Evaluation of the Q Function

A convenient method of evaluating the
Q
function over the parameter space quarter plane is presented. The
Q
function is first expressed as an infinite series. The
N
term truncated series
Q_{N}(a, b)
is used to approximate
Q(a, b)
for
a^{2} + b^{2} \leq R
where the choice of
N
depends on the accuracy desired and
R
is determined by considerations such as computer bit capacity, computational time, and accuracy. For
a^{2} + b^{2} > R
, alternate expressions are used. When
b - a \geq d
, we approximate
Q
by 0, and when
b - a \leq d
, we approximate
Q
by 1. The accuracy is dependent on the choice of the constant
d
. In the reniainder of the quarter plane,
a^{2} + b^{2} > R
and
| b - a < d
, and an efficient expression is used, but it is of limited accuracy (from 10
^-5
to 10
^-9
) near the line
a = b
.