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