On Bit-Wise Decoders for Coded Modulation
Coded modulation is a technique that emerged as a response to the growing demand for high data rates. It was devised to achieve high spectral efficiency with high reliability, potentially approaching Shannon’s capacity. The main idea of coded modulation consists in combining error-correcting coding with higher-order modulation. For fast fading channels, the use of binary codes with a bit-wise interleaver between the encoder and the modulator together with a bit-wise decoder was proposed in order to increase the code diversity. This coded modulation technique is called bit-interleaved coded modulation (BICM) and, due to its flexibility of design and good performance, it gained popularity in various wireless communication systems, e.g., WiFi, LTE, etc.
The key component of a BICM scheme is a demapper that, based on the channel observations, calculates L-values (also known as log-likelihood ratios) for the coded bits. In this thesis, we take a closer look at different properties of L-values and their implications when analyzing the performance of bit-wise decoders for coded modulation systems.
First, the demapper is studied in terms of uncoded bit-error rate (BER) over the additive white Gaussian noise (AWGN) channel, i.e., when hard decisions on the bits are made directly at the output of the demapper. A new expression for the BER is formulated for arbitrary one-dimensional constellations. Next, two demapping strategies are considered: when the demapper calculates exact L-values and when L-values are calculated using the so-called max-log approximation. Closed-form expressions for the BER for 4-ary and 8-ary pulse amplitude modulation constellations with some of the most popular binary labelings are found. The numerical results show that there is no difference between the two strategies in terms of BER for any signal-to-noise ratio of practical interest.
We then study the performance of coded systems when the demapper uses the max-log approximation to calculate L-values. We consider a 16-ary quadrature amplitude modulation constellation labeled with a Gray code over the AWGN channel, as well as flat fading channels. At the receiver, a bit-wise decoder is used for decoding, which finds the maximum correlation between L-values and coded bits. This decoder performs the maximum likelihood decoding for the binary-input AWGN channel, however, it is suboptimal when higher-order modulation is considered. We show that the asymptotic loss in terms of pairwise error probability of such a decoder compared to the maximum likelihood decoder is bounded by 1.25 dB for any flat fading channel (including the AWGN channel as a special case). The analysis also shows that for the AWGN channel, the asymptotic loss is zero for a wide range of linear binary codes.
maximum likelihood decoder
Additive white Gaussian noise
quadrature amplitude modulation.
flat fading channel
pairwise error probability