# Wideband Mixed Signal Separation Based on Photonic Signal Processing
## 2 Interference Management
### Future Perspectives
#### 2.3.2 Separation of Mixed Signal with Progressive Cancellation  
In several widely existing cases, where the mixed signals are difficult to separate, the interference can be progressively removed. The separability of the mixed signals can be quantitatively described by mixing matrix \(A\), where the difficulty in separation is positively correlated to the condition number of matrix \(A\)[54], labeled as \(cond\) (\(A\)). The cases with large \(cond\) (\(A\)) are called ill-conditions [55; 56; 57; 58; 59]. A widely existing ill-condition is that the interference source is much closer to the receiver than the SOI, which is known as the near-far problem [60; 61; 62; 63; 64; 65]. Another common condition is that each antenna of a MIMO device receives similar copies of both the interference and SOI (\(x_{1}\) is similar to \(x_{2}\) in Equation (1)). This problem happens when the receiving antennas are physically close to each other, and often exist in mobile devices, where the spatial dimensions between the antennas are limited. In both cases, the separation system requires a large power dynamic range to process the interference and high-resolution process the SOI with low amplitude or the tiny difference between \(x_{1}\) and \(x_{2}\). With the hybrid system, the mixed signals with ill conditions can be progressively separated. Figure 4: (**a**) 16 QAM signal mixed with Gaussian noise; (**b**) histogram of the original mixed signal; (**c**) histogram of the under sampled signal. The condition numbers in both cases are large (\(\gg 1\)), which results in a challenge to separate the signals in traditional ways. The capability of a BSS system to separate the mixed signals is measured by the cancellation ratio (defined in Section 2.1). Based on equation \(S=A^{-1}X\); the cancellation ratio depends on two factors:  
1. ADC resolution: ADC resolution determines the accuracy of \(X\) by introducing digitization error. In a completely digital system, the cancellation ratio depends on the ADC resolution in the first stage. In the ill-condition cases, the SOI amplitudes are comparable to the ADC resolution, and the signal information is lost in the first at ADC and cannot be recovered. For example, in an 8-bit ADC system, the digitization error is \(\frac{1}{2^{9}}\) of the signal peak amplitude. In the first ill-condition case, the received signal peak amplitude is defined by the interference. If the signal is not pre-separated in an analog way, the digitization error is comparable to or larger than the amplitude of the SOI, which causes unsuccessful separation. The analog system is different from the digital system in a way that the analog system maintains all the signal information, and the cancellation ratio in a multi-stage system can be progressively improved. 2. Weight tunability: Weight tunability determines the accuracy of \(A^{-1}\). The de-mixing matrix is implemented by adding weights to the received signal. By applying multiple stages with coarse and fine adjustment of the weights, the mixed signals can be progressively separated at each stage with the cancellation ratio from each of the stages multiplied. If the cancellation ratio between interference and SOI is less than 30 dB, the wideband interference (3 GHz) can be separated from SOI with one photonic circuit. Other combinations of bandwidth and cancellation ratio can also be achieved in this case with guidelines described in Table 1. This applies to the receiver on mobile devices with limited power and spaces for the integrated photonic chip. A single photonic circuit corresponds to half of the de-mixing matrix \(S=A^{-1}X\), and only recovers the SOI. With two photonic circuits, the full version of the de-mixing matrix is achieved by the photonic hardware, and both interference and SOI are recovered.