A Joint Source-Channel Coding Modulation Scheme for the Transmission of Gaussian Sources
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摘要: 该文提出一种面向高斯信源传输的联合信源-信道编码调制方案。在所提方案中，首先采用Lloyd-Max量化器对高斯信源序列进行量化，输出多元符号序列；针对该多元量化符号序列，构造与之匹配的多元傅里叶变换对码，并选取适配的多元脉冲幅度调制完成符号调制。特别地，调制后的多元符号序列采用分组马尔可夫叠加方式实现传输。此外，为进一步获取成形增益，该文提出星座几何成形方案。仿真实验结果表明：(1) 可依据目标性能指标选取适配的传输方案；(2) 所提几何成形方案能够实现约0.3 dB的误符号性能增益；(3) 在高信噪比区域，该几何成形方案具备更优的失真性能。
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关键词:
- 高斯信源 /
- 傅里叶变换对码 /
- 分组马尔可夫叠加传输 /
- 几何成形
Abstract:Objective The Separated Source-Channel Coding (SSCC) scheme has been proven to incur no performance loss when the source block length tends to infinity. However, SSCC usually requires a large buffer and causes long delay. It may also lead to error propagation when a single symbol error occurs in the communication channel. To alleviate these issues, Joint Source-Channel Coding (JSCC) schemes have been studied for Gaussian source transmission. In this paper, a Joint Source-Channel Coding Modulation (JSCCM) scheme is proposed for Gaussian sources. A Gaussian source reconstruction scheme and its reconstruction expression are also provided. Methods The Gaussian source sequence is quantized into an M-ary symbol sequence by a Lloyd-Max quantizer. For the M-ary quantized symbol sequence, a matching M-ary Fourier Transform Pair (FTP) code is constructed. The corresponding M-ary Pulse Amplitude Modulation (M-PAM) scheme is adopted for modulation. The modulated M-ary symbol sequence is transmitted using Block Markov Superposition Transmission (BMST), forming a BMST-FTP code. In addition, a Geometric Shaping (GS) scheme is proposed to obtain shaping gain. In the proposed source reconstruction scheme, the system output is the weighted average of the representative elements of the Lloyd-Max quantizer, rather than a single representative element. Results and Discussions Simulations are conducted over Additive White Gaussian Noise (AWGN) channels with M-PAM modulation and BMST-FTP codes over Galois Field (GF) orders 3 and 5, denoted GF(3) and GF(5). For FTP codes with random mapping, the Word Error Rate (WER) approaches the Union Bound (UB) at high Signal-to-Noise Ratio (SNR). Similarly, FTP codes with m repeated transmissions show WER performance close to the corresponding UBs. The WER performance of BMST-FTP codes with memory m also approaches the UBs in the high SNR region ( Fig. 6 ). In terms of Symbol Error Rate (SER), the GF(3) BMST-FTP code outperforms the GF(5) BMST-FTP code (Fig. 7(a) ). For the GF(5) BMST-FTP code, GS provides an SER performance gain of approximately 0.3 dB (Fig. 8(a) ). In terms of distortion performance, the GF(3) BMST-FTP code performs better in the