# A High Throughput Hardware Architecture for Parallel Recursive Systematic Convolutional Encoders
## 3 Results
### Impact of the Parallelism Degree on the Data Rate: Case Study  
In this section, we present a case study that permits us to estimate the trend \(R_{IN}[k]\) for the RSC encoder described by the generators shown in Equation (24) by applying the analysis reported in Section 2.4. \[G(x)= \begin{pmatrix}1&;&\frac{1+x+x^{3}}{1+x^{2}+x^{3}}\end{pmatrix} . \tag{24}\]  
For such aim, it is necessary to estimate the dependency of the maximum clock frequency \(f_{clk_{MAX}}[k]\). The first difficulty is that the critical path might involve a different register--logics--register path for every different value of \(k\). Even if this problem is real, in Section 2.4 we illustrate that for increasing values of \(k\), only the networks relative to \(B^{\prime}\) and \(D^{\prime}\) matrices are increasing in complexity. In view of that, it is reasonable to deduce that for sufficiently high values of \(k\) the critical path is in one of the networks relatives to the rows of the matrices \(B^{\prime}\) and \(D^{\prime}\). Such hypothesis is confirmed by the FPGA implementation results shown in Table 2.  
Table 2 shows that for \(k\geq 5\) the critical path is included in the networks relative to \(B^{\prime}\) and \(D^{\prime}\) matrices. \begin{table}
\begin{tabular}{c c c|c c c} \hline \hline
**Parallel.** & \(f_{clk_{MAX}}\) & **Matrix Containing** & **Parallel.** & \(f_{clk_{MAX}}\) & **Matrix Containing** \\
**(k)** & **(MHz)** & **the Critical Path** & **(k)** & **(MHz)** & **the Critical Path** \\ \hline
1 & 784.9293564 & B & 7 & 523.5602094 & B \\
2 & 545.2562704 & A & 8 & 489.2367906 & B \\
3 & 564.6527386 & C & 9 & 366.9724771 & B \\
4 & 548.5463522 & C & 10 & 329.7065612 & D \\
5 & 510.9862034 & D & 11 & 387.4467261 & D \\
6 & 605.3268765 & D & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the case study synthesis on Zynq 7000 xC7z010 clg400-1 field programmable gate array (FPGA). Figure 6: RSC encoders bit error rate \((BER)=BER(\frac{E_{k}}{N_{0}})\) curves. Even if it is probable that for different values \(k>11\) the critical path is not definitely included in the networks relative to a single matrix, it is necessary to consider that the both the \(B^{\prime}\) and \(D^{\prime}\) networks are implemented by using the same topology, whose propagation delay/parallelism grade trend can be described by Equation (22). For such reason, it is possible to approximate the function \(f_{clk_{MAX}}[k]\) by using \(f_{clk_{MAX}}[k]\). The latter describes the dependency of the maximum clock frequency as function of \(k\) of the paths relative to a generic matrix \(M^{\prime}\). For such reason, \(f_{clk_{MAX}}[k]\) can be estimated by using the expression described by Equation (25):  
\[f_{clk_{MAX}}[k]\approx f_{clk_{MAX}_{M^{\prime}}}[k]=\frac{1}{T_{A}+T_{B} \cdot\lceil log2(\Omega_{M^{\prime}}[k])\rceil}, \tag{25}\]  
where, \(T_{A}\) and \(T_{B}\) are parameters to determine, and where, similarly to Equation (22), the term \(\Omega_{M^{\prime}}[k]\) models the maximum number of unitary elements among the all networks relative to the matrix \(M^{\prime}\) for a fixed value of \(k\) (note the absence of the subscript \(i\)), as described by Equation (26). \[\Omega_{M^{\prime}}[k]=max\{\Omega_{M^{\prime}i}[k]\}. \tag{26}\]  
Parameters \(T_{A}\) and \(T_{B}\) of Equation (25) can be estimated by using data shown in Table 2 through a mean square error (MSE) interpolation. In particular, the trend \(f_{clk_{MAX}}[k]\) was approximated with trend of the networks relative to the matrix \(D^{\prime}\). This is due to the fact the maximum number of unitary elements among the rows \(D^{\prime}_{i}\) is always higher than the maximum number of unitary elements among the rows \(B^{\prime}_{i}\) for each value of \(k\in(1,...,100)\). It was demonstrated through a Matlab(r) simulation which calculated the maximum number of unitary elements for \(B^{\prime}_{i}\) and \(D^{\prime}_{i}\) depending on different values of \(k\).