563489578934

Sec. 6–8 Matched Filters 473
r(t)=s(t)+n(t)
Delay
T
Delay
T
Delay
T
Delay
T
a 1 a 2 a 3 a N
r 0 (t)=s 0 (t)+n 0 (t)
Figure 6–20
Transversal matched filter.
or
(6–170)
where R n (t) is the autocorrelation of the input noise. Thus, the output-peak-signal-to-averagenoise-power
ratio is
N
2
s 2 c
0 (t 0 ) a a k s(t 0 - (k - 1)T) d
(6–171)
n 2 0 (t) = k = 1
N
a a k a l R n (kT - lT)
We can find the a k ’s that maximize this ratio by using Lagrange’s method of maximizing the
numerator while constraining the denominator to be a constant [Olmsted, 1961, p. 518].
Consequently, we need to maximize the function
(6–172)
where l is the Lagrange multiplier. The maximum occurs when 0M/0a i = 0 for all the i = 1,
2,...,N. Thus,
0M
0a i
M(a 1 , a 2 , . . . , a N ) = c a
N
= 0 = 2c a
N
n 2 N
0 (t) = a
k =1
a k s(t 0 - (k - 1)T) d s(t 0 - (i - 1)T)
N
- 2l a a k R n (kT - iT)
k=1
N
a
k = 1 l = 1
N
a
k = 1 l = 1
k = 1
-l a
N
a k a l R n (kT - lT)
2
a k s(t 0 - (k - 1)T) d
N
a
k = 1 l = 1
a k a l R n (kT - lT)
(6–173)