CRC-Aided Short Convolutional Codes and RCU Bounds for Orthogonal Signaling

TL;DR

This paper designs $Q$-ary CRC-aided convolutional codes optimized for orthogonal signaling over AWGN channels, deriving bounds and achieving performance within 0.6 dB of the RCU limit.

Contribution

It extends convolutional CRC code design to $Q$-ary alphabets for orthogonal signaling, optimizing distance spectrum and deriving performance bounds.

Findings

Best code within 0.6 dB of RCU bound at FER 10^{-4}

Derived saddlepoint approximation of RCU bound for noncoherent channels

Compared code performance to theoretical limits

Abstract

We extend earlier work on the design of convolutional code-specific CRC codes to $Q$-ary alphabets, with an eye toward $Q$-ary orthogonal signaling. Starting with distance-spectrum optimal, zero-terminated, $Q$-ary convolutional codes, we design $Q$-ary CRC codes so that the CRC/convolutional concatenation is distance-spectrum optimal. The $Q$-ary code symbols are mapped to a $Q$-ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We focus on $Q = 4$, rate-1/2 convolutional codes in our designs. The random coding union bound and normal approximation are used in earlier works as benchmarks for performance for distance-spectrum optimal convolutional codes. We derive a saddlepoint approximation of the random coding union bound for the coded noncoherent signaling channel, as well as a normal approximation for this channel, and compare the performance of our codes to these limits. Our best design is within $0.6$ dB of the RCU bound at a frame error rate of $10^{-4}$.