184  Spatial Sound

then the loudspeaker signals are given as

Similar to Equation (4.3.40) for the first-order Ambisonics, for arbitrary Q-order Ambisonic signals given in Equation (4.3.53), although the virtual source direction is independent from

Atotal, Atotal determines the overall sound pressure or power in reproduction. For the constant amplitude normalization similar to Equation (4.3.41), Atotal can be obtained from the first

equation in Equation (4.3.56)

This equation is identical to the result of the first-order Ambisonics. For the constant power normalization similar to Equation (4.3.43), the following equation can be obtained:

For κq = κ0 = 1 given in Equation (4.3.62),

For κq expressed in Equation (4.3.77),

4.3.4  Discussion and implementation of the horizontal Ambisonics

From the point of multichannel loudspeaker signals, Ambisonics is a global amplitudebased mixing or panning method. From the point of information representation, Ambisonic encodes the spatial information of a sound field into a set of independent and universal signals, which are independent from the loudspeaker configuration in reproduction. Diverse and equivalent forms of independent signals are available. The number and configuration of loudspeakers for Ambisonic reproduction are flexible. Loudspeaker signals are derived from independent signals by decoding equations or matrices. Ambisonics can recreate the spatial information of sound field at the central listening position and up to a certain frequency limit. Therefore, Ambisonics is a universal and flexible system from the point of signal transmission and reproduction.

From the point of physics, Ambisonics is a series of systems with various orders that are based on the principle of spatial harmonics decomposition and each order approximation of the sound field. In Section 1.9.1, Ambisonics is a typical example of a gradual transition

Horizontal surround with regular loudspeaker configuration  185

from the approximate to the exact reconstruction of the target sound field. The lower-order spatial harmonics in Ambisonics represent the rough information of the spatial sound field, and the higher harmonics represent the detailed information of the spatial sound field. When the capacity of signal transmission (and storage) is limited, transmitting the information in the order from rough to detailed is reasonable. Ambisonics reduces the redundancy among the transmitted information and enables efficient use of the finite transmission capacity of the system because the independent information of the spatial sound field is extracted through spatial harmonic decomposition.

The lower-order Ambisonics can reconstruct the target sound field within a very small spatial region and limited frequency range; therefore, appropriate psychoacoustic methods should be applied to create a virtual source and other spatial auditory perceptions. The perception performance of the lower-order (e.g., the first-order) Ambisonics is limited. As the order increases, the spatial region and frequency range of the accurate transmission and reproduction of the information of the spatial sound field are extended gradually. Consequently, the spatial resolution and perceived performance of the reproduced sound field improve, and the listening region widens. However, the required number of independent signals and loudspeakers also increases as the order increases. Therefore, Ambisonics is a series of hierarchical systems in which the performance and complexity increase as the order increases. When the form of independent signals, such as the universal form of spatial harmonics given in Equation (4.3.49), is appropriately chosen, the HOA can be implemented by adding new independent signals to the lower-order Ambisonics and combining into the decoding equation. This characteristic enables the upward and downward compatibilities among different order Ambisonics. Previous studies also suggested some methods for the compatible transmission of mono, stereophonic, and first-order Ambisonic signals (Gerzon, 1985; Xie X.F., 1982).

The aforementioned features are held for horizontal and spatial Ambisonics (Section 6.4). Q-order horizontal Ambisonics involves an odd number of (2Q + 1) independent signals and requires Mmin = (2Q + 1) or more appropriately (2Q + 2) reproduction channels and loudspeakers; that is, the number of loudspeakers should be equal to or at least slightly larger than the number of independent signals. Further increasing the number of loudspeakers may improve the uniformity of the reproduced sound field but may cause new problems.

Regular horizontal loudspeaker configurations are discussed in the preceding discussion, which is true for most early domestic surround sounds. For irregular loudspeaker configurations, if loudspeakers are arranged front-back and left-right symmetrically so that each pair of loudspeakers is respectively located in two opposite directions alone the diameter of a circle, loudspeaker signals can also be derived easily. For example, the azimuths of four loudspeakers in a rectangular configuration are given as

For the first-order Ambisonics, the signals or decoding equation of the four loudspeakers is derived similarly to that in Section 4.3.2 (Gerzon, 1985):

where Ai(θS) with i = 0, 1, 2, 3 denotes ALF, ALB, ARB, and ARF, respectively. When θ0 ≠ 45°, the overall power of the four loudspeaker signals is no longer a constant; instead, it depends on the target source azimuth.