noise so that at the sampling location (xo,yo), the output signal
level will be as large as possible with respect to the output noise.
The signal-to-noise ratio is useful in the evaluation of system
performance, particularly in linear systems. In the matched filter,
the criterion of optimization is that the output signal-to-noise power
be a maximum. The input consists of a known signal s(x,y) and an
additive random noise n(x,y). The system is linear and space
invariant with impulse response h(xo,yo). To optimize the system or
filter, maximize the expression
Ro = So2(oyo)/E{no2(x,y)} (2.25)
where E{no2(x,y)} =ff no2(x,y) dx dy
at some point (xo,Yo). The problem is then to find the system h(x,y)
that performs the maximization of the output signal-to-noise ratio.
The output signal so(x,y) is
so(x,y) = //si(xo,Yo)h(x-xo,y-yo) dxo dyo (2.26)
and the output noise no(x,y) power is
ff ino(x,y) 2 dx dy = f !No(u,v) 2 du dv
= If Ni(u,v)l2 IH(u,v)|2 du dv. (2.27)
The signal-to-noise output power ratio becomes
|ff si(xo,yo)h(x-xo,y-yo) dxo dyo 2 (2.28)
R =
!Ni(u,v) 2 IH(u,v) 2
Thus to complete the maximization with respect to h(x,y), the power
spectral density or some equivalent specification of the input noise