176 Spatial Sound
Table 4.1 Parameters and characters for the first-order horizontal Ambisonics with regular loudspeaker configurations and different decoding methods.
Criteria for optimization |
Frequency range |
Listening region |
Normalization |
|
Atotal |
b |
rv = 1 |
Low |
Small |
Amplitude |
1/M |
2 |
|
rv = 1 |
Low |
Small |
Power |
1/ |
3M |
2 |
Maximize rE |
Mid and high |
Small |
Power |
1/ |
2M |
2 |
In-phase |
Full |
Large |
Power |
|
2/ 3M |
1 |
|
|
|
|
|
|
|
4.3.3 The higher-order horizontal Ambisonics
Sound field signals in Ambisonics are extended from the first order to higher orders to improve the accuracy in spatial information reproduction, which are termed higher-order Ambisonics (HOA; Xie X.F., 1978b; Xie and Xie, 1996; Bamford and Vanderkooy, 1995; Daniel et al., 1998, 2003). For the first-order horizontal Ambisonics, the signal of the loudspeaker at azimuth θi is given in Equation (4.3.15) and can be written as the linear combination of a target-azimuthal-independent component W = 1 and a pair of first-order target-azimuthal harmonics X = cosθS and Y = sinθS:
Ai S Atotal W D11 i X D12 i Y
Atotal 1 D11 i cos S D12 i sin S ,
(4.3.45)
where
D11 i bcos i |
D12 i bsin i. |
(4.3.46) |
For the second-order horizontal Ambisonics, two additional second-order target-azimuthal harmonics components are supplemented to the signals expressed in Equation (4.3.45):
U cos 2 S V sin 2 S. |
(4.3.47) |
Then, the normalized signal amplitude of the loudspeaker at θi becomes a linear combination of five independent components or signals W, X, Y, U, and V:
|
1 |
2 |
|
1 |
i U |
2 |
i V |
|
|
|
Ai S Atotal W D1 |
|
i X D1 |
i Y D2 |
D2 |
|
. (4.3.48) |
||||
|
1 |
|
|
2 |
i sin S |
1 |
|
|
2 |
|
Atotal 1 |
D1 |
i cos S D1 |
D2 |
i cos 2 S D2 |
i sin 2 S |
|||||
Generally, the independent signals of the Q-order horizontal Ambisonics with Q ≥ 1 consist of the preceding (2Q + 1) azimuthal harmonics up to the Q order:
1, cos q S, sin q S q 1, 2. Q. |
(4.3.49) |
Horizontal surround with regular loudspeaker configuration 177
The normalized signal amplitude of the loudspeaker at θi is a linear combination of (2Q + 1) independent components:
Q |
1 |
2 |
|
|
|
|
|||
Ai S Atotal[1 Dq |
i cos q S Dq |
i sin q S . |
(4.3.50) |
|
q 1
Equation (4.3.50) can also be written in the matrix form as
A Atotal D2D S. |
(4.3.51) |
where A = [A0(θS), A1 (θS),…AM−1 (θS)]T is an M × 1 column matrix or vector composed of M normalized loudspeaker signals; the superscript “T” denotes the matrix transpose; S = [1,
cosθS, sinθS, cos2θS, sin2θS…, cosQθS, sinQθS]T is a (2Q + 1) × 1 column matrix composed of (2Q + 1) normalized independent signals; [D2D] is an M × (2Q + 1) decoding matrix with entries 1, Dq1 i , Dq2 i , q 1, 2 Q.
