U. Glaeser

Distance δ min,C can be bounded as follows [30]:
© 2002 by CRC Press LLC
(34.40)
where M = max n,x∈C Σ mx mg n−m, i.e., M is the maximum absolute value of the interference. Note that M =
Σ n|g n|. We will assume that M < 1. The bound is achieved if and only if there exists an �, d(�) = d min, C,
for which Σm�mhn−m ∈{−1, 0, 1} for all n, and there exists an x ∈C such that Σmxmgn−m = + M whenever
Σm�mhn−m = ±1.
An Example
Certain codes provide gain in minimum distance on channels with ITI and colored noise, but not on
the AWGN channel with the same transfer function. This is best illustrated using the example of the
partial response channel with the transfer function h(D) = (1 − D)(1 + D) 2 known as EPR4. It is well
2
known that for the EPR4 channel dmin = 4. Moreover, as discussed in the subsection on “Constraints
for ISI Channels,” the following result holds:
Proposition 1. Error events �(D) such that
take one of the following two forms:
or
Therefore, an improvement of error-probability performance can be accomplished by codes which
eliminate the error sequences � containing the strings −1 +1 −1 and +1 −1 +1. Such codes were extensively
studied in [20].
In the case of ITI (Eq. 34.37), it is assumed that the impulse response to the reading head from an adjacent
track is described by g(D) = αH(D), where the parameter α depends on the track to head distance. Under
2 2
this assumption, the bound (34.40) gives ( 1– 4α)
The following result was shown in [30]:
2
≥
.
Proposition 2. Error events �(D) such that
take the following form:
δmin,C ≥ 1 – M
d 2 �
�( D)
�( D)
( )d min,C
( ) = dmin = 4
For all other error sequences for which d 2 (�) = 4, we have min x�C δ 2 (�, x) = 4(1 − 3α) 2 .
=
δ min
min
x∈C δ 2 2
( �,x ) = δmin �( D)
=
=
l−1
∑
i=0
k−1
∑
j=0
2
D 2j , k ≥ 1
( – 1)
i
D i
, l ≥ 3
d min
2
dmin ( 1– 4α)
2
= =
l−1
∑
i=0
( – 1)
i
D i
, l ≥ 5
4( 1– 4α)
2