For the right front quadrant the matrix elements are identical to the rear elements in the
’89 patent. The corrected elements as in the AES paper were used in the version of March
1997.
First consider the dip in the sum of the squares along the cs=0 axis. This dip exists
because of the use of G(lr) in LRR. This choice was entirely arbitrary – although it makes
implementation in analog circuitry easy. Ideally we would like to have a function GR(lr)
in this equation, and choose GS(lr) and GR(lr) in such a way as to keep the sum of the
squares of LRL and LRR constant along the cs=0 axis, and keep the output zero along the
boundary between left and center. This can be done. We would also like to be sure the
matrix elements are identical to the matrix elements in the right front quadrant along the
lr=0 axis. Thus we assume
LRL = cos(cs) – GS(lr)
LRR = -sin(cs) – GR(lr)
We want the sum of the squares to be one along the cs=0 axis,
(1-GS(lr))^2 + (GR(lr))^2 = 1
and the output to be zero to a steered signal, or as t varies from zero to 45 degrees,
LRL*cos(t) + LRR*sin(t) = 0
These two equations result in a messy
quadratic equation for GR and GS, which
is solved numerically in Figure 15. Use of
GS and GR as shown results in a large
improvement along the lr=0 axis, as
intended. However, the peak in the sum of
the squares along the boundary between
left and center remains. In a practical
design it is probably not very important to
compensate for this error, but we can
attempt to do so with the following
strategy. We will divide both matrix
elements by a factor, which depends on a
new combined variable based on lr and cs.
Call the new variable xymin. ( In practice
the divide can be replaced by a multiply by
the inverse of the factor described below.)
Figure 15: the numerical solution for GS and GR for
constant power level along the cs=0 axis, and zero
output along the boundary between left and center