low SNR region, whereas the GF(5) BMST-FTP code performs better in the high SNR region (Fig. 7(b) ). Compared with other work, the GF(3) BMST-FTP code with m = 1 achieves similar performance, whereas the GF(5) BMST-FTP code with m = 1 achieves better performance (Fig. 7(b) ).Conclusions This work proposes a JSCCM scheme for Gaussian source transmission. In the proposed scheme, two types of BMST-FTP codes are constructed. Each code is matched with a corresponding Lloyd-Max quantizer and M-PAM modulator. A Gaussian source reconstruction scheme and its reconstruction expression are also provided. Simulation results show that an appropriate transmission scheme can be selected according to the target performance. The proposed GS scheme provides an SER gain of approximately 0.3 dB and improves distortion performance in the waterfall region. -
1 Lloyd-Max量化算法
输入：代表元个数$ M $ 输出：代表元集合$ G $ (1) 初始化：将集合$ G $和大小为$ M $的非空集合$ G' $中的元素随机初始化为不同的数值，初始化$ \varepsilon $。 (2) 迭代：当$ \underset{b\in \mathbf{G}}{\max }\underset{b'\in \mathbf{G}'}{\min }\left|\left|b-b'\right|\right| \gt \varepsilon $时，$ G\rightarrow G' $ (a)根据式(6)计算边界点$ {a}_{i}\left(0\leq i\leq M\right) $； (b)根据式(7)计算代表元$ {b}_{j}\left(0\leq j \lt M\right) $。 
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2 多元量化BMST-FTP码的编码算法
输入：$ {\boldsymbol{s}}^{\left(t\right)}\left(0\leq t\leq L-1\right) $ 输出：$ {\boldsymbol{c}}^{\left(t\right)}\left(0\leq t\leq L-1\right) $ (1) 初始化：当$ t \lt 0 $时，令数据块$ {\boldsymbol{u}}^{\left(t\right)}=\mathbf{0}\in \mathrm{GF}(q) $。 (2) 循环：当$ t=0,1,\cdots ,L-1 $时 (a) 量化高斯信源$ {\boldsymbol{s}}^{\left(t\right)} $得到多元信息序列$ {\boldsymbol{u}}^{\left(t\right)} $； (b) 使用基本码$ {\mathcal{C}}_{\mathrm{FTP}}\left[2n,n\right] $的编码算法对信息序列$ {\boldsymbol{u}}^{\left(t\right)} $编码得到码字$ {\boldsymbol{v}}^{\left(t\right)}\in \mathrm{GF}(q) $； (c) 当$ 1\leq i\leq m $时，码字$ {\boldsymbol{v}}^{\left(t\right)} $经第i个符号交织器$ \displaystyle\prod\nolimits_{i} $交织得到序列$ {\boldsymbol{w}}^{\left(i\right)} $； (d) 计算$ {\boldsymbol{c}}^{\left(t\right)}={\boldsymbol{v}}^{\left(t\right)}+\displaystyle\sum\nolimits_{1\leq i\leq m}{\boldsymbol{w}}^{\left(i\right)} $，得到码字$ {\boldsymbol{c}}^{\left(t\right)} $。 (3) 结尾：当$ t=L,L+1,\cdots ,L+m-1 $时，令$ {\boldsymbol{u}}^{\left(t\right)}=\mathbf{0}\in \mathrm{GF}(q) $并按照步骤(2)计算$ {\boldsymbol{c}}^{(t)} $。
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3 多元量化BMST-FTP码的迭代滑窗译码算法
输入：$ {\boldsymbol{y}}^{\left(t\right)}\left(0\leq t\leq L-1\right) $ 输出：$ {\hat{\boldsymbol{s}}}^{(t)}\left(0\leq t\leq L-1\right) $ (1) 全局初始化：假设$ {\boldsymbol{y}}^{\left(t\right)}\left(0\leq t\leq d-1\right) $已接收。只考虑信道约束，对于$ 0\leq t\leq d-1 $，由接收向量$ {\boldsymbol{y}}^{\left(t\right)} $计算后验概率
