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

7 .5 Digital Receivers and Regenerative Repeaters 36 1
The samples of p 0 (t) at t = kT b are
N
Po (kT b ) = L CnP r (kT b - nT b ) k = 0, ±1, ±2, ±3, . .. (7.51a)
n=-N
By using a more convenient notation P r (k] to denote P r (kT b ) and p 0 [k] to denote p 0 (kTb ),
Eq. (7.5 1a) can be expressed as
N
Po[k] = L Cnp,[k - n]
n=-N
k = 0, ±I, ±2, ±3, ...
(7.51b)
Nyquist's first criterion requires the samples p 0 [k] = 0 for k -:/= 0, and p 0 [k] = I for k = 0.
Upon substituting these values in Eq. (7.51 b ), we obtain a set of infinite simultaneous equations
in terms of 2N + 1 variables. Clearly, it is not possible to solve all the equations. However, if
we specify the values of p 0 [k] only at 2N + I points as
!I k = 0
Po[k] = 0 k = ± 1, ± 2, . .. , ±N
(7.52)
then a unique solution exists. This assures that a pulse will have zero interference at sampling
instants of N preceding and N succeeding pulses. Because the pulse amplitude decays rapidly,
interference beyond the Nth pulse is not significant for N > 2, in general. Substitution of the
condition (7 .52) into Eq. (7 .51 b) yields a set of 2N + I simultaneous equations for 2N + 1
variables. These 2N + 1 equations can be rewritten in the matrix form of
C-N
0 C-N+l
p r [0] p,[-1] p,[-2N + l] Pr [-2N]
p,[I] p,[0] P
0
r [-2N + 2] p,[-2N + 1]
C- )
1 co
0
CJ
P r [2N - 1] P r [2N - 2] P r [0] Pr [-1]
Pr[2N] p,[2N - 1] p,[l] p,[0]
0 CN- 1
'-,,-' P,
CN
Po
.__,__,
C
(7.53)
In this compact expression, the (2N + 1) x (2N + 1) matrix P r has identical entries along all
the diagonal lines. Such a matrix is known as the Toeplitz matrix and is commonly encountered
in describing convolutive relationships. A Toeplitz matrix is fully determined by its first row
and first column. It has some nice properties and admits simpler algorithms for computing its
inverse (see, e.g., the method by Trench 7 ). The tap gain q can be obtained by solving this set