## 5 Simulation Results and Analyses
The actual SNR can be the synthesis of the real AWGN noise and the equivalent AWGN noise caused by the inter symbol interference (ISI) due to the false symbol sampling. It can be improved via our proposed method in the several timing parameter updates in Equations (36) and (37). As long as the LDPC decoding is convergent within the range of decoding threshold, the proposed iterative timing recovery is surely to be globally convergent too, thus guarantee the effectiveness of the proposed timing recovery. In addition, it can also turn to the help of an additional Mueller-Muller algorithm [16] for better symbol synchronization, where it plays a similar role of timing synchronization tracking. Furthermore, the proposed algorithm can be further simplified by accumulating the metrics from LDPC variable nodes with high degrees, which holds reliable metrics in the LDPC decoding. It has been verified by numerical simulations with the similar method, except that the calculation of Equation (25) just includes the more reliable variable nodes of high degree. Because the results are exactly similar for the further simplified algorithm with reduced variable node calculation, we will not repeat the redundant result here for the purpose of saving space. In addition, the portion of the low degree variable nodes, that do not take part in the accumulation of Equation (25), should be less than 20% via several simulations. Otherwise, there will be too many fluctuations to affect the approximately convex of the objective function \(\Psi(\tau,v)\) shown in Figure 2, which greatly affect the effectiveness of the proposed iterative timing recovery. Suppose the portion of the high degree variable nodes to the whole code length is \(p\), the whole complexity can be further reduced to the \(p\) of the proposed algorithm. In our work, we mainly discuss the least complex BPSK modulation scheme. We mainly focus on the development of iterative timing recovery itself, so we just use it to illustrate our idea. Actually, long distances in satellite communications and the corresponding low signal-to-noise rate (SNR) transmission environments require power-efficient low-order modulation, such as BPSK, quadrature phase shift keying (QPSK) and some variations of it (e.g., offset quadrature phase shift keying (QQPSK), \(\pi/4\)-differential quadrature phase shift keying (\(\pi/4\)-DQPSK), etc.). Our scheme can be simply modified to satisfy these modulations, since the QPSK modulation series are actually the orthogonal implementation of two group of BPSK, and each branch of BPSK can adopt our proposed timing recovery scheme to accomplish the timing synchronization. The timing recovery scheme can be further improved with the well-known Costas loops for good implementation in practice. In these schemes, we obtain the metrics from the modulated signals in an orthogonal manner to feed the LDPC decoder, and the timing recovery parameters are iteratively calculated in LDPC decoding similar to those in our scheme. In addition, we also verify the roughly convex of the sum of the squares of selected LDPC decoding metrics with QPSK modulation similar as that shown in Figure 2. The efficient approximate stochastic gradient descent algorithm is also feasible. The complexity growth is still acceptable since only two branch of BPSK synchronization loops are required in the QPSK related schemes other than one branch in our scheme. The complexity increment is rather trivial. For more complex modulations, such as the Mary-quadrature amplitude modulation (\(M\)-QAM), Mary-phase shift keying (\(M\)-PSK) and even the Mary-amplitude phase shift keying (\(M\)-APSK) modulations, they can also be divided as the In-phase and Quadrature-phase mode of amplitude-shift-keying (ASK) modulations to properly implement the proposed timing recovery scheme. The complexities mainly lie in the metric extraction from the high-order modulations. They can be the further works in this aspect of research for low complex approach.