Page 43 - Introduction to Information Optics

28 1. Entropy Information and Optics

1.5.1. SIGNAL DETECTION

It is well known that the signal-to-noise ratio at the output end of a
correlator can be greatly improved by matched filtering. Let us consider the
input excitation to a linear filtering system to be an additive mixture signal s(t)
and n(t); that is,

f(t) =s(t) +n(t), (1,93)

in which we assume that the noise is a stationary process. If we denote s 0(t) to
be the corresponding output signal and n 0(t) to be the output noise, then the
output signal-to-noise ratio, at f = 0, would be

H(v)S(v)dv
(1.94)
N ff 2
' \H(v)\ N(v)dv

2
where a is the mean-square value of the output noise and H(v) and JV(v) are
the filter transfer function and input noise power spectral density, respectively.
In view of Schwarz's inequality, the preceding equation can be shown as

( (1 90
'

in which the equality holds if and only if the filter function is

S*(v)
"<v)-K , (1-96)

where K is a proportional constant and the superasterisk represents the
complex conjugation. We note that, if the noise is stationary and white, then
the optimum filter function is

H(v) = XS*(v), (1.97)

which is proportional to the conjugate of the signal spectrum. This optimum
filter is also known as matched filter, since the transfer function H(v) matches
S*(v).