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 $10^{-4}$.
翻译:我们把先前设计革命代码特定CRC代码的工作扩大到Q美元字母,着眼于Q美元正方位信号。从远程频谱优化、零终止、Q美元共振代码开始,我们设计了Q美元共振代码,以使CRC/革命混合为远程频谱最佳。$美元代码符号被映射到Q美元正方位信号中,并发送到一个不兼容接收的AWGN频道。我们侧重于在设计过程中的$Q=4美元、1/2速率方位代码。随机编码联盟约束和正常接近在早期工作中用作远程频谱最佳共振代码的性能基准。我们从随机编码联盟连接和正常接近中得出一个随机组合连接连接为代码非相连接信号频道的马垫点近点,以及该频道的正常近点,并将我们代码的性能与这些限度进行比较。我们的最佳设计是在RCU的10-4约束率框架误差值为0.6美元方位的RCU-B。