Loudspeaker signals depend on entries Dq1 i and Dq2 i . When a loudspeaker configuration is regular and θi of each loudspeaker is given in Equation (4.3.14a), similar to Equation (4.3.46), Dq1 i and Dq2 i take the forms
Dq1 i 2 q cos q i |
Dq2 i 2 q sin q i q 0 b 2 1. |
(4.3.52) |
Then,
|
Q |
|
2 q cos q i |
Ai S Atotal 1 |
|
|
q 1 |
Q
Atotal 1 2 q cos q S
q 1
|
|
|
|
|
|
cos q S q sin q i sin q S |
|
|
|
(4.3.53) |
|
|
||
|
||
i . |
|
|
|
|
|
|
|
The decoding parameter κq specifies the relative proportion of each azimuthal harmonics in loudspeaker signals. A set of κq can be regarded as an azimuthal harmonic window applied to truncate the azimuthal harmonics up to order Q. The loudspeaker signal magnitude maximizes when the target source direction is exactly consistent with the loudspeaker direction, i.e., θS = θi, then
Ai S max |
|
Q |
|
|
|
|
|
(4.3.54) |
|
Atotal 1 |
2 q . |
|||
|
|
q 1 |
|
|
Similar to the case of the first-order Ambisonics, for a regular loudspeaker configuration with θi given in Equations (4.3.14a), (4.3.53), (4.3.16) to (4.3.18) verify that when
M Q 2, |
(4.3.55) |
178 Spatial Sound
the following equation is obtained:
M 1
Ai S MAtotal
i 0 |
|
(4.3.56) |
|
M 1 |
M 1 |
||
|
|||
Ai S cos i 1MAtotal cos S |
Ai S sin i 1MAtotal sin S. |
|
|
i 0 |
i 0 |
|
Equation (4.3.56) is only related to the parameter or proportion κ1 of the first-order azimuthal harmonic component and independent from the second or higher-order azimuthal harmonic component. The perceived virtual source direction is evaluated from Equations (3.2.7) and (3.2.9). For a fixed head,
|
M 1 |
|
|
sin I |
Ai S sin i |
1 sin S, |
|
i 0 |
(4.3.57) |
||
M 1 |
|||
|
Ai S |
|
|
i 0
For the head oriented to the virtual source,
M 1
Ai S sin i
|
i 0 |
|
|
tan I |
|
tan S. |
(4.3.58) |
M 1 |
|||
|
Ai S cos i |
|
|
i 0
The condition of the head oriented to the virtual source in Equation (4.3.58), or the direction of the velocity localization vector does not place restrictions on κq, and the condition for a fixed head in Equation (4.3.57) only limits κ1. When
1 1 |
or b 2 1 2, |
(4.3.59) |
the following equation is obtained:
sin I sin S. |
(4.3.60) |
Equations (4.3.58) and (4.3.60) yield
I I S. |
(4.3.61) |
In this case, the perceived virtual source azimuth matches with that of the target source within the full horizontal direction of −180° ≤ θS ≤ 180°, and the results of the head oriented to the virtual source are consistent with those of a fixed head. Equation (3.2.29) proves that the velocity vector magnitude is rv = 1 when κ1 = b/2 =1. In other words, for Q > 1 order horizontal Ambisonics, the optimized velocity localization vector only requires the decoding parameter of the first-order azimuthal harmonics to be κ1 = b/2 = 1 without restriction on κq for the second or higher-order azimuthal harmonics.
Horizontal surround with regular loudspeaker configuration 179
κq can also be derived from other physical criteria. In Section 9.1, under the criterion of spatial harmonic decomposition and each order approximation of the target sound field, the parameters should be
q b/2 1 |
q 1, 2 Q. |
(4.3.62) |
Loudspeaker signals with the parameter given in Equation (4.3.62)are the conventional or fundamental solution of Ambisonic signals to which a rectangular azimuthal harmonic window is applied to truncate azimuthal harmonic components up to Q.
The two second-order azimuthal harmonic components of the second-order Ambisonics are given in Equation (4.3.47). These two components are equivalent to the signals captured by a pair of the second-order directional microphones, and the polar patterns are illustrated in Figure 4.16. The responses of the second-order azimuthal harmonics vary faster than those of the first-order azimuthal harmonics. Similarly, in comparison with the (Q − 1)-order reproduction signals, two additional Q-order target-azimuthal harmonic components are supplemented to the Q-order reproduction signals. These additional components are equivalent to the signals captured by a pair of the Q-order directional microphones. In practice, the realization of the higher-order directional microphones is difficult, but the higher-order azimuthal harmonic components can be derived from the outputs of a spherical microphone array (Section 9.8). In addition, higher-order azimuth harmonic components can be easily simulated by signal processing.
