must be known. Once the input noise is specified, the filter function

h(x,y) is the only unknown. Equation (2.28) becomes


E{no2(xoo) aso2(xo,yo)} > 0


 Ni2(u,v)H2u,v) du dv alff si(x,y)h(x-xo,y-yo) dxo dy0 2 > 0


where Ro max = 1/a


and the maximum signal-to-noise ratio at the output is obtained when

H(u,v) is chosen such that equality is attained. This occurs when


ff ni2(x,y) h(x-xo,y-yo) dxo dy = si(x,y).


(2.29)


(2.30)


Taking the Fourier transform of both sides and rearranging gives


 S(-u,-v)
H(u,v) ,

 INi(u,v) 12


exp -j(ux0+vy0)


Thus in an intuitive sense, the matched filter emphasizes the signal

frequencies but with a phase shift and, attenuates the noise

frequencies. This becomes clear when the additive noise is white. In

this case the noise power is constant at all frequencies and thus has

a power spectral density of


INi(u,v)12 = N/2


where N is a constant.


(2.32)


From equation 2.32 the form of the matched filter for the case of

white noise is


H(u,v) = Si(-u,-v)exp -j(uxo+vyo)


(2.33)


= S*i(u,v) exp -j(uxo+vyo)


(2.31)