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)