## 4 Results and Discussion
iterations of the IBPSO algorithm, the sparse AHs signal sets \(\sum_{QPSK}\) and \(\sum_{16 QAM}\) of the two signals are statistically obtained by:  
\[ \begin{array}{ll}\sum_{QPSK}=&\{x_{1},x_{2},x_{3},x_{4},x_{7},x_{8},x_{9},x_{1 0},x_{12},x_{13},x_{14},x_{15},x_{17},x_{18},x_{19},x_{20},x_{21},x_{22},x_{23}, x_{24},x_{25},x_{30},x_{32},\\ &x_{33},x_{37},x_{39},x_{44},x_{47},x_{49},x_{52},x_{54},x_{55},x_{56},x_{57},x_ {59},x_{60},x_{63},x_{66},x_{67},x_{68},x_{70},x_{78},x_{79},x_{81},\\ &x_{82},x_{83},x_{85},x_{88},x_{90},x_{91},x_{93},x_{96},x_{98},x_{99},x_{100} \}\end{array}  \tag{9}\]  
\[ \begin{array}{ll}\sum_{16 QAM}=&\{x_{1},x_{2},x_{3},x_{4},x_{7},x_{9},x_{10},x_{11},x_{12},x_{13},x_{15},x_{16},x_{17},x_{18},x_{19},x_{20},x_{22},x_{23}, x_{25},x_{26},x_{27},x_{28},\\ &x_{30},x_{33},x_{36},x_{37},x_{41},x_{44},x_{45},x_{48},x_{49},x_{51},x_{52},x_ {53},x_{55},x_{57},x_{59},x_{60},x_{61},x_{63},x_{64},x_{66},\\ &x_{67},x_{68},x_{69},x_{70},x_{71},x_{72},x_{73},x_{74},x_{75},x_{76},x_{77},x_ {78},x_{80},x_{81},x_{83},x_{86},x_{87},x_{92},x_{93},x_{94},\\ &x_{97},x_{99},x_{100}\}\end{array}  \tag{10}\]  
where the dimensions of the two sparse vectors are 55 and 65, respectively. The sparse amplitude histogram of two signals corresponding to different OSNRs in Figures 2 and 3 are shown in Figure 9. Although the sparse amplitude histogram has only a limited number of bin counts, the figure shows the main characteristics of the complete amplitude histogram. To contrast the effect of training and testing the PSO-DNN network structure with the original and the sparse input vector on the OSNR estimation accuracy, we first optimize the hidden layer neuron nodes with the grid search method when the sparse input vector is the input of the two signals,  
Figure 9: The sparse amplitude histogram of (**a**) PM-RZ-QPSK signal and (**b**) PM-NRZ-16 QAM signal obtained by IBPSO. and obtain the optimal neuron network structure as 55-47-37-1 and 65-47-43-1 for PM-RZ-QPSK and PM-NRZ-16 QAM signals, respectively. Then, we trained and tested the original and sparse input vector of the two signals, respectively, on the structure of their neural networks. The simulation results of OSNR for PM-RZ-QPSK and PM-NRZ-16 QAM signals are shown in Figures 10 and 11. It is clear from the Figures 10 and 11 that OSNR estimates are quite accurate and the mean estimation errors for PM-RZ-QPSK and PM-NRZ-16 QAM signals with 100 bin numbers are 0.29 dB and 0.39 dB, respectively, and the mean estimation errors with 55 and 65 bin numbers for the two signals are 0.29 dB and 0.37 dB. It should be noted that the maximum estimated error of all results does not exceed 0.5 dB. It can be concluded from the above results that the sparse and complete AHs datasets can achieve relatively consistent estimation accuracy in the DNN-based estimation system, however, the structure of neural network corresponding to the sparse dataset after dimensionality reduction is simpler, and the cost and hardware requirements are reduced. The effectiveness of the IBPSO method is proven, correspondingly. In order to prove the high accuracy of the proposed method in the OSNR estimation field, the same datasets are respectively input into the SVM and ANN systems, where SVM and ANN adopt the same training set and test set ratio. The simulation results and average estimation errors of the two signals in different OSNR estimation systems are shown in Figure 12 and Table 1. The maximum errors for the different OSNRs of the two signals are also shown in Table 1. It can be seen from the results that the OSNR estimation errors based on IBPSO-DNN can reach a higher accuracy as compared with the OSNR estimation system based on ANN and SVM for PM-RZ-QPSK and PM-NRZ-16 QAM signals. Figure 10: True versus estimated OSNRs and errors for PM-RZ-QPSK signal with (a) 100 bin numbers and (b) 55 bin numbers using PSO-DNN in different OSNRs.