The normalized signals of Q-order horizontal Ambisonics can be obtained by substituting Equation (4.3.62) into Equation (4.3.53):
|
|
Q |
|
|
|
|
Ai S Atotal |
|
2 cos q i cos q S |
|
|||
1 |
sin q i sin q S |
|||||
|
|
q 1 |
|
|
|
|
|
|
Q |
|
|
|
|
Atotal |
|
|
|
|
|
(4.3.63) |
1 |
2 cos q S i |
|||||
|
|
q 1 |
|
|
|
|
|
|
|
1 |
|
|
|
|
sin Q |
S i |
|
|
||
Atotal |
|
|
2 |
|
. |
|
|
sin |
S i |
|
|
||
|
|
|
|
|||
|
|
2 |
|
|
||
|
|
|
|
|
|
|
(a) U = cos2θS |
(b) V = sin2θS |
Figure 4.16 Polar patterns of a pair of the second-order directional microphones (a) U = cos2θS; (b) V = sin2θS.
180 Spatial Sound
(a) Q = 1 (b) Q = 2 (c) Q = 3
Figure 4.17 Polar patterns of the preceding three orders Ambisonic signals Ai(θS) = Ai(θS − θi) (a) Q = 1;
(b) Q = 2; (C) Q = 3
The recording and reproduction of Ambisonic signals are analyzed to obtain insights into the physical nature of Equation (4.3.63). Figure 4.17 illustrates the polar pattern of the preceding three orders Ambisonic signals Ai(θS) = Ai(θS − θi). In Figure 4.17, the maximal magnitude of the signal is normalized to a unit. From the point of signal recording, Equation (4.3.63) is equivalent to a signal captured with a microphone with an appropriate directivity and main axis direction. The main lobe of the signal is centered at θ′i = (θS − θi) = 0°. The response |Ai(θS − θi)| maximizes at the on-axis direction θ′i = 0° and then decreases as | θ′i| increases. As | θ′i | further increases, the responses exhibit the side and rear lobes (in-phase or out-of-phase) and null points of the polar patterns at some azimuths. As order Q increases, the width of the main lobe and the responses of the side and rear lobes decrease, sharpening the directivity of the resultant signal.
From the point of reproduction, the reproduced sound field in Ambisonics is a superposition of the pressures caused by M loudspeakers. Equation (4.3.63) represents the signal for the ith loudspeaker with θi being the azimuth of the loudspeaker and θS being the azimuth of the target source. As Q of reproduction signals increases, the relative signal magnitude of the loudspeaker nearest the target source direction increases, and the relative signal magnitudes (crosstalk) of the other loudspeakers decrease. Consequently, the perceived performance of the virtual source improves, and the listening region widens. However, these improvements occur at the cost of increasing the complexity of the system. For example, Figure 4.18 illustrates the virtual source position for Q = 1, 2, and 3 order horizontal Ambisonic reproduction. The results are evaluated from Equation (3.2.6) for an eight-loudspeaker configuration (Figure 4.12) and a fixed head. The frequency is f = 0.7 kHz, and the precorrected head radius is a′ = 1.25 × 0.0875 m. Figure 4.18 illustrates the results within 0° ≤ θS ≤ 90° only because of symmetry. Ideally, the perceived virtual source direction should be consistent with that of the
Figure 4.18 Virtual source position for Q = 1, 2, and 3 order horizontal Ambisonic reproduction with an eightloudspeaker configuration and a fixed head.The frequency is f = 0.7 kHz.
Horizontal surround with regular loudspeaker configuration 181
target source direction. The figure also indicates that the movement of the virtual source with frequency decreases as the order increases, and the upper frequency limit for the accurate reproduction of spatial information increases. This problem is addressed in Section 9.3.1.
