## 2 QPSK-Oriented All-Optical Matching System and Noise Analysis
### Noise in the QPSK-Oriented All-Optical Matching System
Beat noise is also a type of quantum noise because it primarily originates from the amplitude and phase fluctuations of light waves during the mixing process, which also involves the wave and quantum properties of light. Quantum noise is a special type of noise that originates from the uncertainty principles of quantum mechanics and the wave properties of photons. The frequency-difference noise is usually not classified as typical quantum noise, although its generation process also involves the quantum properties of light. This is because its characteristics are more related to optical nonlinearity, such as amplitude and phase fluctuations in nonlinear optics. Therefore, frequency-difference noise is considered a type of nonlinear noise. For the QPSK-oriented all-optical matching system, ASE noise from all EDFAs is mainly considered for the following reasons:  
(1) Fundamental noise source: ASE noise is a fundamental noise source in EDFAs and is generated by the spontaneous emission of photons from erbium ions in the doped fibers as they transition from higher to lower energy states. This process is random and inherent to the amplification mechanism, making ASE noise an unavoidable component of EDFA signal amplification. (2) Impact on SNR: ASE noise directly affects the SNR of the amplified signal. As ASE noise accumulates over multiple matching cycles, it reduces the quality of the optical signal, thereby increasing the BER and reducing the performance and capacity of the matching system. Maintaining a high SNR is essential for achieving reliable and efficient optical matching; thus, ASE noise in EDFAs must be managed efficiently. (3) Gain saturation effects: ASE noise contributes to gain saturation in EDFAs, where an increase in the input signal power does not proportionally increase the output power due to the depletion of the available gain medium. ASE noise consumes a portion of the gain that would otherwise be dedicated to signal amplification, thereby necessitating meticulous gain management and EDFA design to achieve optimal performance. The intensity of the ASE noise is usually indicated by the power spectral density. The power spectral density of continuous broadband noise is expressed in Equation (3) [30]:  
\[S_{ASE}(\
u)=n_{sp}h\
u(G-1) \tag{3}\]  
where \(S_{ASE}(\
u)\) is the power spectral density of the ASE noise in watts per hertz (W/Hz ), \(h\) is the Planck constant (approximately 6.626 \(\times\) 10\({}^{-34}\) J-s), \(\
u\) is the frequency (Hz), \(n_{sp}\) is the photon number density (Photons/m\({}^{3}\)), and \(G\) is the amplifier gain. This equation describes the relationship between the ASE noise and the amplifier gain, photon number density, and carrier frequency of the amplified signal. The power spectral density of ASE noise increases with frequency because ASE noise is continuous within the operating frequency range of optical communication systems. When the EDFA gain increases, the ASE noise it generates also increases. Therefore, it is necessary to reduce the use of high-gain EDFAs in the system. It should be noted that due to design limitations in the original system [16], the interference between the PSA-processed signal and the local target sequence cannot completely eliminate power at the positions of mismatched symbols. Residual pulses with power equal to one-quarter of the \"1\"-level power remain, which we define as mismatch noise. Figure 3 illustrates the process of generating mismatch noise. Assume the input QPSK optical signal takes the sequence \(\{0,\pi/2,\pi,3\pi/2\}\), with both the input optical signal and the local target sequence signal maintaining an identical power level \(P\). If the target symbol is \(3\pi/2\), the ideal interference matching should produce a high-power pulse exclusively at the fourth position (\(3\pi/2\)) while suppressing all other positions to near-zero power levels.