where Er := Ro2 t and Es = So2 t are the reference and signal exposures
respectively. Written in this form, the first term in brackets does
not depend upon the signal strength. This term produces a linear
reproduction as in the ideal case. The second term in brackets
contains all contributions depending on the signal strength which
would produce a non-linear result. Thus, in this form, the first term
represents the signal coefficient and the second term the non-linear
noise coefficient. This provides a convenient method to determine the
signal-to-noise ratio for a given signal and reference exposure.
Several interesting cases are apparent from inspection. When Er
is large, the signal portion is dominant. The coefficient of Eac is
maximized when Er = Es. There is a trade-off between the need for
adequate signal and minimizing signal-to-noise ratio. This choice
depends upon factors such as the laser power available and the noise
level which can be tolerated. If Eac is maximized by setting Er = Es
and also set the sum Er + Es = E', the exposure value at the
inflection point, the values of Er and Es are determined uniquely.
The result is Er = Es = -C2/6C3. By substituting these values into
equation 6.12, it is seen that the noise term is zero. This choice of
values is not very useful for matched filtering because the zero noise
condition holds only if Es does not vary. No information can be
stored in this condition. If Es is allowed to vary, the noise
coefficient will increase. The worst case condition is when Es = Er/2
when the signal-to-noise ratio is 12.4. For many applications, this
noise is tolerable and this choice of Es and Er will be the preferred
choice.