Useful Mathematical Tools for Capacity Approaching Codes Design

Abstract

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

Performance Study of a Class of Irregular LDPC Codes Based on Their Weight Distribution Analysis

“…Consider an ensemble of random codes with edgeperspective degree distributions λ(x) and ρ(x) given in Tables I and II. A custom software based on [23] (also used in [24] and [25] to design well performing rate compatible puncturing patterns for LDPC codes 1 on the basis of the results of [28] and [29]) was employed to simulate their performance over an AWGN channel, assuming a BPSK (Binary Phase Shift Keying) modulator. The belief propagation algorithm, also called message passing or sum-product algorithm, commonly employed for LDPC decoding, has been adopted, employing soft decision.…”

Nanosatellite-5G Integration in the Millimeter Wave Domain: A Full Top-Down Approach

“…This situation can be modeled by the average information rate [7], which considers, for each packet, the SINRs s 1 , ..., s K experienced by its K segments. To this aim, rather than using the ideal, and hence optimistic, Shannon bound, which might require the introduction of upper limits on the SINR to avoid possible overestimations of the performance [36], one may more realistically consider a quadrature phase shift keying (QPSK) modulation, for which a reliable capacity expression exists [38]. Accordingly, defining the coefficients a 1 ∼ = −1.29, a 2 ∼ = 0.93, a 3 ∼ = 0.01, and a margin b = 2.5 dB to account for the non-ideality of the code, the average information rate λ may be derived by first evaluating the sequence of rates [38]:…”

Impact of Segmentation and Capture on Slotted Aloha Systems Exploiting Interference Cancellation

“…Since an actual modulation is now employed, the usage of the Shannon bound in (7) to model the relationship between the equivalent rate and the SINR for the generic segment of a packet might result too optimistic. To overcome this limitation, we adopt a more suitable relationship, which has been specifically derived for binary and quadrature PSK modulations in [28]. According to this formulation, the rate Λ l achievable in the presence of a SINR X l when a QPSK modulation is used is approximated, for l = 1, ..., L, by [28]:…”

“…To overcome this limitation, we adopt a more suitable relationship, which has been specifically derived for binary and quadrature PSK modulations in [28]. According to this formulation, the rate Λ l achievable in the presence of a SINR X l when a QPSK modulation is used is approximated, for l = 1, ..., L, by [28]:…”