68
D. Rocchesso: Sound Processing
Vector Base Amplitude
Panning
VBAP
3D panning
The most popular and easy way to spatialize sounds using loudspeakers is am-
plitude panning. This approach can be expressed in matrix form for an arbitrary
number of loudspeakers located at any azimuth though nearly equidistant from
the listener. Such formulation is called Vector Base Amplitude Panning (VBAP)
[72] and is based on a vector representation of positions in a Cartesian plane
having its center in the position of the listener. In the two-loudspeaker case
l
L L
R R
g
g
l
l
u
Figure 12: Stereo panning
(figure 12), the unit-magnitude vector u pointing toward the virtual source can
be expressed as a linear combination of the unit-magnitude column vectors l
L
and l
R
pointing toward the left and right loudspeakers, respectively. In matrix
form, this combination can be expressed as
u = L · g =
l
L
l
R
g
L
g
R
.
(35)
Except for degenerate loudspeaker positions, the linear system of equations (35)
can be solved in the vector of gains g. This vector has not, in general, unit mag-
nitude, but can be normalized by appropriate amplitude scaling. The solution of
system (35) implies the inversion of matrix L, but this can be done beforehand
for a given loudspeaker configuration.
The generalization to more than two loudspeakers in a plane is obtained by
considering, at any virtual source position, only one couple of loudspeakers, thus
choosing the best vector base for that position.
The generalization to three dimensions is obtained by considering vector
bases formed by three independent vectors in space. The vector of gains for
such a 3D vector base is obtained by solving the system
u = L · g =
l
L
l
R
l
Z
g
L
g
R
g
Z
.
(36)
Of course, having more than three loudspeakers in a 3D space implies, for any
virtual source position, the selection of a local 3D vector base.
As indicated in [72], VBAP ensures maximum sharpness in sound source
location. In fact: