# Hybrid Window Decoding for Joint Source Channel Anytime Coding System
## 4 Simulation and Analysis  
In this section, we explore the decoding complexity, anytime properties and error correction performance of the proposed JHWD algorithm with both numerical analyses and simulations. The benchmark algorithms are the joint expanding window decoding (JEWD) algorithm and the partial joint expanding window decoding (PJEWD) algorithm for JSCAC systems in [17; 18]. We also compared the proposed algorithm with the traditional joint source channel decoding (JSCD) for block D-LDPC-based JSCC systems [10] and the SC-LDPC-based JSCC scheme with sliding variable-window decoding (SVWD) [16]. The additive white Gaussian noise (AWGN) channel and the binary phase-shift keying (BPSK) modulation are adopted. The simulation parameters are set as follows: \(Q^{sc}=4\), \(A^{sc}=12\), \(Q^{cc}=3\), \(A^{cc}=3\), \(R^{sc}=0.25\), \(R^{cc}=0.5\), \(\lambda^{sc}=0.5\), \(\lambda^{cc}=0.1\), \(I_{\max 1}=30\), \(I_{\max 2}=5\), \(p=0.02\), and \(E_{b}/N_{0}=2\) dB for anytime properties analyses. We first analyze the asymptotic average message updates of the SC-RA-based JSCAC system as the decoding delay using different decoding schemes. As shown in Figure 3, as the decoding delay increases, the average message updates of all the three decoding schemes show slow and nearly linear growth trends, and their differences become more significant. The right Y-axis of Figure 3 shows the average message updates reduction ratio of PJEWD and the proposed JHWD compared to the traditional JEWD algorithm. As \(t\) is greater than 600, the growth rates tend to flatten out. In the decoding delay region of [100, 1000], the message update reduction ratio of PJEWD to JEWD increases from 26% to 48%; while the reduction rate of the proposed JHWD to JEWD increases from 58% to 94%. In other words, the message updates reduction ratio of the proposed JHWD is about two times higher than the PJEWD algorithm. In Figure 4, the asymptotic anytime properties of SC-RA-based JSCAC with different decoding schemes are analyzed. The anytime properties of a single block in the first sliding window region is observed from \(t=14\). It can be seen that with the same coupling length of \(\gamma=5\), the bit error probability curves of the JEWD and PJEWD algorithm have similar descent rates, while the curve slope of the proposed JHWD algorithm is slightly slower. Figure 4 also shows the error probabilities of the proposed JHWD with different \(\gamma\). It can be seen that the smaller the coupling length, the larger the curve slope. However, in the finite length regime (as seen in Figure 5), a smaller coupling length tends to lead a higher error floor. Correspondingly, as shown in Figure 3, the greater the coupling length, the higher the decoding complexity. These results indicate that the anytime property and decoding complexity of the proposed JHWD are related to the coupling length \(\gamma\), and a proper coupling length should be selected to trade off the error performance and decoding complexity. In Figure 5, the simulation performances of a single message block is observed as a function of decoding delay within the first sliding window region, from \(t=14\), with a source block length of \(N^{sc}=64\) bits and \(E_{b}/N_{0}\) is 2 dB. As shown in Figure 5a, the bit error rate (BER) curve of the proposed JHWD algorithm is located between the JEWD and PJEWD algorithms, which is consistent with the numerical results in Figure 4; and it lastly reaches the error floor at \(t=26\) (also within the sliding window size of \(3\gamma-2\)). Moreover, the error floor of the proposed JHWD algorithm is closer to that of the PJEWD algorithm. When the coupling length is larger, i.e., \(\gamma=8\), the slops of the BER curves are slightly smaller than those of \(\gamma=5\) before the time slot \(t=5\), but can achieve lower BER performance. when \(t=21\), the BER of the proposed JHWD algorithm is lower than \(10^{-6}\). Figure 5b shows the average message updates of a single message block as the decoding delay.