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

12.4 Linear Fractionally Spaced Equalizers (FSE) 687
If A1 (z) and A2(z) do not share any common root, then they are called coprime. The Bezout
identity states that if A1 (z) and A2 (z) are coprime, then there must exist two polynomials
such that
M
B1 (z) = I)uz -; and B2 (z) = L b2,;z -i
M
i=O
B1 (z)A 1 (z) + B2 (z)A2(z) = 1
The order requirement is that M :::: L - 1. The solution of B1 (z) and B2 (z) need not be unique.
It is evident from the classic text by Kailath 8 that the ZF design requirement of Eq. (12.51) is
an m-channel generalization of the Bezout identity. To be precise, let {H; (z), i = I, 2, . .. , m}
be a set of finite order polynomials of z-1 with maximum order L. If the m-subchannel transfer
functions {H; (z)} are coprime, then there exists a set of filters {F; (z)} with orders M :::: L - I
such that
m
L F; (z)H; (z) = z - u
i=I
(12.52)
where the delay can be selected from the range u = 0, I, ... , M + L - l. Note that the
equalizer filters {F; (z)} vary with the desired delay u. Moreover, for each delay u, the ZF
equalizer filters {F;(z)} are not necessarily unique.
We now describe the numerical approach to finding the equalizer filter parameters. Instead
of continuing with the polynomial representation in the z-domain, we can equivalently find
the matrix representation of Eq. (12.52) as
Ji [0]
!1 [1]
h1 [0] h m [0] 0
h1 [l] h m [l] f1 [M ]
h1 [0] h m [0]
0
1 +-- uth
h1 [L] h1 [I] h m [L] h m [ll 0
fm [0]
h1[L]
H : (L + M) x m(M + 1)
h m [L]
fm[ll
m[M]
'-..-'
m(M+l) x l
0
'--,-'
(M+ )xl L
(12.53)
The numerical design as a solution to this ZF design exists if and only if H has full row
rank, that is, if the rows of H are linearly independent. This condition is satisfied for FSE (i.e.,
m > 1) if M :::: L and {H; (z)} are coprime. 6