U. Glaeser

FIGURE 34.41
FIGURE 34.42
over alphabet A can always be characterized by a finite list of forbidden strings F = { w1,…,
wN}
of symbols
A
in A.
Defined this way, FT systems will be denoted by XF . The RLL constraints form a prominent class
of FT constraints, while DC-balanced constraints are typically not FT.
Design of constrained codes begins with identifying constraints, such as those described in the Introduction,
that achieve certain objectives. Once the system of constrained sequences is specified, information
bits are translated into sequences that obey the constraints via an encoder,
which usually has the form of
a finite-state machine. The actual set of sequences produced by the encoder is called a constrained code
and is often denoted C.
A decoder recovers user sequences from constrained sequences. While the decoder
is also implemented as a finite-state machine, it is usually required to have a stronger property, called
sliding-block decodablility, which controls error propagation [24].
The maximum rate of a constrained code is determined by Shannon capacity.
The Shannon capacity
or simply capacity of a constrained system, denoted by C,
is defined as
where N(
n)
is the number of sequences of length n.
The capacity of a constrained system represented by
a graph G can be easily computed from the adjacency matrix (or state transition matrix)
of G (provided
that the labeling of G satisfies some mildly innocent properties). The adjacency matrix of G with r states
and aij
edges from state i to state j,
1 ≤ i,
j ≤ r,
is the r × r matrix A = A(
G)
= { aij}
r×
r.
The Shannon capacity
of the constraint is given by
where λ(
A)
is the largest real eigenvalue of A.
The state-splitting algorithm [1] (see also [24]) gives a general procedure for constructing constrained
codes at any rate up to capacity. In this algorithm, one starts with a graph representation of the desired
constraint and then transforms it into an encoder via various graph-theoretic operations including
splitting and merging of states. Given a desired constraint and a desired rate p/
q ≤ C,
one or more rounds
of state splitting are performed; the determination of which states to split and how to split them is
q p
governed by an approximate eigenvector, i.e., a vector x satisfying A x ≥ 2 x.
Many other very important and interesting approaches are used to constrained code construction—far
too many to mention here. One approach combines state-splitting with look-ahead encoding to obtain
a very powerful technique which yields superb codes [14]. Another approach involves variable-length
and time-varying variations of these techniques [2,13]. Many other effective coding constructions are
described in the monograph [17].
© 2002 by CRC Press LLC
RLL (1,3) constraint.
DC-balanced constraint.
C
log2N( n)
= lim -------------------n
n→∞ C =
log2λ( A)