In the case of the Q-order horizontal Ambisonic signals given in Equation (4.3.14a) for a regular loudspeaker configuration with the number of loudspeakers,
M 2Q 1 |
(4.3.64) |
Equation (4.3.63) verifies that the signals of other 2Q loudspeakers vanish when the target source is located at the direction of one loudspeaker except for the signal of this loudspeaker. In other words, the virtual source in loudspeaker directions is recreated by the single loudspeaker, the crosstalks among loudspeakers vanish, and localization performance enhances. The first-order reproduction with three loudspeakers discussed in Section 4.2.2 is a special example of this case. Here, the analysis is extended to the case of arbitrary Q-order reproduction. If M > (2Q + 1) loudspeakers are used in the Q-order reproduction, crosstalks among loudspeaker signals exist even if the target source is located at the direction of one loudspeaker. For horizontal Ambisonics with a regular loudspeaker configuration, the reproduced sound field exhibits symmetry against the rotation around the vertical axis.
Similar to the case of the first-order Ambisonics, the energy localization vector discussed in Section 3.2.2 is applicable to the analysis of the secondand higher-order Ambisonics. The mid-and high-frequency criteria for optimizing the energy localization vector can be used to choose κq in Equation (4.3.53) (Daniel et al., 1998).From Equation (4.3.53) and by using Equations (4.3.16) to (4.3.18), when
|
M |
|
2Q 1 , |
|
|
|
|
(4.3.65) |
|
|
|
|
|
|
|
|
|
the following equation is obtained: |
|
|
|
|
|
|
|
|
M 1 |
|
|
|
|
|
Q |
|
|
2 |
|
|
2 |
|
1 |
2 |
|
(4.3.66) |
Pow Ai |
S MAtotal |
|
2 q |
. |
||||
i 0 |
|
|
|
|
|
q 1 |
|
|
Therefore, for regular loudspeaker configurations, the overall free-field sound power at the origin in reproduction is independent from the target source direction θS.
Similar to the case of the first-order Ambisonics, by considering Equation (4.3.53), using Equations (4.3.16) to (4.3.18), and applying some simple formulas of trigonometric functions, when
|
M 2Q 2 , |
(4.3.67) |
|
the following equations are obtained: |
|
|
|
M 1 |
|
Q |
|
Ai2 |
S cos i 2MAtotal2 |
q q 1 cos S, |
(4.3.68) |
i 0 |
|
q 1 |
|
M 1 |
|
Q |
|
Ai2 |
S sin i 2MAtotal2 |
q q 1 sin S, |
(4.3.69) |
i 0 |
|
q 1 |
|
182 Spatial Sound
where κ0 = 1.
According to Equation (3.2.34), the direction θE of the energy localization vector satisfies the following equations:
|
M 1 |
|
|
|
|
|
Q |
|
|
|
Ai2 |
S cos i |
|
|
2 q q 1 cos S |
||||
rE cos E |
i 0 |
|
|
|
|
|
q 1 |
|
, |
M 1 |
|
|
|
|
Q |
|
|||
|
Ai2 |
S |
|
|
1 2 q2 |
|
|
||
|
i 0 |
|
|
|
|
|
q 1 |
(4.3.70) |
|
|
M 1 |
|
|
|
|
|
Q |
||
|
|
|
|
|
|
|
|
||
|
Ai2 |
S sin i |
|
2 q q 1 sin S |
|
|
|||
rE sin E |
i 0 |
|
|
|
|
|
q 1 |
. |
|
M 1 |
|
|
|
|
Q |
||||
|
Ai2 |
S |
|
|
1 2 q2 |
|
|
||
|
i 1 |
|
|
|
|
|
q 1 |
|
|
Then,
tan E tan S. |
(4.3.71) |
For a regular loudspeaker configuration in the horizontal plane, the direction of the energy localization vector for the Q-order Ambisonic signals given in Equation (4.3.53) matches that of the target source direction and is independent of κq (q = 1, 2…Q). This feature is desirable if the hypothesis of the energy localization vector theorem is valid above 0.7 kHz.
