B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

484 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Figure 9. 15
Wiener-Hopf
filter calculations.
m(t) + n(t)
m(t) + m.(t) + n 0 h(t)
Hopt(f) 1--------
----1 Hopt(f)
m(t)
m(t) + m.(t)
(a)
(b)
where Sm (f) is the signal PSD at the input of the receiving filter. The channel noise power Neb
appearing at the filter output is given by
where Sn (f) is the noise PSD appearing at the input of the receiving filter. The distortion
component acts as a noise. Because the signal and the channel noise are incoherent, the total
noise N 0 at the receiving filter output is the sum of the channel noise Ne b and the distortion
noise Nv,
(9.45a)
Using the fact that IA + Bi 2 = (A + B)(A* + B*), and noting that both S m (f) and Sn lf) are
real, we can rearrange Eq. (9.45a) as
oo
N = J
o
-oo
[IH (f) - Sm lf 2
) 1 S (f) + Smlf )Sn lf) ] d
lf
opt
Srlf)
r
S rlf)
(9.45b)
where S rlf) = S m lf) + Sn lf). The integrand on the right-hand side of Eq. (9.45b) is nonnegative.
Moreover, it is a sum of two nonnegative terms. Hence, to minimize N0 , we must
minimize each term. Because the second term Smlf)Sn lf)/Srlf) is independent of Hopt lf),
only the first term can be minimized. From Eq. (9.45b) it is obvious that this term is minimum
at zero when
_ S m lf)
H opt (f) - Srlf)
S m lf)
= -----
(9.46a)
For this optimum choice, the output noise power N0 is given by
No = J oo
Sm lf)Sn lf) df
-oo S rlf)
oo
Smlf)Sn lf)
= J
df
-oo S m lf) + Sn lf)
(9.46b)