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.