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Random Processes and Spectral Analysis Chap. 6
for i = 1, 2,..., N. But a N K=1 a ks(t 0 - (k - 1)T) = s 0 (t 0 ), which is a constant. Furthermore,
let l = s 0 (t 0 ). Then we obtain the required condition,
s(t 0 - (i - 1)T) = a
N
(6–174)
for i = 1, 2, . . . , N. This is a set of N simultaneous linear equations that must be solved to
obtain the a’s. We can obtain these coefficients conveniently by writing Eq. (6–174) in matrix
form. We define the elements
k=1
a k R n (kT - iT)
s i ! s[t 0 - (i - 1)T], i = 1, 2, Á ,N
(6–175)
and
r ik = R n (kT - iT),
In matrix notation, Eq. (6–174) becomes
s = Ra
where the known signal vector is
i = 1, Á , N
(6–176)
(6–177)
s 1
s 2
s = D T
o
s N
the known autocorrelation matrix for the input noise is
(6–178)
r 11 r 12 Á r 1N
r 21 r 22 Á r 2N
R = D
T
o o o
r N1 r N2 Á r NN
(6–179)
and the unknown transversal filter coefficient vector is
a 1
a 2
a = D T
o
a N
(6–180)
The coefficients for the transversal matched filter are then given by
a = R -1 s
(6–181)
where R -1 is the inverse of the autocorrelation matrix for the noise and s is the (known)
signal vector.