# Improving Data Communication of Enhanced Loran Systems Using 128-_ary_ Polar Codes
## 2 Materials and Methods
### The Proposed 128-ary Polar Coding Scheme for eLoran
#### 2.2.2 Proposed Demodulation Method  
Conventional demodulation methods for the eLoran system, such as Signal Matching Correlation-Pulse Position Detection (SMC-PPD) [7] and Envelope Phase Detection-Majority Decision (EPD-MD) [8], were designed with significant constraints on computational complexity, rendering them suboptimal in terms of error performance. Given the substantial advancements in modern processing capabilities, this work employs an optimal demodulator, the correlation receiver, to achieve optimal error performance. To this end, we model the demodulation process in two hierarchical layers: (1) The first layer consists of the demodulation of each pulse. (2) The second layer consists of the demodulation of each GRI. Based on this two-layer framework, we propose a Soft-Soft (SS, soft demodulation with two layers) demodulation method based on correlation receiver to work with the SCL decoder of the 128-_ary_ polar code. Additionally, to demonstrate the source of the error performance gains in the proposed method, we also design Soft-Hard (SH, the first layer is soft demodulation and the second layer is hard demodulation) and Hard-Hard (HH, hard demodulation in two layers) demodulation methods to work with the RS code. In the first layer of the SS demodulation, the received sample sequence of the \(h\)-th (\(h\in\{1,\,2,\ldots,6\}\)) pulse in the \(m\)-th (\(m\in\{1,\,2,\ldots,M\}\)) received GRI is denoted as \(R_{m,h}=\left(r_{m,h}^{1},r_{m,h}^{2},\ldots,r_{m,h}^{T}\right)\). And a modulated standard eLoran pulse with modulation level \(s\in\{0,1,2\}\)(corresponding to \(1\)\(\upmu\)s advance, prompt and \(1\)\(\upmu\)s delay) is denoted as \(P_{s}=\left(p_{s}^{1},p_{s}^{2},\ldots,p_{s}^{T}\right)\), where \(T\) is the number of samples per pulse. Let \(\widetilde{y}_{m,h}^{s}\) denote the Euclidean Distance of \(R_{m,h}\) and \(P_{s}\). Therefore, under ideal synchronization conditions, \(\widetilde{y}_{m,h}^{s}\) is calculated as follows:  
\[\widetilde{y}_{m,h}^{s}=\sum_{t=1}^{T}\left(r_{m,h}^{t}-p_{s}^{t}\right)^{2}, \tag{12}\]  
where \(t\) represents the index of each sample in \(R_{m,h}\), and \(P_{s}\). Then, in the second layer of SS demodulation, let \(\widetilde{y}_{m}^{\sim}(j)\) denote the Euclidean Distance of the \(m\)-th received GRI with the modulation sequence corresponding to symbol value \(j\in\{0,\,1,\ldots,q-1\}\):  
\[\widetilde{y}_{m}(j)=\sum_{h=1}^{H}\widetilde{y}_{m,h}^{s(h,j)}, \tag{13}\]  
where \(H=6\) is the total number of pulses in a received GRI and \(S(h,j)\in\{0,1,2\}\) (corresponding to \(1\)\(\upmu\)s advance, prompt and \(1\)\(\upmu\)s delay) denote the modulation level of the \(h\)-th pulse (\(h\in\{1,\,2,\ldots,6\}\)) in a standard modulation sequence corresponding the symbol value \(j\in\{0,\,1,\ldots,q-1\}\). The result of the second layer of soft demodulation is denoted as \(\widetilde{y}_{1}^{M}=\left(\widetilde{y}_{1},\widetilde{y}_{2},\ldots, \widetilde{y}_{M}\right)\), in which \(\widetilde{y}_{m}=\left[\widetilde{y}_{m}(0),\widetilde{y}_{m}(1),\ldots, \widetilde{y}_{m}(q-1)\right]\). After SS demodulation, as introduced in Section 2.2.1, \(\widetilde{y}_{1}^{M}\) is de-punctured to reconstruct \(\boldsymbol{y}_{1}^{N}=(\boldsymbol{y}_{1},\boldsymbol{y}_{2},\ldots, \boldsymbol{y}_{N})(N=32)\), in which \(\boldsymbol{y}_{i}=\varnothing\) for \(i\in\{1,17\}\), and \(\boldsymbol{y}_{i}=\widetilde{y}_{k_{i}}(i\in\{1,2,\ldots,N\}\setminus\{1,17\})\), where \(k_{i}\in\{1,2,\ldots,M\}\) maps to the \(i\)-th non-punctured index. Then, let \(W(\boldsymbol{y}_{i}|j)(i\in\{1,\,2,\ldots,N\})\) denote the transition probability of \(\boldsymbol{y}_{i}\) with the condition that symbol \(j\in\{0,\,1,\ldots,q-1\}\) passesthrough an Additive White Gaussian Noise (AWGN) channel. \(W(\mathbf{y}_{i}|j)\) is obtained as follows:  
\[W(\mathbf{y}_{i}|j)=\left(\frac{1}{\sqrt{2\pi}\sigma}\right)^{T\cdot H}\cdot e^{ \frac{\mathbf{y}_{i}(j)}{2\sigma^{2}}}, \tag{14}\]  
where \(\sigma^{2}\) is the noise variance in the AWGN channel.