# Concatenated Coding for GNSS Signals in Urban Environments
## 2 System Model  
A block diagram of the concatenated coding and decoding system for the GNSS messages in urban environments considered in this work is shown in Figure 1.  
The channel code is a concatenation of a non-binary outer encoder \((n1,k1)\) over \(\mathrm{GF}(2^{m})\) and a binary inner encoder \((n2,k2)\). Encoding is performed in two steps. First, a \(k\)-bit binary message \(a=(a_{0},a_{1},\cdots,a_{k-1})\) is grouped into a \(k1\)-symbol message sequence \(\mathbf{u}\mathbf{=}(\mathbf{u}_{0},\mathbf{u}_{1},\ldots,\mathbf{u}_{k1-1})\), where \(\mathbf{u}_{j}\in\mathrm{GF}(2^{m})\), \(\mathbf{u}_{j}=(u_{j,0},u_{j,1},\cdots,u_{j,m-1})\). Then, the \(k1\) symbols are sent to the outer encoder to generate an \(n1\)-symbol codeword \(\mathbf{c}1=(\mathbf{c}1_{0},\mathbf{c}1_{1},\cdots,\mathbf{c}1_{n1-1})\), which is subsequently converted into a \(k2\)-bit binary sequence, where \(k2=n1\times m\). Second, the output \(k2\)-bit binary sequence is encoded by the inner encoder  
Figure 1: Concatenated coding and decoding system for GNSS messages.  
to obtain an \(n2\)-bit codeword \(c2=(c2_{0},c2_{1},\cdots,c2_{n2-1})\). Between the two encoders, an interleaver is implemented to distribute the transmitted RS symbols so that the errors are randomized [25]. After the symbol mapper, the symbols are transmitted through a fading channel. The output of the channel is given by:  
\[y=hs+n \tag{1}\]  
where \(y,h,s\), and \(n\) are the received symbols, the fading channel coefficients, the transmitted symbols, and the independent and identically distributed (i.i.d.) ambient additive noise, respectively.  
In this paper, the two-state LMS model was used to model the fading features in urban environments. This model is a state-oriented channel model based on a Markov chain that defines the reception conditions as good and bad states, where a good state represents a line-of-sight to a moderate shadow condition, and a bad state represents a moderate-to-deep shadow condition. Within each state, the signal variation \(h\) is assumed to follow the Loo distribution:  
\[h\sim\text{Loo}(M,\Sigma,MP) \tag{2}\]  
The Loo distribution assumes that the received signal consists of the direct signal and the multipath signal, and the probability density function is given by:  
\[f_{Loo}(h)=\frac{8.686h}{{\sigma_{L}}^{2}\Sigma\sqrt{2\pi}}\int_{0}^{\infty} \frac{1}{z}\exp\left[-\frac{h^{2}+z^{2}}{2{\sigma_{L}}^{2}}\right]\exp\left[- \frac{(20\log z-M)^{2}}{2\Sigma^{2}}\right]I_{0}\left(\frac{hz}{{\sigma_{L}}^ {2}}\right)dz \tag{3}\]  
where \(z\) is the amplitude of the direct signal, \(M\) and \(\Sigma\) are the mean and standard deviation of the log-normal distribution for the direct signal's amplitude in dB, respectively, and \(2{\sigma_{L}}^{2}\) (\(2{\sigma_{L}}^{2}=10^{MP/10}\)) is the average power of the multipath signal. The parameters of the Loo distribution are constant when the environment scenario and satellite elevation are fixed, and the durations of each state are associated with the terminal speed.  
Additive noise \(n\) is assumed to follow a complex Gaussian distribution with mean zero and standard deviation \(\sigma_{N}\):  
\[n\sim N(0,{\sigma_{N}}^{2}) \tag{4}\]