Focus of this letter is the oldest class of codes
that can approach the Shannon limit quite closely, i.e., lowdensity
parity-check (LDPC) codes, and two mathematical tools
that can make their design an easier job under appropriate
assumptions. In particular, we present a simple algorithmic
method to estimate the threshold for regular and irregular LDPC
codes on memoryless binary-input continuous-output AWGN
channels with sum-product decoding, and, to determine how close
are the obtained thresholds to the theoretical maximum, i.e., to
the Shannon limit, we give a simple and invertible expression
of the AWGN channel capacity in the binary input - soft output
case. For these codes, the thresholds are defined as the maximum
noise level such that an arbitrarily small bit-error probability
can be achieved as the block length tends to infinity. We assume
a Gaussian approximation for message densities under density
evolution, a widely used simplification of the decoding algorithm.
