A Joint Source-Channel Coding Modulation Scheme for the Transmission of Gaussian Sources
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摘要: 提出了一种联合信源信道编码调制的高斯信源传输方案。在所提方案中，Lloyd-Max量化器将高斯信源序列量化为多元符号序列。对于多元量化符号序列，构造了与其相匹配的多元傅里叶变换对码，且调制方式采用与其相适应的多元脉冲幅度调制。特别地，调制后的多元符号序列以分组马尔可夫叠加的方式传输。此外，为了获得成形增益，还提出了星座几何成形方案。仿真结果表明：1)可以根据目标性能选择合适的传输方案；2)所提几何成形方案可以获得约0.3dB的误符号性能增益；3)所提几何成形方案在高SNR区域有更低的失真性能。
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关键词:
- 高斯信源 /
- 傅里叶变换成对码 /
- 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, which will not incur performance loss as long as the source block length goes to infinity. However, the SSCC scheme usually leads to a large buffer and long delay, and may cause error propagation in case of a single symbol error incurred in the communication channel. In order to alleviate these issues, Joint Source Channel Coding (JSCC) schemes have been investigated to transmit Gaussian sources. In this paper, a JSCC modulation scheme for the transmission of Gaussian sources is proposed, and a Gaussian source reconstruction scheme and its reconstruction expression are provided. Methods In this paper, the Gaussian source sequence is quantified as a sequence of M-ary symbols by a Lloyd-Max quantizer. For the M-ary quantization symbol sequence, the matching M-ary Fourier Transform Pair (FTP) code is constructed, and the modulation mode adopts the corresponding M-ary Pulse Amplitude Modulation (M-PAM). In particular, the modulated M-ary symbol sequences are transmitted in a block Markov superposition way, which constructs the Block Markov Superposition Transmission FTP (BMST-FTP) code. In addition, in order to obtain the shaping gain, the constellation Geometry Shaping (GS) scheme is also proposed. For the proposed source reconstruction scheme, the system output is the weighted average of the representative elements of the Lloyd-Max quantizer, which replaces the representative elements. Results and Discussions The simulations are conducted over M-PAM modulated AWGN channels using GF(3) and GF(5) BMST-FTP codes. For FTP codes employing random mapping, the WER approaches the Union Bound (UB) at high SNR. Similarly, the FTP codes with m repeated transmissions exhibit WER performances that approach the corresponding UBs. Furthermore, the WER performance of BMST-FTP codes with memory m matches UBs in the high SNR region ( Fig. 6 ). For 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, the GS can achieve an SER performance gain of approximately 0.3 dB (Fig. 8(a) ). In terms of distortion performance, the GF(3) BMST-FTP code outperforms the BMST-FTP code in the low SNR region, whereas in the high SNR region, the GF(5) BMST-FTP code performs better (Fig. 7(b) ). Furthermore, compared with other work, the GF(3) BMST-FTP code with m=1 has a similar performance, and the GF(5) BMST-FTP code with m=1 performs better (Fig. 7(b) ).Conclusions This work has proposed a joint source-channel coding modulation scheme for the transmission of Gaussian sources. In the proposed scheme, two types of BMST-FTP codes were constructed, each matched with a corresponding Lloyd-Max quantizer and M-PAM modulator. Additionally, a Gaussian source reconstruction scheme and its reconstruction expression were provided. Simulation results demonstrate that the appropriate transmission scheme can be selected according to the aim performance. The proposed GS scheme can obtain a gain of SER of about 0.3dB, which can improve the distortion performance of the waterfall area. -
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' $ 2.1. 根据公式(6)计算边界点$ {a}_{i}\left(0\leq i\leq M\right) $； 2.2. 根据公式(7)计算代表元$ {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 $时 2.1. 量化高斯信源$ {\boldsymbol{s}}^{\left(t\right)} $得到多元信息序列$ {\boldsymbol{u}}^{\left(t\right)} $； 2.2. 使用基本码$ {\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) $； 2.3. 当$ 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)} $； 2.4. 计算$ {\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) $ ● 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) $。正规图
上连接着第0到$ 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 $。● 迭代滑窗译码：对于t=0, 1, ···,L–1 1 局部初始化：如果$ 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 $层的其他边上的消息都按照均匀分布进行初始化。2 迭代：对于$ 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)} $。如果满足熵终止条件[24]，则把$ {\hat{\boldsymbol{u}}}^{(t)} $作为高斯信源
量化值的结果并退出迭代。● 解量化：根据$ {\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 4.6359, 1.8730, 0 1.4660, 0.5923, 0 1.6339
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