# Design and Implementation of DSLMS Algorithm Based Photoelectric Detection of Weak Signals
## 2 Related Work
### Principle of the Pair of Eliminators  
Noise-counteracting techniques originated in the mid-1960s by a team of researchers at Stanford University and are now widely used in various environments. The basic principle of noise counter-cancelation is shown in Figure 2[26; 27; 28].  
Noise cancelation systems are a vital signal-processing technique that operates through two main inputs: the primary input, \(d_{j}\), and the reference input, \(x_{j}\). The primary input signal, \(d_{j}\), combines the desired signal, \(s(n)\), and the accompanying noise, \(v_{0}(n)\). The  
Figure 2: Noise canceller schematic diagram.  
reference input signal, \(v_{1}(n)\), which is usually obtained from an ambient noise source, is associated with the noise component, \(v_{0}(n)\), in the primary input but is independent of the desired signal, \(s(n)\). In a noise cancelation system, a reference signal, \(v_{1}(n)\), generates an output designed to match and neutralize the primary signal's noise component, \(v_{0}(n)\). The error signal \(e_{j}=d_{j}-x_{j}\) near-clean signal \(s(n)\) is obtained by subtracting the estimated noise, \(y_{j}\), from the primary input signal, \(d_{j}\).  
The noise pair cancelation technique can identify and cut down unwanted noise components employing a reference noise signal. Although the original signal, \(d_{j}\), is interfered with by the noise, \(v_{0}(n)\), the pair cancelation system ensures that the final error signal (\(e_{j}\)) maximally reflects the desired signal, \(s(n)\), by accurately estimating and cutting the reference noise, \(v_{1}(n)\), which improves the signal-to-noise ratio of the desired signal [29].  
Assuming that the signals \(s(n)\), \(v_{0}(n)\), and \(v_{1}(n)\) are smooth, the expectation of the square of the output error mode \(\left|e_{j}\right|\) of the filter is  
\[E\left[\left|e_{j}\right|^{2}\right]=E\left[\left|s+v_{0}-v_{1}\right|^{2} \right]=E\left[\left|s\right|^{2}\right]+E\left[\left|v_{0}-y_{j}\right|^{2} \right]+2E\left\{Re\left[s\times\left(v_{0}-y_{j}\right)\right]\right\} \tag{5}\]  
Since the useful signal \(s(n)\) is uncorrelated with the noise signal \(v_{0}(n)\) and the reference signal \(v_{1}(n)\), the  
\[E\left[\left|e\right|^{2}\right]=E\left[\left|s\right|^{2}\right]+E\left[\left| v_{0}-y_{j}\right|^{2}\right] \tag{6}\]  
Because the power of the sound signal is not affected by the filter power vector, the moment when \(E\left[\left|v_{0}-y_{j}\right|^{2}\right]\) reaches its minimum value is the moment when \(E\left[\left|e\right|^{2}\right]\) reaches its minimum value. When \(v_{0}-y_{j}=0\), i.e., when \(v_{0}=y_{j}\), the minimum value is obtained.  
\[E_{min}\left[\left|e_{j}\right|^{2}\right]=E\left[\left|s\right|^{2}\right] \tag{7}\]  
Equations (5)-(7) show that an ideal noise counter-cancelation system can eliminate the difference between the estimated noise and the actual noise to a large extent, thus leaving only the effective signal. However, in real dynamically changing noise environments, the counter-cancelation system may not effectively respond to changes in noise characteristics due to its lack of real-time adjustability, which results in incomplete noise cancelation and prevents the system from adapting quickly when noise characteristics change.