and Furthermore, the use of microphones of the family of cardioids simplifies expressions (1.10) and (1.14) in a way that, separating spatial dependence from frequency dependence, one has:

(1.14) in a way that, separating spatial dependence from frequency dependence, one has:

(

)

(

( )

)

(

))

0 0 00 ,, 11 m m mm mmnn m m nn mm

p

W

B

B Y

σ σ

Y

σ σ σ σ

θ

ω

θ

∞ ∞ = = ≤≤ ≤≤ ==±± = =

∑

_(1.16)_(1.16) where

where

W

_m_m

( ( ))

ω

is the weighting factor: is the weighting factor:

(

( )

)

mm

( (

(

( ))

))

(

( )

1

1 ) (

''

( ))

m m mm MMIICC mm MMIICC

W

ω

= =

jj

α

jj

kkrr

− −

jj

−−

α

jj

kkr

r

_(1.17)_(1.17) Equation (1.17) highlights how the recording field is dependent on the frequency (in fact,

Equation (1.17) highlights how the recording field is dependent on the frequency (in fact, k k is the wave is the wave number

number

k

2

2 π

π

λ

=

= _{, where}_{, where} λ λ _{is the wavelength}_{is the wavelength}

cc

f

λ

== _,_{, cc is the speed of sound and}_{is the speed of sound and f}_{f is the frequency).}_{is the frequency).}

Basically, when miking a source, besides considering source directivity, we must consider the Basically, when miking a source, besides considering source directivity, we must consider the microphone polar characteristic with respect to the frequency.

microphone polar characteristic with respect to the frequency.

In light of the above, recorded ambisonic signals correspond to coefficients

B

T T weighted by functionweighted by function

( ( ))

m m

W

ω

which depends on frequency. Equation (1.17) was obtained by weighting Equation (1.10) by which depends on frequency. Equation (1.17) was obtained by weighting Equation (1.10) by a cardioid characteristic function of the kind

a cardioid characteristic function of the kind

G

( )

(

θ

θ α

)

= =

α

+ +

( ))

(

1

−−

α θ

α

cco

oss

θ

_{. Remember that a cardioid}_{. Remember that a cardioid} microphone is generated by the superimposition of an omnidirectional microphone (responsive to microphone is generated by the superimposition of an omnidirectional microphone (responsive to

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1.3 1.3 Ambisonic Formats

Ambisonic Formats

As we have seen, in Ambisonics, sound directional components are encoded vectorially in a set of As we have seen, in Ambisonics, sound directional components are encoded vectorially in a set of spherical harmonics. This paragraph shows how audio signals are recorded and processed in spherical harmonics. This paragraph shows how audio signals are recorded and processed in Ambisonics. Actually, Ambisonics is not limited to a particular number of channels: a greater number Ambisonics. Actually, Ambisonics is not limited to a particular number of channels: a greater number of channel provides a higher dire

of channel provides a higher directional resolution.ctional resolution.

In Ambisonics several formats exist for microphone recording, broadcasting and reproduction of In Ambisonics several formats exist for microphone recording, broadcasting and reproduction of recorded signals.

recorded signals.

-- A-Format: A-Format: suitable suitable for for miking miking with with specific specific microphone microphone (e.g. (e.g. Soundfield Soundfield mic);mic); -- B-Format: B-Format: suitable suitable for for miking miking and and processing processing with with studio studio equipment;equipment;

-- C-Format/UHJ: suitable C-Format/UHJ: suitable for for mono, mono, stereo, stereo, 3-channel 3-channel systems systems and and broadcasting;broadcasting; -- D-Format: D-Format: suitable suitable for for decoding decoding and and playback playback through through array array of of speakers;speakers; -- G-Format: G-Format: alike alike D, D, but but decoder decoder is is not not required;required;

A-Format A-Format

A-Format is achieved from the recording of four signals using a microphone equipped with four A-Format is achieved from the recording of four signals using a microphone equipped with four sub-cardioid capsules mounted on the faces of a tetrahedron and oriented as shown in

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A sub-cardioid capsule is characterized by a polar diagram of A sub-cardioid capsule is characterized by a polar diagram of the type shown in Figure 1.8 and it is described by the equation: the type shown in Figure 1.8 and it is described by the equation:

( ( ))

0 0..7

7 0

0..3 c

3 co

oss

ρ

ρ θ

θ

= = ++

θ

where

where θ θ _{is the angle of incidence of the acoustic wave.}_{is the angle of incidence of the acoustic wave.}

The summary table below includes microphones from the family of cardioids, their polar The summary table below includes microphones from the family of cardioids, their polar characteristics and the equations describing them:

characteristics and the equations describing them: POLAR

POLAR DIAGRAM DIAGRAM TYPE TYPE OF OF MICROPHONE MICROPHONE EQUATIONEQUATION Family of cardioids

Family of cardioids

(general equation)

ρ

ρ θ

( )

(

θ

)

α

(

( ))

1

1 α

α

cco

oss

θ

= = + + −− OMNIDIRECTIONAL OMNIDIRECTIONAL

_ρ

_{ρ θ}

_{( ( ))}

_θ

==

₁

SUB-CARDIOID SUB-CARDIOID

_{ρ θ}

_ρ

_{( ( ))}

_θ

= =

_0..7

₀

_{7 0}

++

_{0..3 c}

_{3 co}

_oss

_θ

Figure 1.9 Polar

Figure 1.9 Polar diagram of a diagram of a sub- sub-cardioid capsule.

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B-Format B-Format

B-Format consists o

B-Format consists of four signals called f four signals called W, X, Y, Z. W, X, Y, Z. As already mentioned abovAs already mentioned above, signal W ise, signal W is relative to the pressure component of the sound field in all directions, while X, Y, Z refer to the relative to the pressure component of the sound field in all directions, while X, Y, Z refer to the horizontal components of velocity on the horizontal plane (X, Y) and the vertical component (Z) of horizontal components of velocity on the horizontal plane (X, Y) and the vertical component (Z) of velocity. Microphone takes in B-Format are achieved using three figure-of-8 microphones for signals velocity. Microphone takes in B-Format are achieved using three figure-of-8 microphones for signals X, Y, Z and an omnidirectional microphone for signal W.

X, Y, Z and an omnidirectional microphone for signal W.

The axis pointing 0° in microphone X points the source (it is equivalent to MID in MS), The axis pointing 0° in microphone X points the source (it is equivalent to MID in MS), microphone Y is rotated by 90° (the 0° axis points leftwards), with respect to X (it is equivalent to microphone Y is rotated by 90° (the 0° axis points leftwards), with respect to X (it is equivalent to SIDE in MS). Microphone Z is oriented along the orthogonal plane with respect to the plane described SIDE in MS). Microphone Z is oriented along the orthogonal plane with respect to the plane described by the axes X and Y (the 0° axis points upwards).

by the axes X and Y (the 0° axis points upwards). Figure 1.10Figure 1.10 shows the microphone layout just shows the microphone layout just depicted in words:

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