# Analysis of TDMP Algorithm of LDPC Codes Based on Density Evolution and Gaussian Approximation
## 5 Simulation Results
### The Simulation Results of Density Evolution of the BP Algorithm and the TDMP Algorithm  
The performance curves of the BP algorithm and the TDMP algorithm at six code rates are shown in Figure 2a,b respectively. The values of the signal-noise ratio (SNR) of rate 1/2, rate 2/3A (or 2/3B), rate 3/4A (or 3/4B) and rate 5/6 are 0 to 2.5(dB), 0.5 to 3.0(dB), 1.0 to 3.5(dB), and 1.5 to 4.0(dB), respectively.  
As can be seen from Figure 2, as the SNR increases, the BER curve of the TDMP algorithm decreased faster than the BP algorithm. The following is a comparison of the convergence of the two algorithms under a single code rate. The code rate 2/3B and SNR 2.0(dB) were chosen in discussion. The relationship between BER and the iterations is shown in Figure 3. Obviously, the BER convergence speed of the TDMP algorithm was faster than the BP algorithm. When the two algorithms achieved the same BER, such as \(10^{-5}\), the number of iterations of the BP algorithm was 13, but the number of the TDMP algorithm was only 7.  
When expected BER reaches to \(10^{-5}\) at code rate 2/3B under different SNRs, iterations needed for the BP algorithm and the TDMP algorithm are shown in Figure 4. The BP algorithm and the TDMP algorithm cannot reach to expected BER within the maximum iterations, when SNR are 0, 0.5, 1.0 and 1.5 (dB), respectively. When SNR is greater than 1.5 (dB), the TDMP algorithm needs fewer iterations compared with the BP algorithm.  
Figure 3: The relationship between BER and iterations.  
Figure 2: Bit error rate (BER) performance comparison at six code rates of density evolution (DE); (a) Belief propagation (BP) algorithm; (b) TDMP algorithm.  
From the simulation results of Figures 3 and 4, we can prove that decoding convergence speed of the TDMP algorithm was faster than the BP algorithm from the perspective of density evolution theory.  
In order to verify the results, we simulated and analyzed the LDPC codes with 1/2 code rate under the 802.16 standard and the DTMB standard, respectively. The degree distributions of check nodes and variable nodes under the two standards are as follows:  
Under the 802.16 standard, the degree distribution of check nodes and variable nodes are shown in Equations (34) and (35)  
\[\lambda(x)=0.2895x+0.3158x^{2}+0.3947x^{5} \tag{34}\]  
\[\rho(x)=0.6316x^{5}+0.3684x^{6} \tag{35}\]  
Under the DTMB standard, the degree distribution of check nodes and variable nodes are shown in Equations (36) and (37)  
\[\lambda(x)=0.3382x^{2}+0.2618x^{3}+0.4x^{10} \tag{36}\]  
\[\rho(x)=0.1273x^{6}+0.8727x^{7} \tag{37}\]  
Figures 5 and 6 compare the BP and TDMP algorithms under the two standards. When the expected BER reaches to \(10^{-5}\), the number of iterations required under different SNRs is shown in Figure 5. When the SNR is 2 dB, the relationship between BER and iteration is shown in Figure 6.  
By comparing the simulation results under the two standards in Figures 5 and 6, it can be seen that when the SNR is 2 dB, the convergence speed of TDMP algorithm was faster than that of BP algorithm under the two standards. In addition, as can be seen from Figure 5, when the was greater than 1.5 dB, the number of iterations required by the TDMP algorithm is about 1/2 of the BP algorithm.