$ \Pr \left\{C_{j}^{\left(t\right)}=i|{y}^{\left(t\right)}\right\}\propto \dfrac{1}{\sqrt{2\text{π} }}\exp \left\{-\dfrac{{\left|\left|y_{j}^{\left(t\right)}-{x}_{i}\right|\right|}^{2}}{2}\right\},i\in \mathrm{GF}(q),{x}_{i}\in \mathcal{X}\left(0\leq j\leq N-1\right) $，与节点$ \boxed{+} $相连的半边上的消息$ P_{C_{j}^{\left(t\right)}}^{\left(|\rightarrow +\right)}\left(k\right) $初
始化为$ \Pr \left\{C_{j}^{\left(t\right)}=k|{y}^{\left(t\right)}\right\} $。节点$ \boxed{\text{Q}} $与节点$ \boxed{\text{FTP}} $相连边上的消息初始化为$ P_{u_{j}^{(t)}=i}^{\left(\text{Q}\rightarrow \text{FTP}\right)}=\dfrac{1}{\sqrt{2\text{π} }}\displaystyle\int \nolimits_{{a}_{i}}^{{a}_{i+1}}\exp \left\{-\dfrac{{x}^{2}}{2}\right\}\text{d}x,i\in \mathrm{GF}(q) $。正规图
上连接着第0～$ d-1 $层的其他边上的消息都按照均匀分布进行初始化。设置最大迭代次数$ {I}_{\mathrm{max}} \gt 0 $。(2) 迭代滑窗译码：对于t=0, 1, ···,L–1 (a) 局部初始化：如果$ t+d\leq L+m-1 $，由接收向量$ {\boldsymbol{y}}^{\left(t+d\right)} $计算后验概率$ P_{C_{j}^{\left(t+d\right)}}^{\left(|\rightarrow +\right)}\left(i\right),{x}_{i}\in \mathcal{X},k\in \mathrm{GF}(q) $。与节点$ \boxed{\text{Q}} $与节点$ \boxed{\text{FTP}} $相连
边上的消息初始化为$ P_{u_{j}^{(t)}=i}^{\left(\text{Q}\rightarrow \text{FTP}\right)}=\dfrac{1}{\sqrt{2\text{π} }}\displaystyle\int \nolimits_{{a}_{i}}^{{a}_{i+1}}\exp \left\{-\dfrac{{x}^{2}}{2}\right\}\text{d}x,i\in \mathrm{GF}(q) $。正规图上连接着第$ t+d $层的其他边上的消息都按照均匀分布
进行初始化。
(b) 迭代：对于$ I=0,1,2,\cdots ,{I}_{\text{max}} $ 2.1 前向递归：对于$ i=0,1,\cdots ,\min \left(d,L+m-1-t\right) $，在正规图的第$ t+i $层按以下顺序执行消息传递算法。 $ \boxed{+}\rightarrow \boxed{\prod }\rightarrow \boxed{=}\rightarrow \boxed{\text{FTP}}\rightarrow \boxed{=}\rightarrow \boxed{\prod }\rightarrow \boxed{+} $ 2.2 后向递归：对于$ i=0,1,\cdots ,\min \left(d,L+m-1-t\right) $，在正规图的第$ t+i $层按以下顺序执行消息传递算法。 $ \boxed{+}\rightarrow \boxed{\prod }\rightarrow \boxed{=}\rightarrow \boxed{\text{FTP}}\rightarrow \boxed{=}\rightarrow \boxed{\prod }\rightarrow \boxed{+} $ 2.3 硬判决和提前终止：对$ {\boldsymbol{U}}^{\left(t\right)} $的消息进行硬判决，得到高斯信源量化值的估计$ {\hat{\boldsymbol{u}}}^{(t)} $。如果满足熵终止条件[19]，则把$ {\hat{\boldsymbol{u}}}^{(t)} $作为高斯信源
量化值的结果并退出迭代。(3) 解量化：根据$ {\boldsymbol{U}}^{\left(t\right)} $的外信息和公式(8)计算高斯信源的重构$ {\hat{\boldsymbol{s}}}^{(t)} $，并把$ {\hat{\boldsymbol{s}}}^{(t)} $作为结果输出。
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表 1 不同SNR 下GS-5-PAM 星座集、归一化星座集和互信息
SNR (dB) 星座能量 星座集${\boldsymbol{\chi}}_{\mathrm{GS}} $ 归一化星座集 互信息 3 2.00 2.1762, 0.5024, 0 1.5406, 0.3557, 0 0.7852 4 2.51 2.4206, 0.6486, 0 1.5273, 0.4092, 0 0.8944 5 3.16 2.7028, 0.7750, 0 1.5199, 0.4358, 0 1.0098 6 3.98 2.9829, 1.0271, 0 1.4950, 0.5148, 0 1.1299 7 5.01 3.3469, 1.1524, 0 1.4950, 0.5148, 0 1.2539 8 6.31 3.7079, 1.4233, 0 1.4761, 0.5666, 0 1.3792 9 7.94 4.1603, 1.5970, 0 1.4761, 0.5666, 0 1.5064 10 10.00 4.6359, 1.8730, 0 1.4660, 0.5923, 0 1.6339
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