28-02-2013, 10:15 AM
Linear LLR Approximation for Iterative Decoding on Wireless Channels
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Abstract
On a fading channel with no channel state information
at the receiver, true log-likelihood ratios (LLR) are
complicated functions of the channel output. It is assumed in
the literature that the power of the additive noise is known and
the expected value of the fading gain is used in a linear function
of the channel output to find approximate LLRs. This approach,
however, is not optimal in the sense of bit error rate performance.
In this paper, we introduce a measure of accuracy for the
approximate LLRs based on their probability density function
and we show that this measure provides a very convenient tool
for finding good approximate LLRs. Assuming that the power of
the additive noise is known, and using the proposed measure, we
find a linear LLR approximation whose performance is extremely
close to that of the true LLR calculation on an uncorrelated
Rayleigh fading channel. These results are then extended to the
case that the noise power is also unknown and a performance
almost identical to the previous case is obtained.
INTRODUCTION
IN recent years, there have been many advances in iterative
decoding techniques and it has been shown that using
graphical codes such as LDPC codes [1] and turbo codes [2]
associated with iterative decoding, the Shannon limit on many
channels can be approached, e.g., see [3]. Therefore, these
codes have also been proposed for wireless fading channels
[4].
Application of LDPC codes on the Rayleigh fading channel
is pioneered in [4], where a detailed study of performance
and code design is conducted. This work is later extended to
time-selective complex fading channels [5], to Rician fading
channels [6], and also to Rayleigh block fading channels [7].
The application of turbo codes on Rayleigh fading channels
is also studied in [8].
OPTIMUM LINEAR LLR CALCULATION
To optimize the linear LLR approximating function and
obtain good BER performance, different methods can be
applied. The most trivial method is the direct exhaustive search
using Monte Carlo simulation which can be done for various
choices of for a given code. Then, the value of which gives
rise to the best BER curve is chosen. In the case of LDPC
codes, instead of Monte Carlo simulation, one may use density
evolution [3]. Although this approach of finding the optimum
is exact, it is too complex to be practical. Therefore, indirect
methods based on different measures of accuracy of LLRs will
be investigated.
CODE DESIGN UNDER MCLA
In this part we show that good LDPC codes can be
optimized for uncorrelated fading channels in the absence of
SI based on MCLA.
In general, two LDPC code design processes associated with
two measures of performance can be defined. One can be
defined as maximizing the threshold of the code over its degree
distributions given a target code rate [3], [4] and another one,
can be defined as maximizing the code rate over its degree
distributions given the channel LLR pdf [26]. In our case,
this pdf depends on the noise power, the pdf of