Efficient detection and decoding of q-ary LDPC coded signals over partial response channels
Andrea Marinoni
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Pietro Savazzi
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Paolo Gamba
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Dipartimento di Ingegneria Industriale e dell'Informazione,
University of Pavia
, Pavia,
Italy
In this study, we consider the ways for concatenating the intersymbol interference (ISI) detector with a q-ary lowdensity parity-check (LDPC) decoder for transmissions over partial response (PR) channels. LDPC codes allow achieving performance close to the channel capacity in additive white Gaussian noise channels, while designing receivers employing these codes for transmissions over channels affected by ISI is still an open issue. Turbo equalization schemes are considered with a novel joint message-passing-based receiver, which is derived from a recently proposed joint algorithm for binary LDPC codes. Simulation results provide performance evaluation of these systems over three different PR channels, together with an analysis of the trade-off between error-rate performance and complexity.
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equal weight, while irregular LDPC codes do not exhibit
this property. Non-binary (or q-ary) LDPC codes have
codewords (and also a PC matrix) whose symbols are
elements of the finite field GF(q), with q > 2. These
non-binary LDPC codes can allow significantly enhanced
performance with respect to the binary case, by
increasing the finite field dimension [7]. However, the decoding
complexity is O(Ntq2), where N is the block length, t is
the average column weight, and q is the alphabet size [7,8].
Recent papers reveal that q-ary LDPC may be applied to
magnetic recording channels, allowing reduced
complexity schemes [9] and robustness to burst errors [10], making
the q-ary solution comparable with the binary case.
Historically, considering an LDPC-coded system, the
most popular scheme to improve error-correction
performance over channels affected by ISI has been to serially
concatenate a soft-output detection algorithm and the
binary LDPC decoder. However, a greater performance
improvement can be achieved by incorporating the
channel detector in the iterative decoder: this implies a turbo
concatenation of the two system blocks and several papers
in literature call that configuration turbo equalization
(TE). Further, in [11-13], an LDPC-coded
detection-anddecoding system implemented by a joint algorithm based
on the message-passing (MP) algorithm is addressed.
In this article, we extend the joint
detection-anddecoding scheme to q-ary LDPC codes. Furthermore, we
compare the proposed approach to TE, paying particular
attention to the properties of the detector and the decoder
selected for each one. Some preliminary results of this
study are presented in [14].
The article is organized as follows. The first sections
introduce the system model for the different architectures
that are discussed in the article, namely TE, first turbo
equalization iteration (FTEI), and the proposed joint
MPbased architecture. For the FTEI scheme, no extrinsic
information is exchanged, since only the first iteration is
performed by doing soft separate detection and decoding.
Further, the computational complexity of each receiver
scheme considered is discussed in a dedicated section.
Finally, simulation results are given, and some remarks
about future research development conclude the article.
2 System model
In this section, we analyze the performance of a novel
receiver algorithm for q-ary LDPC-coded signals over PR
channels, comparing its performance with those obtained
by TE and FTEI schemes.
The basic system model is shown in Figure 1. The
qary LDPC encoder output is a length-N codeword =
[ i]i=1,...,N , i GF(q = 2p) i = 1, . . . , N such that
where H = {Hij}i=1,...,M,j=1,...,N , Hij GF(q = 2p)
is the parity-check matrix. The binary representation of
each codeword { }n is transmitted by an antipodal binary
pulse-amplitude modulator through a PR channel having
a memory of length bits.
Each receiver architecture that has been taken into
account employs a q-ary LDPC decoder and a BCJR
detector that can either be bit-based (BB) or
symbolbased (SB).
2.1 BB detector
The BB detector is represented by the standard BCJR
algorithm [3]. The symbol-wise a posteriori probabilities
(APPs) are approximated applying the BCJR algorithm to
the PR channel and then multiplying the APPs that form
a symbol [15,16]. The symbol-wise APPs are passed to the
q-ary LDPC decoder to initialize the a priori probabilities.
2.2 SB detector
The SB detector [15,17] modifies the way the
probability functions are updated when compared to the original
BCJR algorithm. Hoeher [17] develops a method, called
optimal subblock-by-subblock detector, in order to
calculate the APP of a block of p consecutive bits. Cheng
et al. [15] show that simplifications of the algorithm can
be made for the case of the binary-input ISI channels,
specifically for p .
Let a GF(q = 2p) be an information symbol. It is
possible to map each symbol in GF(q = 2p) to a distinct
sequence of p bits; in other terms, the binary
representation of a is b0p1 (...truncated)