# Adaptive Modem Based on LSTM-AutoEncoder with Vector Quantization
## 5 Results
### Effectiveness and Complexity
#### 5.4.2 Computational Complexity  
In this section, we study the computational complexity of the VQ-LSTM-AE model and compare it with existing techniques. Due to the offline training mode, generation of the vector quantization codebook can be obtained offline, and only the best matching value needs to be queried, so the training complexity can be ignored, and the focus is on the prediction complexity. Specifically, the complexity of the proposed solution is determined by the number of autoencoder layers, the number of neurons in each layer, the dimension of the input data, and the number of codewords in the codebook. Therefore, the computational complexity can be expressed as Formula (15), where \(L_{e}\) and \(L_{d}\) represent the number of  
Figure 8: Perplexity of VQ-LSTM-AE during training. (**a**) Second-order perplexity; (**b**) fourth-order perplexity; (**c**) eighth-order perplexity; (**d**) sixteenth-order perplexity; (**e**) thirty-second-order perplexity. layers of the encoder and decoder, \(Z_{e}\) and \(Z_{d}\) represent the size of the hidden layer of the encoder and decoder, \(C\) and \(D\) represent the dimension and size of the codebook, and \(I_{e}\) and \(I_{d}\) represent the dimension of the input data of the hidden layer of the encoder and decoder, respectively. \[C_{VQ-LSTM-AE}=L_{e}\cdot\left(I_{e}\cdot Z_{e}+{Z_{e}}^{2}\right)+C\cdot D+L_{d }\cdot\left(I_{d}\cdot Z_{d}+{Z_{d}}^{2}\right) \tag{15}\]  
Formula (15) shows that the complexity of VQ-LSTM-AE can be simplified to \(O\left(L_{e}\cdot\left(I_{e}\cdot Z_{e}+{Z_{e}}^{2}\right)\right)\). For the existing multi-layer perceptron-based model [6], its complexity depends on the number of hidden layers and the dimensions of input/output. Therefore, the complexity of the existing DAN-AE can be given by Formula (16), where \(L_{e}\) and \(L_{d}\) represent the number of layers of the encoder and decoder, \(Z_{e}\) and \(Z_{d}\) represent the size of the hidden layers of the encoder and decoder, and \(I_{e}\) and \(I_{d}\) represent the dimensions of the input data of the hidden layers of the encoder and decoder, respectively, which can be expressed as \(O(L_{e}\cdot I_{e}\cdot Z_{e})\). The complexity of the two is at the same order of magnitude, but VQ-LSTM-AE has an advantage because it has a smaller number of parameters. \[C_{DAN-AE}=O(L_{e}\cdot I_{e}\cdot Z_{e})+O(L_{d}\cdot I_{d}\cdot Z_{d}) \tag{16}\]  
Table 3 compares the number of parameters of DAN-AE and VQ-LSTM-AE. Regarding two, four, eight, and sixteen orders, VQ-LSTM-AE reduces the number of parameters by 70%, 66%, 58%, and 22%, respectively, compared with DAN-AE, but is greater than DAN-AE at 32 orders. Therefore, the method proposed in this paper is suitable for scenarios such as military satellite communications that pursue stability and a low bit error rate, in which the total amount of resources is limited and it is necessary to improve resource utilization. We also tested the effectiveness of our proposed method from the perspective of computing time. Computation time is an important indicator for evaluating classifier performance, especially in the era of big data, where real-time classification requires a large amount of data. Table 4 shows the training time of the VQ-LSTM-AE algorithm and DAN-AE. We observed that the VQ-LSTM-AE algorithm consumes significantly less time in training than the DAN-AE method, with an average training time of only 7.4% that of DAN-AE, and this trend becomes more obvious as the order increases. The average number of parameters of VQ-LSTM-AE in five cases is only 79.6% that of DAN-AE. This characteristic of de-linear correlation between computational time and the order of input data will show greater advantages and appeal when it is expanded to 64 or 128 orders in the future. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order** & **VQ-LSTM-AE(s)** & **DAN-AE** \\ \hline
2 & 528 & 1812 \\
4 & 656 & 1942 \\
8 & 1008 & 2442 \\
16 & 2096 & 2722 \\
32 & 5808 & 3762 \\ Average & 2019 & 2536 \\ \hline \hline \end{tabular}
\end{table}