## 5 Performance Analysis and Discussion  
In this section, we employ numerical simulations to quantify the system performance in BER. The BLS for generating the extended SAC labels is centered at \(\lambda_{c}\) and has a flat spectrum over bandwidth \(v\). OBPFs for de-multiplexing \(M\) WDM channels have a bandwidth of \(v\)/\(M\) and central frequency of \(\lambda_{c}\)-\(v\) (2/\(1\)-\(M\))/\(2M\), where \(j=1\), \(2\), \(\ldots\), \(M\). AWGs with the channel spacing of \(v\)/\(MN\) and free spectral range (FSR) of \(v\)/\(M\) are used to generate \(N\) labels in each channel. We consider the packet -switching process of decoding label \(\mathbf{E}(1,1)\) from the label stack. Using the definition in reference [22] and the correlation property of (2), the photocurrent \(I\) generated from decoding the optical label is expressed as follows. \[I\ =\ \big{[}\mathbf{E}(1,1)-\overline{\mathbf{E}}(1,1)\big{]}\mathbb{S} \sum_{i=1}^{\kappa}\mathbf{E}(i,1)\ =\ \frac{RP_{\text{sr}}}{MN}\cdot\frac{(N+1)}{2} \tag{5}\]  
where \(R\) denotes the photodiode responsivity, \(P_{sr}\) is the received optical power, and \(\kappa\) is the effective label number. When a label is converted from an optical to electrical signal by a balanced detector, PIN, thermal noise, and shot noise are the dominant performance degradations. The mathematical expression of PIN variance \(\sigma_{\text{PINN}}^{2}\) is deducted as follows. \[\begin{array}{c}\sigma_{\text{PINN}}^{2}\ =\ \frac{R^{2}P_{  
where \(B\) is the electrical bandwidth. The thermal noise and shot noise variance are defined as:  
\[\sigma_{\text{Th}}^{2} = S_{\text{Th}}B \tag{7}\] \[\sigma_{\text{Shot}}^{2} = 2 eBI = \frac{eBRP_{\text{sr}}(N+1)}{MN} \tag{8}\]  
where \(S_{\text{Th}}\) is the PSD of thermal noise in W/Hz and \(e\) is the elementary charge of a single electron. When **E**(1,1) is decoded, only the extended SAC labels with the index \(i\) = 1 enter the decoder, while the OBPF removes the labels of \(i\) \(\
eq\) 1. The relation between the effective label number and the total label number \(K\) is described as follows. \[\kappa = \left\lfloor\frac{K}{M}\right\rfloor+1 \tag{9}\]  
where \(\lfloor.\rfloor\) denotes the floor function. Since the payload bits are modulated in OOK, the label stack only presents when bit \"1\" is sent. The BER \(P_{\text{B}}\) is written as  
\[P_{B} = \frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{t/2}e^{-(x-I)^{2}/2 \sigma^{2}}dx = \frac{1}{2}\text{erfc}\bigg{(}\frac{I}{2\sqrt{2}\sigma}\bigg{)} \tag{10}\]  
where \(\sigma^{2}\ =\ \sigma_{\text{PINI}}^{2}+\sigma_{\text{Th}}^{2}+\sigma_{\text{Shot}}^ {2}\), which denotes the total noise variance. Figure 8 shows the BER comparisons of two labeling scenarios in GMPLS networks. Parameters used for numerical simulations are \(P_{\text{sr}}\) = -10 dbm, \(R\) = 0.95 A/W, \(B\) = 200 MHz, \(\lambda_{c}\) = 193.4 THz, \(v\) = 3.75 THz, and \(S_{\text{Th}}\) = 6.4 \(\times\) 10\({}^{-24}\) W/Hz. For a given stacked label number \(K\), the BER values for extended SAC labels are smaller than MLS labels. The reason is that the number of overlapping chips between labels is reduced by adding zeros to extend the code vector. Furthermore, varying the code length only causes a slight difference in system performance. However, the labels with a long code length are suitable for switching packets in a large-scale network, as the number of path connections in the network is relatively large. Figure 9 shows the relationship between BER and the received optical power at the core node's decoder for \(K\) = 15. At low power levels, both MLS and extended labels suffer from thermal noise and the performance of MLS labels is slightly better due to a relatively large photocurrent. When the power level increases, the BER improvement of extended SAC comes from the small average cross-correlation, which has a less contribution to PINI. Under similar code lengths, the labels with  
Figure 8: Bit-error-rate (BER) versus the number of stacked labels for MLS and extended SAC labels.