Optimization of GDLD Codes

An optimum GDLD code is one that results in the lowest error-rates and computational complexity for the JVDD system. An analytical optimization method can be employed to obtain an optimum set of design parameters for a GDLD code at a fixed N and R. This analytical method is known as Response Surface Methodology (RSM) [76].

The RSM is a DOE technique that establishes a relationship between a set of control parameters, xi, and response parameter, y, of a complex system as depicted in Figure 4.29 [77].

It attempts to find a best-fit second order model to a set of experimental data and subsequently an optimum solution that maximizes/minimizes the target. The best-fit second order model of RSM can be expressed mathematically in terms of its control parameters as follow:

𝑦(𝑥₀, 𝑥₁, … , 𝑥_𝐿−1) = 𝐴₀+ 𝐴₁𝑥₀+ 𝐴₂𝑥₁+ 𝐴₃𝑥₂ + ⋯ + 𝐴_𝐿𝑥_𝐿−1+

𝐴_𝐿+1𝑥₀𝑥₀+ 𝐴_𝐿+2𝑥₀𝑥₁+ ⋯ + 𝐴_𝑀−1𝑥_𝐿−1² (4.23) The second-order model in equation (4.20) can be re-written in vector form given by (4.24).

𝑦(𝒙) = ∑ 𝐴_𝑘𝑥′_𝑘 = 𝒂^T

𝑘

𝒙́ (4.24)

where 𝑦(𝒙) = 𝑦(𝑥₀, 𝑥₁, … , 𝑥_𝐿−1), 𝒂^T = [𝐴₀ 𝐴₁ 𝐴₂ ⋯ 𝐴_𝑀−1] and 𝒙́ = [1 𝑥0 𝑥₁⋯ 𝑥_𝐿−1² ]^T. The best-fit second order RSM model is one which minimizes the cost function, 𝐽, as follow:

Figure 4.29: DOE technique attempts to optimize a response parameter given a set of control parameters.

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Alternatively, equation (4.25) can be fully expressed in terms of the vector and matrix elements as:

The partial derivative of 𝐽 with respect to 𝐴_𝑘 can be computed as follow:

90 requires equating the partial derivative of 𝐽 with respect to 𝒂 to 0. This solves for the optimum 𝐴_𝑘 values in the best fit second order model. To solve for the optimum set of control parameters of the complex system in Figure 4.29, the best-fit value of 𝐴_𝑘 can be substituted back into (4.27). The optimum 𝑥_𝑖 values, 𝑥_𝑖^∗, are computed by equating the partial derivative of 𝑦(𝒙) with respect to each 𝑥_𝑖 to 0 and subsequently solving 𝐿 equations with 𝐿 unknowns.

Table 4.3: DOE table consisting of control and response parameter sets

Run offset wrow σ FC

To establish a relationship between the design parameters of GDLD code and the performance and complexity of JVDD, offset, wrow and σ of the GDLD are used as control parameters. As we wish to obtain a set of code parameters that gives JVDD its best performance (lowest FER) and lowest complexity (smallest average number of survivors), the response parameter is defined as a metric, log 𝐹𝐶, in this optimization problem. This metric is shown in equation (4.28). The logarithmic function is used in the metric definition to avoid arriving at an invalid solution where the FER or average number of survivors is < 0.

log 𝐹𝐶 = log (𝐹𝐸𝑅 ∗ 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑜. 𝑜𝑓 𝑠𝑢𝑟𝑣𝑖𝑣𝑜𝑟𝑠) (4.28) Table 4.3 shows a sample set of control and response parameters used in the DOE for optimizing GDLD code at N = 512 and R = 0.90. The DOE was carried out in multiple phases. In

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Phase 1, the DOE was performed over a wide range of control parameters given by Table 4.4.

The optimum values of x0, x1 and x2 are obtained by computing the partial derivative of y with respect to x to 0 as shown in (4.27). The mathematical expression is then equated to 0 and the optimum set of control parameters are obtained as shown in (4.29).

Table 4.4 Range of control

Solving (4.30), the optimum set of control parameters in Phase 1 of the DOE is given by Table 4.5. A plot comparing the analytical values of the response parameter computed from Phase 1 of the DOE and experimental values gathered from computer simulations is shown in Figure 4.30(a). A close fit between the two sets of values is evident. The solution of the DOE can be further refined by improving the range of the control parameters based on the optimum solution obtained in Phase 1. The DOE can be repeated in multiple phases until eventually the best solution is obtained. For the example illustrated in this section, the DOE was carried out until Phase 4. The improved range for the control parameters used in this phase is shown in Table 4.6 and the corresponding set of optimum design parameters of GDLD code is presented in Table 4.7. A comparison of the analytical and experimental values for Phase 4 of the DOE is shown in Figure 4.30(b). Comparing the plots of Figure 4.30, a better fit is seen between the

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analytical and experimental curves in Phase 4 than in Phase 1. In Phase 4, J was also found to be lower than that in Phase 1 (J = 6.806 in Phase I and J = 0.772 in Phase 4).

Table 4.6 Range of control parameters used in Phase 4 of DOE

Table 4.7 Optimum set of control parameters obtained in Phase 4 of DOE

x0 (offset) 0, 100, 125 *x0 (*offset) 89.85

x1 (wrow) 50, 100, 150, 200 *x1 (*wrow) 98.65

x2 (σ) 100, 200, 225 *x2 (*σ) 162.28

(a) (b)

Figure 4.30: Comparison of experimental and analytical values obtained in (a) Phase 1 and (2) Phase 4 of DOE.

To confirm that the optimum code found in Phase 4 of the DOE is indeed an optimum one for a fixed N and R, its performance and complexity was compared against other good performing GDLD codes (found experimentally) having a different set of design parameters.

This is shown in an FER vs complexity plot in Figure 4.31. This plot shows the relationship between the performance and complexity of the JVDD when used with a fixed code. The performance-complexity relationship is obtained by varying Ʈ of the JVDD. On this plot, the curve that is closest to the origin signifies the best curve as it achieves the best performance with the lowest complexity. Therefore from Figure 4.31, it can be concluded that the GDLD code that has been optimized analytically performs best. This code optimization technique was subsequently used to optimize GDLD codes at various N and R values.

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Figure 4.31: FER vs Complexity plot of optimum GDLD code obtained analytically from DOE against other code design parameters.