For the Q-order horizontal Ambisonics with a regular loudspeaker configuration, the condition that the overall power is given in Equation (4.3.65) is target-direction-independent requires M ≥ (2Q + 1) reproduction channels and loudspeakers. The result of the energy localization vector theorem given in Equation (4.3.67)requires one more channel and loudspeaker than that of Equation (4.3.65), i.e., it requires (2Q + 2) channels and loudspeakers at least. Therefore, using (2Q +2) loudspeakers at least is appropriate for the Q-order horizontal Ambisonics to consider the requirement of the energy localization vector. This number of channels and loudspeakers is minimal for the Q-order horizontal Ambisonic reproduction. The same conclusion is made in Section 4.3.2 for the first-order reproduction, and the conclusion is extended to the arbitrary Q-order reproduction. Therefore, Q = 1, 2, 3, and 4-order horizontal Ambisonics require 4, 6, 8, and 10 loudspeakers, respectively. This conclusion is consistent with the results derived from an evaluation of the width of the directivity of the signals for arbitrary Q-order reproduction (Xie and Xie, 1996). As stated in Section 4.2.2, a further increase in the number of loudspeakers may decrease the perceived difference between a virtual source in and off loudspeaker directions, but it may also cause some other problems.
The energy vector magnitude of the Q-order reproduction is evaluated from Equation (4.3.70) or more directly from Equation (3.2.36):
|
Q |
|
|
|
|
2 q q 1 |
|
||
rE |
q 1 |
|
. |
(4.3.72) |
|
Q |
|||
|
1 2 q2 |
|
||
q 1
Horizontal surround with regular loudspeaker configuration 183
For the conventional solution of the Ambisonic signals with κq = κ0 = 1 given in Equation (4.3.62), Equation (4.3.72) yields
rE |
|
2Q |
. |
(4.3.73) |
1 |
|
|||
|
2Q |
|
||
Q = 1, 2, and 3-order Ambisonics have the resultant rE of 0.667, 0.800, and 0.857, respectively. As Q increases, rE gradually approaches the unit value, i.e., it approaches the case of ideal reproduction.
If the hypothesis of the energy localization vector theorem is valid above 0.7 kHz, the criterion of maximizing the energy vector magnitude can be applied to choose κq in Equation (4.3.53). In Equation (4.3.72). According to the condition:
|
rE |
0 |
q 1, 2 Q, |
(4.3.74) |
|
|
|
||||
|
q |
|
|
|
|
a set of equations for κq is obtained |
|
|
|
||
q 1 2rE q q 1 |
0 |
q 1, 2 Q 1 |
(4.3.75) |
||
Q 1 2rE Q 0. |
|
||||
|
|
||||
The solution for these equations is expressed as
|
q |
|
|
|
q cos |
|
|
q 1, 2 Q. |
|
2Q 2 |
||||
|
|
|
The maximum energy vector magnitude is
rE max cos 2 2 .
Q
(4.3.76)
(4.3.77)
Q = 1, 2, and 3-order horizontal Ambisonics have (rE)max of 0.707, 0.866, and 0.924, respectively. As Q increases, (rE)max approaches the unit value. However, Equation (4.3.73) indicates that rE approaches the unit value as Q increases even for the conventional solution of Ambisonic signals. Therefore, the optimization of energy vector magnitude at mid and high frequencies may be unnecessary for choosing κq in the HOA.
Similar to the case of the first-order Ambisonics, the in-phase solutions for the secondor higher-order horizontal Ambisonic signals can be derived to restrain the out-of-phase crosstalks from the opposite channels. Various in-phase solutions are provided for the second- and higher-order horizontal Ambisonics. Some additional criteria are applied to derive the in-phase solutions, resulting in maximum rE, maximum front–back ratio, a maximum integrated front–back ratio, smooth and first-order extended solutions (Monro, 2000). If the following κq is chosen for the in-phase solution (Daniel, 2000; Neukom, 2007),
Q! 2 |
|
q Q q ! Q q ! , |
(4.3.78) |